Modular Invariance in Superstring On Calabi-Yau N-Fold With: D E Singularity
Modular Invariance in Superstring On Calabi-Yau N-Fold With: D E Singularity
Modular Invariance in Superstring On Calabi-Yau N-Fold With: D E Singularity
www.elsevier.nl/locate/npe
Abstract
We study the type II superstring theory on the background Rd1,1 Xn , where Xn is a Calabi
Yau n-fold (2n + d = 10) with an isolated singularity, by making use of the holographically dual
description proposed by Giveon et al., 1999. We compute the toroidal partition functions for each
of the cases d = 6, 4, 2, and obtain manifestly modular invariant solutions classified by the standard
ADE series corresponding to the type of singularities on Xn . Partition functions of these modular
invariants all vanish due to theta function identities and are consistent with the presence of space-time
supersymmetry. 2000 Elsevier Science B.V. All rights reserved.
PACS: 11.25.-w; 11.25.Hf; 11.25.Pm
1. Introduction
String theory on singular backgrounds has been recently receiving much attentions from
various view points [113]. An important feature of a string propagating near singularities
is the appearance of light solitons originating from the branes wrapped around some
vanishing cycles. This is a typical non-perturbative effect in string theory which is difficult
to be worked out from the world-sheet picture of perturbative string theory, even when a
decoupling limit gs ( e() ) 0 is taken. In fact, no matter how small gs is, the VEV
of dilaton will blow up at the location of Rsingularity. As was first pointed out in [14], the
vanishing world-sheet theta angle (ws B = 0) is essential in the appearance of such a
non-perturbative effect. Several authors demonstrated [3,7,15,16] that the conformal theory
on string world-sheet becomes singular in this situation. On the other hand, the ordinary
1 eguchi@hep-th.phys.s.u-tokyo.ac.jp
2 sugawara@hep-th.phys.s.u-tokyo.ac.jp
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 5 0 - 4
1
1
Q
1
T = (Y )2 ()2 2 ( + + ),
2
2
2
2
1
Q
(2.1)
G = (iY ) ,
2
2
J = + QiY,
which generates the N = 2 superconformal algebra (SCA) with c(
3c ) = 1 + Q2 .
Since we have a linear dilaton background, () = Q
2 , the theory is weakly coupled
in the near boundary region +. On the other hand, in the opposite end
(near the singularity) the string coupling blows up, and hence the perturbative approach
does not make sense. As is discussed in [1], one must add the Liouville potential (or the
cosmological constant term) to the world-sheet action of the Liouville sector,
SL SL + S+ + S ,
Z
1
def
S = d 2 z e Q (iY ) ,
(2.2)
in order to prevent the string from propagating into the dangerous region . S are
actually the screening charges in the sense that they commute with all the generators of
SCA (2.1). So, we shall carry out all the computations as a free conformal theory on the
(2.5)
2/Q2
is replaced by
In Refs. [2,3] the negative power term in the superpotential W x
the KazamaSuzuki model for SL(2, R)/U (1) with the level k 0 Q22 + 2. The equivalence
between such a non-compact KazamaSuzuki model and the N = 2 Liouville theory (2.1)
( corresponds to the cosmological constant in (2.2)) was discussed in [5] and it
was pointed out that both theories are related by a kind of T-duality. We argue for this
equivalence from the point of view of the free field realization in the Appendix B.
2
0 with Q /2 /gs
where 1 + i2 is the modulus of the torus (d 2 /22 is the modular invariant measure).
ZGSO denotes the part of the partition function which consists of those contributions on
which the GSO projection acts non-trivially. We write the remaining part as Z0 .
Obviously Z0 includes only the contributions from the transverse non-compact bosonic
coordinates Rd2 R . The Liouville sector R is slightly non-trivial because of the
existence of the background charge. We should bear in our mind that only the normalizable
states contribute to the partition function. The normalizable spectrum (in the sense of the
delta function normalization because the spectrum is continuous) in Liouville theory has
the lower bound h = Q2 /8 [29,44,45]. Since this lower bound is non-zero, we must be
careful in the integration over the zero-mode momentum. The result, however, turns out to
be the same as that of the standard non-compact free boson without background charge,
1
ZL (, ) = Q
| n=1 (1 q n )|2
1
= Q
| n=1 (1 q n )|2
=
1
1/2
2 |( )|2
iQ
2 +
cL
1 2 i
dp exp 42 p + pQ
2
2
24
iQ
2
Z+
1
cL
1
dp exp 42 p2 + Q2
2
8
24
(3.2)
where cL = 1 + 3Q2 . It is well-known [44,45] that the effective value of the Liouville
central charge ceff,L is equal to
ceff,L cL 24
Q2
= 1,
8
(3.3)
irrespective of the value of Q and the dependence on the background charge disappears
from the net result.
In this way we obtain
Z0 (, ) =
1
(d1)/2
2
|( )|2(d1)
(3.4)
m
+ n(n Z),
N
l(l + 2) m2
+ n,
n 12 Z>0 ,
h=
4N
m 1
(R)
(l + m 1 (mod 2)): q = + n(n Z),
Hlm
N
2
l(l + 2) m2 1
+ + n,
n Z>0 .
h=
4N
8
(NS)
Hlm
(l + m 0 (mod 2)): q =
(3.5)
1
We also consider the (left-moving) Fock
H space Hp of the bosonic coordinate Y of S
constructed on the Fock vacuum |pi, iY |pi = p|pi. Values of the momenta p are
chosen so that they are compatible with the GSO projection condition.
The total N = 2 U (1)R -charge is given by
m
(NS)
pQ,
J0 = F + FMN +
N
(3.6)
m 1
pQ,
J0(R) = F + FMN +
N
2
where F denotes the fermion number of Rd2 (R S 1 ) sector and FMN denotes the
fermion number of the MN sector.
After these preparations conditions for the GSO projection (the conditions for the mutual
locality with the spacetime SUSY charges) can now be written as
NS-sector
m
(3.7)
F + FMN + pQ 2Z + 1,
N
R-sector
m
(3.8)
F + FMN + pQ 2Z.
N
Let us now compute the trace over the left-moving Hilbert space. For instance, let us
1
m
2n + 1 + N
suppose F + FMN = even, and consider the NS-sector. Then we have p = Q
(n Z), and the sum over momenta becomes,
X
X 1 2 X N
m 2
m+N 2
q 2p =
q 4 (2n+1+ N ) =
q N(n+ 2N ) = m+N,N ( ).
(3.9)
n
When combined with factors coming from oscillator modes and the minimal model MN ,
NS-sector partition function becomes
3
3
3
4
1
(NS) m+N,N
(NS)
.
chlm
chlm +
2
2
P n2
P
P 1 (n 1 )2
Q
n
n n2
2
2
2 ,
Here = q 1/24
n q , 4 =
n (1) q , 2 =
nq
n=1 (1 q ) and 3 =
P N(n+ m )2
(NS)
(NS)
2i
lm ( )
2N
(q e
) are the standard theta functions. chlm ( ), ch
m,N = n q
denote the irreducible characters of N = 2 minimal model for NS-sector. (We summarize
the definitions of these functions in the Appendix A.) Similarly we can calculate the trace
in other sectors, and obtain (we omit the factors of -functions for simplicity),
Gl =
l=0
N2
1 X X 3 (NS)
(NS)
3 chlm (m,N + m+N,N ) 43 ch
lm (m,N m+N,N )
2
l=0 mZ2N
(3.10)
23 ch(R)
lm (m,N + m+N,N ) .
The above sum (3.10), however, counts each state twice due to the symmetry Gl =
GN2l . To avoid this double counting, we may define
X
def 1
3 (NS)
3 (R)
m,N 33 ch(NS)
(3.11)
Fl =
lm 4 chlm 2 chlm
2
mZ2N
and have
Gl = Fl + FN2l .
(3.12)
(3.13)
l,l=0
Thanks to the branching relation (A.12) we may rewrite Fl in the following simple form
[3]:
1
(3.14)
Fl ( ) = 34 44 24 l(N2) ( ),
2
c k character of the spin l/2 representation. Hence the
where l(k) ( ) denotes the SU(2)
expression (3.13) is manifestly modular invariant when Nl,l is chosen to be one of the
(N2)
c N2 theory which fulfill the conditions,
of SU(2)
modular invariants L
l,l
l + 2)
l(
l(l + 2)
(k)
(k)
Z, Lkl,kl = Ll,l ,
4(k + 2) 4(k + 2)
r
X (k) (k) (k)
(l + 1)(l 0 + 1)
2
(k)
(k) def
sin
.
Ll,l Sll 0 Sll0 = Ll 0 ,l0 , Sll 0 =
k+2
k+2
(k)
Ll,l = 0,
unless
(3.15)
l,l
Furthermore, Fl (3.14) identically vanishes by virtue of the Jacobis abstruse identity: this
is consistent with the existence of space-time SUSY. 4
(N2)
The general solutions Ll,l
of (3.15) were completely classified by the ADE
series in Ref. [31,32]. In these solutions the values of spin l/2 are in a one-to-one
correspondence with the exponents of ADE Lie algebra, and hence to each of the
relevant deformations of the singularity F = 0. In this way we obtain the modular
invariants classified by the ADE series corresponding to the singularity type of Xn
[3,30].
The appearance of the affine SU(2) character in the expression (3.14) is quite expected.
One may relate the background of degenerate K3 surface to a collection of NS5-branes
4 As discussed in [28,29], we only have the SUSY along the boundary Rd1,1 , and no SUSY in the whole
bulk space including the throat sector. Nevertheless we can conclude that the partition function should vanish in
all the genera. See [29] for the detail.
10
by means of T-duality [3,33], and it is well-known [34,35] that the world-sheet conformal
field theory in the NS5 background contains the SU(2) WZW model.
3.2. d = 2 case
In the case of d = 2 the GSO conditions are given as follows:
NS-sector
m
(3.16)
F + FMN + pQ 2Z + 1,
N
R-sector
m
(3.17)
F + FMN + pQ 2Z + 1.
N
We again determine the spectrum of the momenta p by imposing these conditions, and the
trace for the left-movers is calculated as follows,
N2
X
Gl =
1 X n
(NS)
3 chlm m , N + m+N , N
N+1 N+1
N+1 N+1
2
N2
X
l=0
l=0
mZ2N
(NS)
4ch
lm
2ch(R)
lm
N
m
N+1 , N+1
m
N
N+1 , N+1
m+N ,
N
N+1 N+1
+ m+N ,
N
N+1 N+1
o
.
ch
ch
m , N 3 ch(NS)
Fl =
m+N
N
4 lm
2 lm ,
lm
N+1 N+1
N+1 , N+1
2
(3.18)
(3.19)
mZ2N
Gl = Fl + FN2l ,
(3.20)
to avoid the double counting of states. However, it is easy to see that Fl here does not have
a good modular transformation property. This is because the theta functions appearing in
(3.19) have fractional levels, and they do not close among themselves under the modular
transformation.
In order to avoid this difficulty we further decompose Fl into a set of functions Fl,r (r =
0, 1, . . . , 2N + 1) with the help of the formula (A.5) as
X
Fl,r ,
(3.21)
Fl =
rZ2(N+1),
l+r0 (mod 2)
def
Fl,r =
1 X
(R)
l+r
(NS)
(N+1)m+Nr,N(N+1) 3 ch(NS)
4 ch
lm 2 chlm . (3.22)
lm (1)
2
mZ2N
rZ2(N+1),
l+r1 (mod 2)
Fl,r .
(3.23)
(3.24)
11
It turns out that the functions Fl,r ( ) have the good modular properties as:
l(l+2)
3+(1)l+r
N2
r2
Fl,r ( + 1) = e2i 4N 8N 4(N+1) + 8
Fl,r ( ),
N2
2 X X
1
=
i
S(l,r) (l 0,r 0 ) Fl 0 ,r 0 ( ),
Fl,r
0
0
(3.25)
(3.26)
l =0 r Z2(N+1)
(l,r) (l 0 ,r 0 )
def
(1)l+r
rr 0
+ (1)l +r (N2)
1
Sll 0
e2i 2(N+1) .
2
N +1
(3.27)
Therefore, it seems reasonable to regard the functions Fl,r ( ) as the basic conformal blocks
of our partition function. Note that due to (3.23) the two sets {Fl,r ; l + r 0 (mod 2)} and
{Fl,r ; l + r 1 (mod 2)} are not independent and we should choose one of these as the
building block of the theory. Let us take the set with l + r 0 (mod 2). This restriction is
consistent since (3.27) implies that the modular transformations act separately for each set.
Thanks to the transformation laws (3.25), (3.26) we can now construct the modular
invariant partition function in the following form,
ZGSO (, ) =
=
N(l,r),(l,
r)
N2
X
1
4
|( )|
N(l,r),(l,
r ( ),
r ) Fl,r ( )Fl,
(3.28)
r, r Z2(N+1)
l,l=0
(N2) (N+1)
Ll,l Mr,r .
c N2 , and M (k)
again denotes one of the ADE modular invariants of SU(2)
Here L(N2)
r,r
l,l
is the modular invariant of the theta system which satisfies the following conditions,
r 2
r2
(k)
(k)
(k)
Z, Mr+k, r +k = Mr,r ,
Mr,r = 0, unless
4k
4k
X (k) (k) (k)
rr 0
(k)
(k) def 1
Mr,r Rr,r 0 Rr ,r 0 = Mr 0 ,r 0 , Rr,r 0 = e2i 2k .
2k
r,r
(k)
(k)
(3.29)
(N2)
The simplest solution for Mr,r is, of course, given by Mr,r = r,r (or Mr r
and the most general solution is given by [36]
X
1
(k)
r,x+y r,xy ,
Mr,r =
2
= r,r ),
(3.30)
xZ2 , yZ2
o
32 42 (, z)0,1 (, 2z) 22 + 12 (, z)1,1 (, 2z) ,
(3.31)
12
X
rZ2(N+1),
l+r1 (mod 2)
1 (N2)
r,N+1 (, 0)Fl,r (, z) = l
(, 0)
2
o
32 + 42 (, z)1,1 (, 2z) 22 12 (, z)0,1 (, 2z) ,
def
Fl,r (, z) =
(3.32)
1 X
2z
(N+1)m+Nr,N(N+1) (, )
2
N
mZ2N
l+r
lm 2 ch(R)
4 ch
3 ch(NS)
lm (1)
lm
(R)
l+r
(3.33)
(Note Fl,r ( ) Fl,r (, z = 0).) It is known [46] that the combination of theta
functions in the right-hand side of (3.31),(3.32) vanishes identically
(3.34)
32 42 (, z)0,1 (, 2z) 22 + 12 (, z)1,1 (, 2z) = 0,
2
2
2
2
(3.35)
3 + 4 (, z)1,1 (, 2z) 2 1 (, z)0,1 (, 2z) = 0.
Thus the sum of functions Fl,r (3.31), (3.32) in fact vanishes identically. Then these
equations imply that the functions Fl,r themselves should vanish separately for each
|r| since the level-(N + 1) theta functions r,N+1 (, 0) are functionally independent
for different |r|. We have explicitly verified by M APLE that Fl,r (, z) + Fl,r (, z)
in fact vanishes in lower orders in q e2i , y e2iz and y 1 , for every l, r and
N = 2, 3, 4.
We conjecture that the identity
Fl,r (, z) + Fl,r (, z) 0
(3.36)
NS-sector
F + FMN +
m
pQ 2Z + 1,
N
13
(3.38)
R-sector
1
m
pQ 2Z + .
N
2
In this case the trace over the left-moving Hilbert space is given by,
F + FMN +
N2
X
Gl =
l=0
N2
X
l=0
(3.39)
1 X n 2 (NS)
3 chlm 2m , 2N + 2(m+N) , 2N
N+2 N+2
N+2
N+2
2
mZ2N
(NS)
42 ch
lm
2(m+N) ,
2m
2N
N+2 , N+2
22 ch(R)
lm 2m+N ,
N+2
N+2
2N
N+2
+ 2mN ,
N+2
2N
N+2
2N
N+2
o
.
(3.40)
Again Gl consists of theta functions with fractional levels and we have Gl = GN2l . We
introduce Fl = 12 Gl and expand Fl into a set of functions Fl,r (r = 0, 1, . . . , 2N + 3),
def 1 X
(N+2)m+Nr,2N(N+2)
Fl,r =
4
mZ4N
r+m
2 (R)
(NS)
2 2 ch
32 ch(NS)
4 lm 2 chlm
lm (1)
(3.41)
(l + r 0 (mod 2)),
def
(3.42)
rZ2(N+2)
(3.43)
Note that (3.43) is consistent with the definition Fl,r 0 (l + r 1 (mod 2)), since l + r
l + r + 2(N + 2) (N 2 l) + (r + N + 2) (mod 2).
Fl,r possess the following modular transformation properties:
n
2i
l(l+2) N2
r2
1
4N 8N 4(N+2) + 2
Fl,r ( + 1) = e
X
3 N2
1
i
=
Fl,r
Fl,r ( ),
S(l,r) (l 0,r 0 ) Fl 0 ,r 0 ( ),
(3.44)
(3.45)
l =0 r 0 Z2(N+2)
rr 0
1
e2i 2(N+2) .
2(N + 2)
It is now easy to construct modular invariant partition functions
def
(N2)
S(l,r) (l 0 ,r 0 ) = Sll 0
ZGSO (, ) =
N(l,r),(l,
r) =
N2
X
1
6
|( )|
N(l,r),(l,
r ( ),
r ) Fl,r ( )Fl,
r,r Z2(N+2)
l,l=0
(N2) (N+2)
(N2)
(N+2)
1
Mr,r
+ LN2l, lMr+N+2, r ,
2 Ll,l
(3.46)
(3.47)
14
(k)
where L(k)
and Mr,
r are defined as before.
l,l
The solution for the simplest case N = 2 (the case of conifold singularity in CY3 [2])
was first obtained by Mizoguchi from a somewhat different approach [48].
As in the two-dimensional case, we can construct the following combination of the Fl,r
functions
X
r,N+2 (, 0)Fl,r (, z)
rZ2(N+2)
1 (N2)
l
(, 0) 34 (, z) 44 (, z) 24 (, z) + 14 (, z) ,
4
z
def 1 X
(N+2)m+Nr,2N(N+2) ,
Fl,r (, z) =
4
N
mZ4N
r+m
r+m
(R)
2 (R)
(NS)
2 2 ch
2 2 ch
(1)
ch
+
i(1)
32 ch(NS)
4 lm
2 lm
1 lm (, z),
lm
=
(3.48)
(l + r 0 (mod 2)),
def
Fl,r (, z) = 0
(l + r 1 (mod 2)).
(3.49)
We note that the right-hand side of (3.48) vanishes due to Jacobis identity. Then as in the
case of two-dimensional theories, we expect that functions Fl,r should vanish separately
for each |r|, Fl,r (, z) + Fl,r (, z) 0. We have explicitly checked this for lower orders
of q, y, y 1 by M APLE and found that in fact a stronger relation
Fl,r (, z) 0,
(3.50)
holds. We conjecture that (3.50) holds for all l, r, N . In this case all the modular invariant
theories again have vanishing partition functions and are consistent with the presence of
space-time supersymmetry.
We may again read off the modular transformation rule (3.46) from the identity (3.48):
c
character in its index l and like U (1) theta function in
Fl,r transforms like an affine SU(2)
its label r.
We may show
2
(3.51)
Fl,r , z + = (1) q 2 y 2 Fl,r (, z),
2
which implies that the functions Fl,r (, z) are the flow-invariant orbits for each l, r.
4. Conclusions
In this paper we have constructed the toroidal partition functions of the non-critical
superstring theory on Rd1,1 R SY1 MN , which is to provide the dual description
of the singular CalabiYau compactification in the decoupling limit. We have found that
there exists a natural ADE classification of modular invariants associated to the type
of CalabiYau singularities in all cases of d = 6, 4, 2. In cases d = 4, 2, we found that
15
the conformal blocks composing the partition function behave like primary fields of the
parafermionic theory. It will be very interesting if we could identify our conformal blocks
with suitable scaling operators in respective field theories and elucidate their dynamical
properties.
As we have discussed at the beginning of Section 3, the presence of the background
charge in the Liouville sector creates a gap h > Q2 /8 in the CFT spectrum. In particular
the graviton (which corresponds to h = 0) does not appear in the modular invariant partition
function. Thus the system in fact describes some non-gravitational theory and the theory is
interpreted as being at the decoupling limit of type II superstring. It is quite reassuring to
us that one can construct modular invariant amplitudes for string propagation even in such
a singular situation where some of the conventional world-sheet technology may break
down and non-perturbative effects play an important role.
LandauGinzburg theory has the disturbing feature of the appearance of a negative
power piece in the superpotential when applied to describe singular (non-compact) Calabi
Yau manifolds. It is not clear how to treat the negative power operator within the framework
of the standard N = 2 SCFT. It now appears, however, the negative power term may
be handled properly by means of the Liouville degrees of freedom with an appropriate
background charge. The appearance of the gap and the continuous spectrum above the
gap in string propagation in singular CalabiYau manifold are reproduced exactly by the
dynamics of the Liouville field. It will be extremely useful if we have a better understanding
on the relationship between the singular geometry and the dynamics of Liouville field.
It will be interesting to consider more general class of N = 2 models instead of N = 2
minimal model (LandauGinzburg orbifolds [50,51] or Gepner models [49], etc.). Quite
recently, in Ref. [6], LandauGinzburg orbifolds are discussed, relating it to the N = 2
SCFT4 with matter fields [3941]. It may also be interesting to study non-rational Calabi
Yau singularities (collapse of del Pezzo surfaces, etc.). These problems may be regarded as
natural generalizations of the Gepner model, namely, the (orbifoldized) tensor product of
minimal models (compact models) with the Liouville theory (non-compact models).
Construction of modular invariants for such models will provide important consistency
checks of their dynamics.
Acknowledgement
We would like to thank especially Dr. S. Mizoguchi whose talk at Univ. of Tokyo
stimulated the present investigation. Y.S. would also thank Drs. K. Ito and A. Kato for
useful comments, and Prof. I. Bars and his theory group for kind hospitality at USC. Part
of this work was done while Y.S. was attending the workshop Strings, Branes and Mtheory at CIT-USC Center for Theoretical Physics.
This work is supported in part by Grant-in-Aid for Scientific Research on Priority Area
]707 Supersymmetry and Unified Theory of Elementary Particles from Japan Ministry
of Education.
16
(1)n q (n1/2)
n=
2 exp
2 (, z) =
i
4
i
2 exp
4
3 (, z) =
4 (, z) =
m,k (, z) =
X
n=
2 /2
qn
y n1/2
sin(z)
(1 q m )(1 yq m )(1 y 1 q m ),
m=1
q (n1/2)
n=
2 /2
2 /2
y n1/2
cos(z)
(1 q m )(1 + yq m )(1 + y 1 q m ),
m=1
yn
m=1
2 /2
(1)n q n
yn
n=
m=1
m 2
q k(n+ 2k ) y k(n+ 2k ) .
(A.1)
(A.2)
n=
(1 q n ).
(A.3)
n=1
(A.4)
rZk+k0
0
0 0
zz
kz+k z
where we set u = k+k
0 , v = k+k 0 .
The following identity is often used (p is an integer):
X
m+2kr,pk (, z/p).
m/p,k/p (, z) = m,k (/p, z/p) =
rZp
(A.5)
17
l+1,k+2 l1,k+2
(, z).
1,2 1,2
l ( ) is defined by
String function cm
X
(k)
l
cm
( )m,k (, z).
l (, z) =
(A.6)
(A.7)
mZ2k
We introduce
ml,s (, z) =
l
cms+4r
( )2m+(k+2)(s+4r),2k(k+2) , z/(k + 2) .
(A.8)
rZk
l has the following properties:
String function cm
kl
l
l
l
= cm
= cm+2k
= cm+k
,
cm
l
cm
= 0,
(A.9)
ml, s = 0,
(A.10)
y J0 = ml,0 + ml,2 ,
chl,m (, z) TrHNS q L0 c/8
(NS)
l,m
F L0 c/8
(NS)
y J0 = ml,0 ml,2 ,
ch
l,m (, z) TrHNS (1) q
l,m
L0 c/8
y J0 = ml,1 + ml,3 ,
ch(R)
l,m (, z) TrHR q
(A.11)
l,m
F L0 c/8
(R)
y J0 = ml,1 ml,3 .
ch
l,m (, z) TrHR (1) q
l,m
It is easy to prove the following branching relation by means of the product formula of
theta functions (A.4) [49,53]:
X
(k)
ml,s (, z)m,k+2 , w 2z/(k + 2) . (A.12)
l (, w)s,2 (, w z) =
mZ2(k+2)
This relation (A.12) represents the minimal model as the KazamaSuzuki coset for
SU(2)k /U (1).
Appendix B. Equivalence between the KazamaSuzuki model for SL(2; R)/U (1)
and the N = 2 Liouville theory
In this appendix we discuss the equivalence between the N = 2 coset SCFT for
SL(2; R)/U (1) and the N = 2 Liouville theory from the viewpoint of free field
18
realizations. We start from the following free field realization 5 of SL(2; R)-current algebra
with the level k + 2,
r
k+2
u,
J =
2
(B.1)
r
q
r
2
k+2
k
k+2
(u+iX)
J =
iX
e
,
2
2
where X(z)X(0) ln z, (z)(0) ln z, u(z)u(0) ln z are free scalar fields. The
Sugawara stress tensor is given by
1
1
1
1
TSL(2;R) = (X)2 ()2 2 (u)2 ,
2
2
2
2k
(B.2)
H (z)H (0) ln z.
(B.3)
The total stress tensor for the KazamaSuzuki model then reduces to the following form,
1
1
1
1
1
1
T = (X)2 ()2 2 (u)2 (v)2 (H )2 , (B.4)
2
2
2
2
2
2k
where (, ) is the spin (0, 1) ghost system and the BRST charge is given by
r
I r
k+2
k
u + i
v + iH .
(B.5)
QU (1) =
2
2
This stress tensor (B.4) has the correct central charge c = 3(1 + 2k ) and the world-sheet
N = 2 superconformal symmetry is generated by the currents,
r
r
q
1
1
k+2
k
2 (u+iX)iH
iX
e k+2
=
J
=
2
2
k
k
(B.6)
J = + + 2 (J 3 + + ) = k + 2 iH + 2(k + 2) u.
k
k
k
5 We have another familiar free field realization: Wakimoto representation [55] (, , ). The relation
between these free fields and X, , u here is given as follows:
=+
(u + iX),
k+2
! q
r
r
2 (u+iX)
k+2
k
=
,
iX +
e k+2
2
2
2 (u+iX)
k+2
=e
.
19
Now, let us try to reduce the above KazamaSuzuki model to the N = 2 Liouville theory.
For this purpose it is convenient to introduce the following field redefinition,
q
q
!
0
k
2
v
v
def
k+2
k+2
q
q
=
.
(B.7)
H0
H
2
k
k+2
k+2
Clearly we have v 0 (z)v 0 (0) ln z, H 0 (z)H 0 (0) ln z and v 0 (z)H 0 (0) 0. The BRSTcharge (B.5) is rewritten as
r
I
k+2
(B.8)
(u + iv 0 ),
QU (1) =
2
and the stress tensor (B.4) is reexpressed as follows,
1
1
1
1
T = (X)2 ()2 2 (H 0 )2
2
2
2
2k
1
0
(u + iv ) .
+ QU (1) ,
2(k + 2)
(B.9)
r
q
q
r
q k
2
2
0
0
1
k+2
k
i
iX
e
,
G =
2
2
k
r
2
k+2
iH + QU (1) , .
J =
k
k
(B.10)
q
H
2
def k+2
(u+iv 0 )J
v0
def
k+2
k
(B.12)
r
I
2
0
(u + iv )J, T (z) = 0.
k+2
It is also obvious that U QU (1) U 1 = QU (1) , and hence this similarity transformation is
in fact well-defined on the Hilbert space of KazamaSuzuki model. Furthermore it is
convenient to rotate again X, H 0 as,
q
q
k
2
X
Y def k+2
k+2
q
q
=
,
(B.13)
H0
H 00
2
k
k+2
k+2
20
00
and set = eiH . Then we finally obtain (upto QU (1) -exact terms)
1
1
1
1
T 0 = (Y )2 ()2 2 ( + + ),
2
2
2
2k
1
1
0
G = (iY ) ,
k
2
J 0 = +
iY.
k
(B.14)
21
[19] C. Vafa, N. Warner, Catastrophes and the classification of conformal theories, Phys. Lett. B 218
(1989) 51.
[20] W. Lerche, C. Vafa, N. Warner, Chiral rings in N = 2 superconformal theories, Nucl. Phys.
B 324 (1989) 427.
[21] M. Douglas, Enhanced gauge symmetry in M(atrix) theory, JHEP 9707 (1997) 004, hepth/9612126.
[22] D. Diaconescu, M. Douglas, J. Gomis, Fractional branes and wrapped branes, JHEP 9802 (1998)
013, hep-th/9712230
[23] Y. Kazama, H. Suzuki, New N = 2 superconformal field theories and superstring compactification, Nucl. Phys. B 321 (1989) 232.
[24] J. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor.
Math. Phys. 2 (1998) 231, hep-th/9711200.
[25] S. Gubser, I. Klebanov, A. Polyakov, Gauge theory correlators from non-critical string theory,
Phys. Lett. B 428 (1998) 105, hep-th/9802109.
[26] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253, hepth/9802150.
[27] O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, Y. Oz, Large N field theories, string theory
and gravity, hep-th/9905111.
[28] D. Kutasov, N. Seiberg, Noncritical superstrings, Phys. Lett. B 251 (1990) 67.
[29] D. Kutasov, Some properties of (non) critical strings, Lecture given at ICTP Spring School,
Trieste 1991, hep-th/9110041.
[30] D. Diaconescu, N. Seiberg, The Coulomb branch of (4, 4) supersymmetric field theories in two
dimensions, JHEP 9707 (1997) 001, hep-th/9707158.
[31] A. Cappelli, C. Itzykson, J.-B. Zuber, Modular invariant partition functions in two dimensions,
(1)
Nucl. Phys. B (FS 18) 280 (1987) 445; The ADE classification of minimal and A1 conformal
invariant theories, Commun. Math. Phys. 113 (1987) 1.
[32] A. Kato, Classification of modular invariant partition functions in two dimensions, Mod. Phys.
Lett. A 2 (1987) 585.
[33] R. Gregory, J. Harvey, G. Moore, Unwinding strings and T-duality of KaluzaKlein and Hmonopoles, Adv. Theor. Math. Phys. 1 (1997) 283, hep-th/9708086.
[34] C. Callan, J. Harvey, A. Strominger, World-sheet approach to heterotic instantons and solitons,
Nucl. Phys. B 359 (1991) 611.
[35] C. Callan, J. Harvey, A. Strominger, World-brane actions for string solitons, Nucl. Phys. B 367
(1991) 60.
[36] D. Gepner, Z. Qiu, Modular invariant partition functions for parafermionic field theories, Nucl.
Phys. B (FS 19) 285 (1987) 423.
[37] N. Seiberg, Matrix description of M-theory on T 5 and T 5 /Z2 , Phys. Lett. B 408 (1997) 98,
hep-th/9705221.
[38] A. Losev, G. Moore, S.L. Shatashvili, M & ms, Nucl. Phys. B 522 (1998) 105, hep-th/9707250.
[39] P.C. Argyres, M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl.
Phys. B 448 (1995) 93, hep-th/9505062.
[40] P.C. Argyres, M.R. Plesser, N. Seiberg, E. Witten, New N = 2 superconformal field theories in
four dimensions, Nucl. Phys. B 461 (1996) 71, hep-th/9511154.
[41] T. Eguchi, K. Hori, K. Ito, S.-K. Yang, Study of N = 2 superconformal field theories in 4
dimensions, Nucl. Phys. B 471 (1996) 430, hep-th/9603002.
[42] A. Giveon, M. Rocek, Supersymmetric string vacua on AdS3 N , JHEP 9904 (1999) 019,
hep-th/9904024.
[43] D. Bernstein, R.G. Leigh, Spacetime supersymmetry in AdS3 backgrounds, Phys. Lett. B 458
(1999) 297, hep-th/9904040.
[44] D. Kutasov, N. Seiberg, Number of degrees of freedom, density of states and tachyons in string
theory and CFT, Nucl. Phys. B 358 (1991) 600.
22
[45] N. Seiberg, Notes on quantum Liouville theory and quantum gravity, Prog. Theor. Phys.
Suppl. 102 (1990) 319.
[46] A. Bilal, J. Gervais, New critical dimensions for string theories, Nucl. Phys. B 284 (1987) 397.
[47] T. Eguchi, H. Ooguri, A. Taormina, S.-K. Yang, Superconformal algebras and string compactification on manifolds with SU(N) holonomy, Nucl. Phys. B 315 (1989) 193.
[48] S. Mizoguchi, Talk given at Univ. of Tokyo, November 1999: Modular invariant critical
superstrings on four-dimensional Minkowski space two-dimensional black hole, hepth/0003053.
[49] D. Gepner, Exactly solvable string compactifications on manifolds of SU(N) holonomy,
Phys. Lett. B 199 (1987) 380; Space-time supersymmetry in compactified string theory and
superconformal models, Nucl. Phys. B 296 (1988) 757.
[50] C. Vafa, String vacua and orbifoldized LG models, Mod. Phys. Lett. A 4 (1989) 1169.
[51] K. Intriligator, C. Vafa, LandauGinzburg Orbifolds, Nucl. Phys. B 339 (1990) 95.
[52] V. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge University Press.
[53] T. Kawai, Y. Yamada, S.-K. Yang, Elliptic genera and N = 2 superconformal field theory, Nucl.
Phys. B 414 (1994) 191, hep-th/9306096.
[54] F. Ravanini, S.-K. Yang, Modular invariance in N = 2 super conformal field theories, Phys.
Lett. B 195 (1987) 202.
(1)
[55] M. Wakimoto, Fock representations of the affine Lie algebra A1 , Commun. Math. Phys. 104
(1986) 605.
Abstract
We study the expectation value of (the product) of the one-particle projector(s) in the reduced
matrix model and matrix quantum mechanics in general. This quantity is given by the nonabelian
Berry phase: we discuss the relevance of this with regard to the spacetime structure. The case of
the USp matrix model is examined from this respect. Generalizing our previous work, we carry
out the complete computation of this quantity which takes into account both the nature of the
degeneracy of the fermions and the presence of the spacetime points belonging to the antisymmetric
representation. We find the singularities as those of the SU(2) Yang monopole connection as well as
the pointlike singularities in 9 + 1 dimensions coming from its SU(8) generalization. The former type
of singularities, which extend to four of the directions lying in the antisymmetric representations, may
be regarded as seeds of our four-dimensional spacetime structure and is not shared by the IIB matrix
model. From a mathematical viewpoint, these connections can be generalizable to arbitrary odd space
dimensions due to the nontrivial nature of the eigenbundle and the Clifford module structure. 2000
Elsevier Science B.V. All rights reserved.
PACS: 11.25
Keywords: Matrix model; Nonabelian Berry phase; Nonabelian monopole; Spacetime
1. Introduction
Continuing studies in matrix models for superstrings and M-theory [15] indicate
that we are in a stage of obtaining a renewed understanding of old notions such as
compactification and spacetime distribution in this constructive framework. These physical
quantities are obtained after integrations of matrices and are no longer fixed input
parameters or backgrounds. Another feature common to these models is that the actions
This work is supported in part by the Grant-in-Aid for Scientific Research (10640268) and Grant-in-Aid for
Scientific Research fund (97319) from the Ministry of Education, Science and Culture, Japan.
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 4 2 - 5
24
contain terms bilinear in fermions. This is, of course, related to the D-brane in the RR
sector as the absence of these bilinears imply the absence of the RR sector of the model. 1
The major objective of this paper is to enlighten the spacetime structure and the presence
of solitonic objects revealed by the fermionic integrations of the matrix models. These
are represented by the behavior of the spacetime points (or D0-branes in [1]), which
are the eigenvalue distributions or the diagonal elements of the bosonic matrices. The
effective dynamics of the spacetime points is obtained by carrying out the integrations
of the remaining degrees of freedom: we will carry out the half of them represented by the
fermions. Our interest, therefore, lies in a collection of individual fermionic eigenmodes
obtained from the fermionic part (denoted generically by Sfermion ) of the action upon
diagonalization. An object which, we find, plays a role of revealing singularities as those
of the bosonic parameters (in particular, those of the spacetime points) is an expectation
value of the one-particle projector belonging to each of the fermionic eigenmodes (see
Eq. (2.27)).
We see that this expectation value of the projector is generically given by the nonabelian
Berry phase [6,7]. In matrix models of superstrings and M-theory, this result offers
spacetime interpretation: this is because the parameter space, where the connection oneform lives, is that of the spacetime points or D0-branes. An interesting case that we study
as our major example is the USp reduced matrix model [4,5,811]. We will carry out an
explicit evaluation of the nonabelian Berry phase factor for different types of the projectors.
Our computation leads us to the su(2) Lie algebra valued connection one-form known as
the Yang monopole [12] in five spatial dimensions and its su(8) generalization in nine
spatial dimensions. This latter one, to the best of our knowledge, has not appeared in
physics context before. The existence of the nontrivial eigenbundles based on the first
quantized Hamiltonian with gamma matrices and the orthogonal projection operators
ensures that these nonabelian connections are straightforwardly generalized to arbitrary
odd spatial dimensions.
The conclusion derived from our computation in the context of the USp matrix model
is that there exist singularities extending to four of the directions of the spacetime points
which lie in the antisymmetric representation. These singularities are represented by the
Yang monopole. In addition, we find pointlike singularities in 9 + 1 dimensions which are
represented by its SU(8) generalization. The former type of singularities may be regarded
as being responsible for our four-dimensional spacetime structure and is not shared by the
IIB matrix model. 2 It is noteworthy that the requirement of having 8 + 8 supersymmetries
brings us such possibility.
Before moving to the next section, let us mention the procedure to formulate the
expectation value of the one-particle projector in the reduced matrix models (see
1 The reduced matrix model provides a constructive definition to the GreenSchwarz superstrings in the Schild
gauge. A subsector of the state space which contains fermion bilinears is able to see the RR sector of the
superstrings.
2 The signature of the four directions which the Yang monopole is extended to is Euclidean in the model. We
have nothing to say on how to make the signature Minkowskian and on the extension of the spacetime in the v0
direction.
25
Eq. (2.27)). To say things short, these are obtained from short time/infinite temperature
limit of the matrix quantum mechanical system, where a path in the parameter space can
be readily introduced. This is equivalent to imposing a periodicity with period R on one
of the original bosonic matrices, say, v0 and to letting this period go to infinity in the end.
Machinery to deal with these situations has been developed in [13,14]. We mention here
the calculation of [15,16], which is similar in spirit to ours. There the exact computation
of the partition function of the IIB matrix model [2] has been carried out as the limit of
infinite temperature of the path integral of the BFSS [1] quantum mechanical model. This
is the same limiting procedure as ours although we will measure the one-particle projector
instead of unity (see Eq. (2.35)).
In the next section, we describe the formulation and the procedure of our computation
indicated above after a brief review on the USp(2k) matrix model. In Section 3, we
decompose the fermionic part of the action into the bases of the adjoint, antisymmetric
and fundamental representations appropriate to our computation. This amounts to
diagonalizing it for the case that the bosonic matrices are diagonal. Unlike our previous
studies [8,9], we do not set the spacetime points lying in the antisymmetric representation
to zero. This turns out to improve substantially our picture of the spacetime formation
suggested by the model. In Section 4, we compute the nonabelian Berry phase for
three types of actions obtained in Section 3. The connection one-forms we find are the
nonabelian SU(2) Yang monopole in five dimensions and its SU(8) generalization to nine
dimensions. The degenerate state space originating from the spinorial space is responsible
for making these nonabelian gauge fields. In Section 5, we summarize the spacetime
picture emerging from our computation. In Section 6, we discuss the generalization of the
Yang monopole in arbitrary odd space dimensions by clarifying the eigenbundle structure
associated with the nonabelian Berry connection. Detail of the basis decomposition in
Section 3 is collected in the appendix. Some of the papers on fermionic and bosonic
integrations of matrix models are listed in [1719].
26
(2.1)
(2.2)
(2.3)
It is easy to recognize
u(2k) = usp(2k) asym(2k).
A representation of Eq. (2.1) in accordance with the choice (Eq. (2.3)) is
H
B
A
usp(2k),
B H
H = H,
(2.5)
B = B,
(2.4)
(2.6)
B = B.
Let us recall here some aspects of the reduced USp(2k) matrix model which are relevant
to our discussion in what follows. The definition, the criteria and the rationale leading to
the model as descending from type I the superstrings are elaborated fully in Ref. [4,5]. We
will therefore not repeat these here.
b be a projection operator acting on 2k 2k Hermitian matrices. For the ten bosonic
Let
b projects the 0, 1, 2, 3, 4, 7 components onto the adjoint representation and the
matrices,
5, 6, 8, 9 components onto the antisymmetric representation:
vM = (v , vn ),
v usp(2k),
= 0, 1, 2, 3, 4, 7,
vn asym(2k),
(2.7)
n = 5, 6, 8, 9.
(2.8)
where
adj t(1 , 0, 2 , 0, 0, 0, 0, 0, 0, 1, 0, 2 , 0, 0, 0, 0),
asym t(0, 0, 0, 0, 1, 0, 2 , 0, 0, 0, 0, 0, 0, 1, 0, 2 ).
(2.9)
Modulo labelling the indices, these projections are determined by the requirement of
having 8 + 8 supersymmetries. Finally, we add degrees of freedom corresponding to an
27
open string degrees of freedom while preserving supersymmetry. This amounts to adding
nf = 16 of the hypermultiplets in the fundamental representation in the 4d language. We
display these degrees of freedom by the complex 2nf dimensional vectors (see Appendix A
of [10])
f = 1 nf ,
Q(f ) ,
Q
1
e
F Q(f nf ) , f = nf + 1 2nf ,
(
(2.10)
Q(f ) ,
f = 1 nf ,
Q e
Q(f nf ) F, f = nf + 1 2nf .
(
Q
(
Q
Q(f ) ,
f = 1 nf ,
F 1 Q
e(f n ) , f = nf + 1 2nf ,
f
Q(f ) ,
Q
e(f n ) F,
f
f = 1 nf ,
f = nf + 1 2nf .
(2.11)
(2.12)
where
1 1
Tr 4 [vM , vN ] v M , v N 12 M [vM , ]
2
g
is the closed string sector of the model. We denote the fermionic part by
1
SMW 2 Tr M [vM , ] .
2g
The remainder of the action
Sclosed =
(2.13)
(2.14)
(2.15)
consists of the parts which depend on the fundamental hypermultiplet. We only spell out
the parts relevant to our subsequent discussion:
1
(2.16)
Sgf = 2 Q m vm Q + i 2 Q Q i 2 Q Q ,
g
!
X
1
2 Wmatter
C2 C1 + h.c.
SYukawa = 2
C1 C2
g
e
e
(c1 ,c2 )=(Q,Q),(Q,
1 ),(1 ,Q)
1
= 2
g
Here
1
2Q
F ( 2 1 + M) Q + 2 Q F 1 Q + h.c. .
0 I
,
I 0
M diag(m(1) , . . . , m(nf ) , m(1) , . . . , m(nf ) ),
(2.17)
nf
Wmatter =
X
f =1
e(f ) Q(f ) +
m(f ) Q
e(f ) Q(f ) ,
2Q
(2.18)
(2.19)
(2.20)
28
and implies the standard inner product with respect to the 2nf flavour indices. For a more
complete discussion, see [10].
2.2. One-particle projector and nonabelian Berry phase
Let us first imagine diagonalizing some action Sfermion which is bilinear in fermions. In
the example of the last subsection, this is given by the fermionic part of the action
Sfermion SMW + Sgf + SYukawa .
(2.21)
(2.22)
with being the Clifford vacuum which annihilates half of the fermions
(bA , b A ),
which are
written as
(, , Q Q )
` =
bA `A
.
(2.24)
` `
can be
(2.25)
The expectation value is defined through the integrations over the fermionic variables of
the model. A formula that we find in the end (Eq. (2.35)) and the one we use (Eq. (2.38))
in the subsequent sections are obtained from the short time/infinite temperature limit of the
corresponding quantum mechanics, in which the path dependence can be easily introduced.
In order to argue more directly that this quantity can be defined in the reduced models, we
start with imposing a periodicity constraint on one of the directions, say, v0 :
SvM S 1 = vM + R1M.0 .
(2.26)
The size of the matrices is necessarily infinite dimensional in order to permit solutions to
Eq. (2.26). Each matrix divides into an infinite number of blocks. The shift operator S acts
on each block and moves it diagonally by one in our situation.
Let us introduce
Z
0
0
00
b 0 | Z 00 i.
b
(2.27)
[D][DZ]eiS(,Z ;Z ,Z ) hZ 0 | P
P` lim
`
R
29
where I is a subset of all eigenmodes and the case of our interest is the one in which
this subset is over the eigenmodes belonging to the positive eigenvalues. This choice is
motivated by the Dirac sea filling.
In principle, one can diagonalize Eq. (2.21) for general vM and Q. We will, however,
restrict ourselves to the case
vM = XM = diagonal,
Q = 0.
(2.29)
Explicit diagonalization of Eq. (2.21) to the form of Eq. (2.22) in this case will be carried
out in the next section.
Let us convert Eq. (2.27) into the Fourier transformed variables and this helps us
understand the limiting procedure in Eq. (2.27) better:
Z
0
0
b, | z00 i,
b
=
lim
(2.30)
dz0 dz00 F (z0 ; | z00 ; 0) hz0 | P
P
`
`
0
Z
R 0
0
0
0 00
0
(2.31)
F = [D()][Dz()]ei 0 d Lfermion(( ),z( );z ,z ,XM =XM ( )) .
Here
= 2/R
(2.32)
0
0
0
b
= lim Trfermion ()F ei 0 d H ( ) P` .
P
`
(2.35)
Reducing this expression into that of the first quantized quantum mechanics, we find that
this quantity is nothing but the time evolution of the `th degenerate eigenfunction ` .
(Note that this ` is the same as the one appearing in the original expression Eq. (2.25).)
The generic expression is known to consist of the energy dependent dynamical phase and
the nonabelian Berry phase [6,7]. This latter phase factor is given by the path-ordered
exponential of the loop integral of the connection one-form A` (XM ). We obtain
"
d E` XM ( ) i
lim P exp i
#! 0
I
A` (XM )
(2.36)
30
Here
A` (XM ) = i` d` = i
`A
d`A
(2.37)
and the nonabelian gauge field A` (XM ) originates from the degenerate eigenfunction.
Finally letting R large or the time period short, we find
" I
# 0
0
b
= P exp i A` (xM )
,
(2.38)
P
`
(3.1)
The matrices are all u(2k) Lie algebra valued. Here, for brevity, we ignore the coupling
and the minus sign in the action (these will be put back in the next section):
1
xM
0
..
(3.2)
XM =
.
.
N
0
xM
Using the set of bases defined by
(Eab )ij ai bj ,
(3.3)
we can write the bases of the Lie algebra generators of U (N), which are Hermitian
matrices, as
Ha = Ea,a Ea+1,a+1
Sa,b = Ea,b + Eb,a
(a = 1, . . . , N 1),
(a < b),
(a < b).
(3.4)
(3.5)
(3.6)
Here
4 With regard to the last footnote, we have adopted here the normalization of the energies/eigenvalues as
quantum mechanics.
Ha =
1
1
Sa,b =
1
1
31
Ta,b =
i
i
(3.7)
Let us decompose into the diagonal part and the off-diagonal part :
= + .
(3.8)
(3.9)
a<b
(3.10)
a<b
where a, b = 1, . . . , N . Introducing
Uab Sab iTab ,
a
b
Mab = M xM
xM
(3.11)
we find
S=
1 X ab ab ab
U M U Lab Mab Lab .
2
(3.12)
a<b
(3.14)
(3.15)
32
Sab + Sa+k,b+k ,
Tab Ta+k,b+k ,
Sa,b+k Sb,a+k ,
Ta,b+k Tb,a+k ,
(3.16)
The fermions adj and asym are expanded respectively by the bases Eq. (3.15) and
Eq. (3.16).
After some algebraic manipulation which we leave in the appendix, we find that the
fermionic action of USp(2k) reduced model is expressed in terms of three types of actions
(
1 X DU ab
DU ab a
b
LI adj
, asym
; xM , xM
S=
2
a<b
X
DL ab
DL ab
a
b
LI adj
, asym
; xM
, xM
+
+
+
+
a<b
OU ab
OU ab a
b
LII adj
, asym
; xM , xM
a<b
OL ab
OL ab
a
b
LII adj
, asym
; xM
, xM
a<b
ODU a
LIII adj
; xa xa+k
X
a
ODL a
LIII adj
)
; xa + xa+k ,
(3.17)
where
M (xM yM )( + ),
LI (, ; xM , yM ) 2( + )
M xM (yM ) ( + ),
LII (, ; xM , yM ) 2( + )
x .
LIII (; x )
(3.18)
(3.19)
(3.20)
We call LI , LII and LIII type I, type II and type III action respectively. See the appendix
for detail of our notation.
It is obvious that only components from the diagonal blocks D contribute to type I
action, while only components from O contribute to type II action. As for type III action,
only components from OD contribute and they are all in the adjoint representation. We
will find that the contribution from the fermions in the fundamental representation has the
same form as type III action. We see that the part of the adjoint fermions and all of the
antisymmetric fermions form MajoranaWeyl fermions while the remainder of the adjoint
fermions decouple from the spacetime points lying in the antisymmetric representation. We
indicate below the parts of the matrix degrees of freedom of the fermion contributing to
LI , LII and LIII by , , and ?, respectively,
0 ?
0 ?
0 ?
(3.21)
? 0 .
? 0
? 0
33
Apart from the fermions in the closed string sector, we also have the fermions belonging
to the fundamental representation in Eqs. (2.16) and (2.17). The action, up to the prefactor,
reads
(3.22)
SF = Q m Xm Q + 12 Q F (X4 + iX7 + M) Q + h.c. .
As is already discussed in [8,9], this action does not depend on Xn , n = 5, 6, 8, 9, and is
in fact type III:
X
LIII Q(f ) a ; xa m(f ) ,4 .
(3.23)
SF =
a,f,
See [8,9] for more detail. The fermionic part of the action reads
Sfermion S + SF .
(3.24)
where s are the five-dimensional gamma matrices obeying the Clifford algebra. We
take
2 = 2 3,
7 = 12 2 ,
3 = 3 3,
(4.2)
and each one is doubly degenerate. We focus on the two-dimensional subspace of the oneparticle states which belongs to the positive eigenvalue. The nonabelian Berry connection
obtained will be su(2) Lie algebra valued. The eigenstates can be obtained with the help
of projection operators, which are defined by
P 12 (14 y ),
(4.3)
34
where
y
x
|x|
(4.4)
(4.6)
(4.10)
(4.11)
Introducing
C (y dy y dy ),
we obtain
M=
1
1
1
1
dy + C
dy3 P+ ,
1 + y3 2
4
1 + y3
A(yi ) =
where
1
B ,
2(1 + y3 )
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
35
yi =
2zi
,
1 + z2
i = 1, 2, 4, 7,
y3 =
1 z2
,
1 + z2
(4.18)
(4.19)
(4.20)
Here z is a quaternion
z z2 12 + iz .
(4.21)
We obtain
1 y3
2
1
=
1 + z 2
A=
dT T 1 =
z2
dT T
1 + z2
1
(d z z z d z ),
2
(4.22)
which is exactly the gauge connection for the ASD instanton on R4 [24].
In terms of zi , the Berry connection can also be written as
A=
1
1
(4.23)
where
i
ij E[ i , j ] tE,
2
5 In our notation z are real numbers.
i
(4.24)
36
with components
1 0
,
12 =
0 1
0 1
17
,
=
1 0
1 0
,
47 =
0 1
14 =
0
i
i
,
0
(4.25)
0 1
=
,
1 0
0 i
27 =
.
i 0
24
(4.26)
(4.27)
(4.28)
(4.29)
i=1,2,...,9
(4.33)
i=1
This time, the projection operator acts on the Weyl projected sixteen-dimensional space.
The gamma matrices are understood to act on this space and are regarded as 16 16
37
matrices. With the help of this projection operator, the orthogonal eigenvectors can be
constructed. In the case of the positive eigenvalue, the eigenvectors are
1
P+ e ,
N
1
2
2
1+
N N =
2
1
1
N 02 N2 =
2
=
(4.34)
x3
,
|x|
x3
,
|x|
Here the second line is well-defined around the north pole while the third line is welldefined around the south pole.
We focus on the eigenspace well defined around the north pole with the positive
eigenvalue. The eigenspace forms an eight-dimensional vector space. The nonabelian
Berry phase is su(8) Lie algebra valued one form and is given by
1
3
.
A = i .. d(1 , 3 , . . . , 14 , 16 ),
14
16
= EO t E,
(4.35)
where
E t(e1 , e3 , e5 , e7 , e10 , e12 , e14 , e16 ),
(4.36)
and
1 1
P d P+ ,
N + N
1
1
2
d(N
)
P
+
P
dP
.
= i 2
+
+
+
N
2N 2
O i
(4.37)
(4.38)
O = i
1 + y3
(1 + y3 )
1
1
116 dy3 0 i yi dy3 + (1 + y3 ) 0 i dyi
= i
2(1 + y3 ) (1 + y3 )
1 i j
(4.39)
, Cij .
4
Here we have introduced
Cij yi dyj yj dyi ,
(4.40)
38
and used
X
yi dyi = 0.
(4.41)
i=1,2,...,9
Observe that
116 dy3 0 i yi dy3 + (1 + y3 ) 0 i dyi
X
= 116 + 0 3 dy3 +
0 A (dyA + C3A ),
(4.42)
A=1,2,4,5,6,7,8,9
as well as
(
E
116 +
0
dy3 +
)
(dyA + C3A )
0
E = 0,
(4.43)
due to the orthogonality of different eigenvectors. By the same reason, the last line in
Eq. (4.39) with y3 term involved will not contribute to the Berry connection either. We
conclude that only the last term of O with i, j 6= 3 contributes to the nonabelian Berry
phase.
Notice that the vector E satisfies
t
EE = 12 1 + 0 3 ,
(4.44)
0 i t EE + t EE 0 i = 0 i ,
Defining 8 8 matrix
i
ij E i , j t E,
2
we obtain
1
Cij ij .
A=
4(1 + y3 )
for i = 1, 2, 4, 5, 6, 7, 8, 9.
(4.45)
(4.46)
(4.47)
Here the indices i, j run over the eight directions. The generators ij is found to form an
so(8) algebra:
ij
(4.48)
, kl = 2i j k il j l ik ik j l + il j k .
The curvature is
F = dA iA A,
1
1
dy3 Cij ij .
(4.49)
= dyi dyj ij
4
4(1 + y3 )
As in the case of the SU(2) Berry connection, we change our coordinate system to that
on R8 :
2zi
1 z2
,
y3 =
.
(4.50)
yi =
2
1+z
1 + z2
Here zi (i = 1, 2, 4, . . . , 9) are the orthogonal coordinates on R8 . The Berry connection
can be rewritten as
1
(zi dzj zj dzi ) ij .
(4.51)
A=
2(1 + z2 )
39
(4.52)
(4.54)
Tr(F 4 ) = 1.
C4 (P ) = c4 (P ) =
2
4!
Our eigenbundle is in fact nontrivial and has the fourth Chern number 1.
0
b
= P exp i A` (xM )
.
P
`
(5.1)
The results from our computation in the last section are summarized as Eqs. (4.17), (4.23):
1 y3
dT T 1 ,
2
1
1
(zi dzj zj dzi ) ij ,
ASU(2) =
2 1 + z2
i
ij E[ i , j ] tE,
2
for the generic type III action LIII , giving the SU(2) monopole and as Eqs. (4.47), (4.51):
ASU(2) =
1
Cij ij ,
4(1 + y3 )
1
1
(zi dzj zj dzi ) ij ,
ASU (8) =
2 1 + z2
i
ij E i , j tE,
2
for the generic type I, II action LI,II , giving the SU(8) monopole. Putting these together,
we state that the expectation value of the projector of a fermionic eigenmode is given by
the path-ordered exponential of the integration of the connection one-form and that this
factor in the case of USp matrix model is controlled by the SU(2) or the SU(8) nonabelian
monopole singularity sitting at the origin of the parameter space XM . In the case of the
SU(8) monopole, it is a pointlike singularity in nine dimensions while in the case of the
SU(2) Yang monopole it is a singularity which does not depend on the four antisymmetric
ASU (8) =
40
directions. The latter one, viewed as a singularity in the entire space, is not pointlike but is
actually a four dimensionally extended object. The emergence of these interesting objects,
albeit being aposteriori, justifies the study of this expectation value rather than of the
fermionic part of the partition function.
The matrix models in general contain many species of fermions, which couple to
different spacetime points or D0 branes. They provide a collection of nonabelian Berry
phases rather than just one. With this respect, it is more appropriate to consider the
manybody counterpart indicated in Section 2 (Eq. (2.28)):
" I
#
Y X
Y X
Y
b
b
=
=
tr` P exp i A` (xM ) , (5.2)
P
P
`
`I+
`I+
`I+
where the subset I+ of all eigenmodes is taken over the eigenmodes belonging to the
positive eigenvalues.
Actually, Sfermion S + SF of the USp matrix model contains many terms consisting of
the generic LI,II type action as well as the generic LIII type action. Listing the parameters
which they depend on, we obtain from Eq. (3.17)
a
b
a
b
xM
, (xM
xM
),
(i) xM
a
b
a
b )),
(ii) xM (xM ), (xM (xM
a
a
(5.3)
(iii) 2x , 2x .
a
b
Singularities occur when the two points xM and xM collide in the case of the first line, when
a and (x b ) collide in the case of the second line and when x a lies in the orientifold
xM
M
M
surface. Similarly from Eq. (3.23), we obtain
(5.4)
(iv) xa m(f ) 0 .
The singularities occur when xa is away from the orientifold surface in the x4 direction by
m(f ) . The situation is depicted in Fig. 1.
a,b(K)
, K = 1, . . . , 6, and the last two cases
We denote the first six cases of Eq. (5.3) by xM
a(K 0 )
of Eq. (5.3) and the case of Eq. (5.4) (sum over a and f ) by x . In the case where more
than two spacetime points collide, we will obtain an enhanced symmetry which supports a
nonabelian monopole.
Fig. 1.
41
Table 1
As for Eq. (5.2), the subset I+ is taken over the eigenmodes seen in Eq. (5.3) and
Eq. (5.4). To write this more explicitly,
" I
" I
#
#
YY
YY
a,b(K)
a(K 0 )
tr8 P exp i ASU(8) xM
tr2 P exp i ASU (2) x
.(5.5)
K a<b
K0
In the context of a matrix model for unified theory of superstrings, measuring Eq. (2.28),
or Eq. (2.38) will provide means to examine spacetime formation suggested by the model.
The presence of colliding singularities of spacetime points in general means a dominant
probability to such configurations. As we have said, there are two varieties of singularities
we have exhibited in this paper. The singularities of the SU(8) monopoles appear to be
evenly distributed in all directions as long as the off-diagonal bosonic integrations are
ignored. This type is present both in the IIB matrix model and the USp matrix model.
The singularities of the SU(2) monopoles appear to be a string soliton of four-dimensional
extension present in the USp matrix model and is not shared by the IIB matrix model. It
is tempting to think that the four-dimensional structure is formed by a collection of the
SU(2) monopoles. We put the spacetime picture emerging from contributions of various
eigenmodes of the USp matrix model in Table 1.
As for the v0 direction, we have used this to parametrize the path in the remaining nine
directions. The price we have to pay is that we have nothing to say on the distributions
of the spacetime points in this direction and this is closely related to the problem of the
scaling limit. Further progress and understanding of the spacetime formation of matrix
models require overcoming this point.
(6.1)
42
Let
(x) = xi i .
(6.2)
(6.3)
That is, i are in fact gamma matrices. It is said that i give a Cliff(R m+1 ) module
structure to C k .
Let
(6.4)
P = 12 1 (x) , for |x| = 1
be an orthogonal projection onto the 1 eigenspace of (x). Let
= (x, ) S m C k : (x) =
(6.5)
(6.6)
It is obvious that the orthogonal projection defined here is exactly the projection operator
defined from our first quantized Hamiltonian. The Berry connection is just the connection
on eigenbundle + or . It has been shown that the Chern number of this eigenbundle
is
Z
cj + = i j 2j Tr 0 2j .
(6.7)
S 2j
From this formula on the Chern number, we see that when m is odd, Tr( 0 m )
vanishes and the eigenbundle is trivial. When m is even, we find
(6.8)
Tr 0 m = 2m/2 (i)m/2 .
Therefore the m2 th Chern number should be 1. We have proven this explicitly by deriving
the SU(2) Berry connection on R5 (m = 4 case), and the SU(8) Berry connection on
R9 (m = 8 case), and the curvature two-forms associated with them. The U (1) Berry
connection on R3 (m = 2 case) is well known. The Berry connection on R7 (m = 6 case)
may be related to some configuration in the reduced matrix model.
Having formulated the eigenbundles associated with the nonabelian Berry phase in
arbitrary odd dimensions, we can safely state that the connection one form and the
curvature two form on Rm+1 are given by
A=
1
(zi dzj zj dzi ) ij
2(1 + z2 )
F=
1
dzi dzj ij ,
(1 + z2 )2
(6.9)
and
(6.10)
43
Acknowledgements
We thank Toshihiro Matsuo and Asato Tsuchiya for helpful discussions, Soo-Jong Rey
and Arkady Tseytlin for interesting remarks, and Ashoke Sen and Ke Wu for useful
comments.
Appendix
In this appendix, we give some details of the derivation of the component expression
of the action (Eq. (3.17)). Let us first expand adj and asym by the generators stated
respectively in Eq. (3.15) and in Eq. (3.16):
adj =
k
X
ODS a
ODT a
Sa,a+k + adj
Ta,a+k
adj
a=1
X
DS
adj
ab
DT
(Sab Sa+k,b+k ) + adj
ab
(Tab + Ta+k,b+k )
a<b
OS ab
OT ab
(Sa,b+k + Sb,a+k ) + adj
(Ta,b+k + Tb,a+k ) ,
+ adj
X
DS ab
DT ab
(Sab + Sa+k,b+k ) + asym
(Tab Ta+k,b+k )
asym
asym =
a<b
OS
+ asym
ab
OT
(Sa,b+k Sb,a+k ) + asym
ab
(Ta,b+k Tb,a+k ) .
(A.1)
(A.2)
Let us explain our notation more carefully. adj is expanded by Eq. (3.15) which consists
of three sets of generators: the first set of generators (the first line of Eq. (3.15)) is in
the off-diagonal elements of the diagonal blocks; the second set of generators (the second
line of Eq. (3.15)) is in the off-diagonal elements of off-diagonal blocks; and the third
set of generators (the third line in Eq. (3.15)) is in the diagonal elements of the offdiagonal blocks. We, therefore, distinguish these generators by D (the diagonal blocks),
O (off-diagonal elements of the off-diagonal blocks) and OD (diagonal elements in the
off-diagonal blocks) respectively. In each set of the generators, there are contributions
from both the real part S and the purely imaginary part T . We distinguish these by the
superscript S and T .
As for asym , the bases are O type and D type only. In the above expansion (Eqs. (A.1),
(A.2)), the component fields are specified by the superscript specifying the species of the
ODS
) are the
generators and the subscript specifying the representation. For example, (adjoint
component fields of the adjoint fermions which correspond to the expansion coefficients
of the S type OD generators. The same is true for the other components.
Let us denote
(A.3)
i S M T T M S = 2 Im S MT .
The component action can then be written as
Xn
o ab n DT ab
o
DS ab
DS ab
DT ab
+ asym
+ asym
M
adj
S = 4 Im
adj
a<b
44
+ 4 Im
Xn
o a+k,b+k n DT ab
o
DS ab
DS ab
DT ab
+ asym
asym
adj
M
adj
a<b
+ 4 Im
ODS a
ODT a
Ma,a+k adj
adj
+ 4 Im
Xn
OS
adj
a<b
+ 4 Im
Xn
OS
adj
ab
ab
OS
+ asym
OS
asym
(A.4)
a<b
Here Mab = x a x b . Let us redefine the component fields, introducing complex notation
DU
DS
DT
adj
iadj
,
adj
DL
DS
DT
asym
+ iasym
,
asym
OU
OS
OT
adj
iadj
,
adj
DL
DS
DT
adj
adj
+ iadj
,
DU
DS
DT
asym
asym
iasym
,
ODU
ODS
ODT
adj
adj
iadj
,
OL
OS
OT
adj
adj
+ iadj
,
ODL
ODS
ODT
adj
adj
+ iadj
,
OU
OS
OT
asym
asym
iasym
,
OL
OS
OT
asym
+ iasym
.
asym
We obtain
S=
Xn
DU
adj
ab
DU
+ asym
(A.5)
ab o ab n DU ab
o
DU ab
+ asym
M
adj
a<b
Xn
o a+k,b+k n DU ab
o
DU ab
DU ab
DU ab
+ asym
asym
adj
M
adj
a<b
Xn
DL
adj
ab
DL
+ asym
ab o ab n DL ab
o
DL ab
+ asym
M
adj
a<b
Xn
o a+k,b+k n DL ab
o
DL ab
DL ab
DL ab
+ asym
asym
adj
M
adj
a<b
ODU a
ODU a
Ma,a+k adj
adj
ODL a
ODL a
Ma,a+k adj
adj
X
a
Xn
OU
adj
a<b
Xn
OU
adj
a<b
Xn
OL
adj
a<b
Xn
OL
adj
ab
ab
ab
ab
OU
+ asym
OU
asym
OL
+ asym
OL
asym
(A.6)
a<b
(A.7)
LII (, ; xM , yM ) ( + ) M xM (yM ) ( + )
( ) M (xM ) yM ( ),
x .
LIII (; xM )
45
(A.8)
(A.9)
(A.10)
with
H diag(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1).
(A.11)
(A.13)
x .
LIII
(A.14)
(A.12)
It is easy to see that the action S can be written in terms of LI , LII , and LIII , as is seen in
the text (Eq. (3.17)).
References
[1] T. Banks, W. Fischler, S. Shenker, L. Susskind, Phys. Rev. D 55 (1997) 5112, hep-th/9610043.
[2] N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya, Nucl. Phys. B 498 (1997) 467, hepth/9612115.
[3] R. Dijkgraaf, E. Verlinde, H. Verlinde, Nucl. Phys. B 500 (1997) 43, hep-th/9703030.
[4] H. Itoyama, A. Tokura, Prog. Theor. Phys. 99 (1998) 129, hep-th/9708123.
[5] H. Itoyama, A. Tokura, Phys. Rev. D 58 (1998) 026002, hep-th/9801084.
[6] M.V. Berry, Proc. R. Soc. London A 392 (1984) 45.
[7] F. Wilczek, A. Zee, Phys. Rev. Lett. 52 (1984) 2111.
[8] H. Itoyama, T. Matsuo, Phys. Lett. B 439 (1998) 46, hep-th/9806139.
[9] B. Chen, H. Itoyama, H. Kihara, Mod. Phys. Lett. A 14 (1999) 869, hep-th/9810237.
[10] H. Itoyama, A. Tsuchiya, Prog. Theor. Phys. 101 (1999) 1371, hep-th/9812177.
[11] H. Itoyama, A. Tsuchiya, Prog. Theor. Phys. Suppl. 134 (1999) 18, hep-th/9904018.
[12] C.N. Yang, J. Math. Phys. 19 (1978) 320; J. Math. Phys. 19 (1978) 2622.
[13] W. Taylor IV, Phys. Lett. 394B (1997) 283, hep-th/9611042.
[14] O.J. Ganor, S. Ramgoolam, W. Taylor IV, Nucl. Phys. B 492 (1997) 191, hep-th/9611202.
[15] G. Moore, N. Nekrasov, S. Shatshvile, hep-th/9803265.
[16] I. Kostov, P. Vanhove, hep-th/9809130.
[17] H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, T. Tada, Prog. Theor. Phys. 99 (1998) 713, hepth/9802085.
[18] T. Suyama, A. Tsuchiya, Prog. Theor. Phys. 99 (1998) 321, hep-th/9711073.
[19] T. Hotta, J. Nishimura, A. Tsuchiya, Nucl. Phys. B 545 (1999) 543, hep-th/9811220.
[20] M. Claudson, M. Halpern, Nucl. Phys. B 250 (1985) 689.
[21] H. Itoyama, Phys. Rev. D 33 (1986) 3060.
[22] A. Smilga, Nucl. Phys. B 287 (1987) 589.
[23] A.A. Belavin, A.M. Polyakov, A.S. Schwartz, Yu.S. Tyupkin, Phys. Lett. B 59 (1975) 85.
46
Abstract
The orbifold CFT dual to string theory on AdS3 S 3 allows a construction of gravitational
actions based on collective field techniques. We describe a fundamental role played by a Lie algebra
constructed from chiral primaries and their CFT conjugates. The leading terms in the algebra at large
N are derived from the computation of chiral primary correlation functions. The algebra is argued to
determine the dynamics of the theory, its representations provide free and interacting Hamiltonians
for chiral primaries. This dynamics is seen to be given by an effective one plus one dimensional
field theory. The structure of the algebra and its representations shows qualitatively new features
associated with thresholds at L0 = N, L0 = N/2 and L0 = N/4, which are related to the stringy
exclusion principle and to black holes. We observe relations between fusion rules of SU q (2|1, 1) for
q = ei/(N+1) , and the correlation functions, which provide further evidence for a non-commutative
spacetime. 2000 Elsevier Science B.V. All rights reserved.
PACS: 11.25; 11.25.H; 12.40.N
Keywords: AdS/CFT correspondence; Non-commutative geometry; Duality; Large N ; Black holes
1. Introduction
In the context of the AdS/CFT duality [13] we began in [4] (hereafter referred to as
I) a construction of elements of supergravity on AdS3 S 3 based on a simple orbifold
conformal field theory with target space S N (T 4 ) or S N (K3). The novel feature of the
emerging gravitational theory is a stringy exclusion principle [5] which follows from the
CFT. It was seen [4] that this exclusion principle implies a role for a non-commutative
spacetime (a subject reviewed, for example, in [6]). Similar aspects of non-commutativity
1 antal@het.brown.edu
2 mm@het.brown.edu
3 ramgosk@het.brown.edu
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 4 7 - 4
48
in closed string backgrounds were also found recently in [7], and in earlier works on matrix
models [8].
In this paper we study three point functions of chiral primary fields in the orbifold CFT
for a large T 4 . We see that the correlators have a 1/N expansion, and we exhibit the leading
terms in the 1/N expansion as quantities which can be compared with semiclassical gravity
calculations (see [9]). These correlators are essential ingredients in applying collective field
theory [10,11] to derive spacetime actions. The exact three point functions also exhibit
finite N effects which can be interpreted in terms of a non-commutative spacetime along
the lines of I.
The techniques of collective field theory allow, in general, the construction of spacetime
actions starting from the algebra of SN invariant variables. Since the short representations
related to chiral primaries have been matched with gravity on AdS3 S 3 , it is instructive to
focus on the dynamics associated with these representations and their CFT conjugates.
We describe a central role played by a Lie algebra of observables associated to the
chiral primaries. The simplest representations of this Lie algebra involve the Fock space
generated by a certain class of chiral primaries (those which are generators of the chiral
ring). One wants to focus on these representations first because they include states created
by the graviton, whose propagation and dynamics is of interest. We describe this dynamics
in terms of two-dimensional field theory. This represents the simplest representation of the
algebra. Other reps. will be of relevance when we go beyond chiral primaries and study
more stringy states [1214].
The paper is organized as follows. In Section 2, we summarize the field content and
basic symmetries of the CFT, and then describe the computation of some exact correlation
functions involving twisted sector chiral primaries. In Section 3 concentrating on the
untwisted sector we review the algebra of observables given by the creationannihilation
operators and their commutators. We study a Fock space representation of this algebra
where a creation operator is associated to each generating chiral primary. We use a coherent
state description for the Fock space and derive, using the correlation functions of the chiral
primaries, a formula for a symplectic form (equivalently the kinetic term of a Lagrangian)
on the space of coherent states. We also obtain a formula for the Hamiltonian in this
coherent state representation. We outline how these exact formulae can be developed in
a 1/N expansion into a realization of the algebra in terms of free oscillators. In Section 4,
we describe the algebra when twisted sector operators are taken into account. We describe
the leading terms in a systematic 1/N free field representation of the algebra. We study the
form of the Hamiltonian, and observe that free field realizations which differ in detailed
form, lead to the possibility of a quadratic Hamiltonian and an interacting one. We write
a formula for the interacting Hamiltonian and observe similarities with other gravitygauge theory correspondences, which suggest that the dynamics of chiral primaries can
be understood as a reduction of AdS S backgrounds to a (1 + 1)-dimensional system.
In Section 5 we turn to finite N effects, like the stringy exclusion principle, where the
free field realizations start to break down. We outline a non-trivial property of the finite
N Lie algebra which follows from independently known facts about the cohomology of
the instanton moduli spaces. We return to the picture of a non-commutative spacetime
49
(2.1)
(2.2)
J (z) =
1 X a
I I a ,
2
(2.3)
where a is the left moving component of the corresponding fermion, and the ,
a b are used to raise and lower the spinorial indices. The lowest modes of this currents
{L0,1 , Ga1 , J0 } will generate together to their right counterparts the SU(2|1, 1)L
2
SU (2|1, 1)R symmetry which is mapped in the AdS/CFT correspondence to the superisometries of the AdS3 S 3 . In addition, it is possible to construct other symmetries
commuting with the previous set and related to global T 4 rotations, and they are given
by the following currents:
K a b (z) =
1 X a b
I I XIa a XIba ,
2
(2.4)
and similar expressions for the right-movers. This symmetry acts non-trivially on the space
of chiral primaries.
50
2 i
n I
(1...n) (0) + .
(2.5)
(2.6)
The general twist operator for a general conjugacy class of SN is obtained by its
decompostion into cycles. We may define a twisted sector vacuum by |(1 . . . n)i =
(1...n) (0)|0i. Over this vacuum we can write the following mode expansion:
2 i
m
iX
m e n I m z n 1 ,
(2.7)
XI (z) =
n m
where [m , n ] = mm+n,0 and m |(1 . . . n)i = 0 for m > 0. The expression above can be
generalized to any other primary operator constructed from XI , the only difference being
that in the exponent of z, 1 is replaced by the corresponding conformal dimension. This
shows that the SN invariant operators have nonsingular OPE with the twist operator. The
dimension of the twist operator can be calculated by computing the energy momentum
tensor in the state |(1 . . . n)i and it is (for one boson, left-mover):
1
1
n
.
(2.8)
1(1...n) =
24
n
For our system, we will bosonize first the fermions by introducing 8I1,2 (z, z ) = I1,2 (z)+
I1,2 (z), the final theory being one of 6N free bosons and we will construct the chiral
primaries using these bosons:
I+a (z) = eiI (z) ,
a
(2.9)
(2.10)
for right-movers. Using these expressions, the SU(2)L currents are given in the standard
way:
iX
I1 (z) + I2 (z) ,
J3 =
2
I
X
X
1
2
1
2
+
ei(I +I )(z) ,
J =
ei(I +I )(z).
(2.11)
J =
I
Let us construct the Zn twist operators which can be used to build SN invariant chiral
primaries by averaging over SN . These twist operators play a distinguished role in the chiral
ring in the sense that they can be used to generate the rest by using the ring structure. In
the correspondence with gravity in AdS3 S 3 they are in one-to-one correspondence with
51
single particle states. These vertex operators are written in terms of the 6 free scalar fields
and the twist operators. Consider first the fields appearing in the untwisted sector, n = 1:
(0,0)
= 1,
O(1)
a
O(1)
= 1+a ,
a
O(1)
= 1+a ,
aa ,b
= 1+a 1+a 1+b ,
O(1)
0
a,b
O(1)
= 1+a 1+b ,
a,bb
O(1)
= 1+a 1+b 1+b ,
(2,2)
= 1+1 1+2 1+1 1+2 ,
O(1)
(2.12)
where (1) represents the one cycle containing only I = 1. We will also construct vertex
operators out of the simple twists which will correspond to the nontrivial chiral fields
associated with the twist. For that we consider the cycle (1 . . . n) and we define the
following Sn invariant 6-dimensional vector observables (left and right):
1 X
YL (z) =
XI1 L , XI2 L , XI3 L , XI4 L , I1 , I2 (z),
n
I =1,...,n
YR (z) =
1
n
XI1 R , XI2 R , XI3 R , XI4 R , I1 , I2 (z).
(2.13)
I =1,...,n
Let us consider now the following field which consists of the twist operator for all 6
fields and having momenta along the 2 extra 8 dimensions:
(0,0)
O(1...n) (z, z ) = ei
n1 1 n1 2
n1 1
2 z))
z)+ n1
I ( 2n I + 2n I (z)+ 2n I (
2n I (
(2.14)
n1
The dimensions of this field is ( n1
2 , 2 ) and its charges can be shown to be equal to the
dimensions. In comparison with the formulae in I, we have extracted and made explicit
the U (1) charges by exhibiting exponentials of the bosons 8, leaving a twist operator for
both the 8 and X. This field will be used to construct the SN invariant chiral primary
On(0,0)(z, z ) [5]. A more precise way to write which we will further generalize to the other
cases is to write:
(0,0)
(2.15)
n1
n1 n1
where kL = (0, 0, 0, 0, n1
2 , 2 ), kR = (0, 0, 0, 0, 2 , 2 ) represents the left and right
momenta in 6 dimensions. The dimensions of a field of type (2.15) is given by the following
general formula:
1
1
1
n
+ kL2 ,
1O = 6
24
n
2n
1
1
1
O =6
n
+ kR2 ,
(2.16)
1
24
n
2n
and for our ks we obtain the dimensions above. For the charge we can read it from the
momenta on only the two extra dimensions, the twist being uncharged. In order to construct
all the other chiral fields we will combine the construction in untwisted sector with the
twist. We will focus then on the following construction:
(0,0)
A
A
(z, z ) O(1)
(z, z )O(1...n) (z, z ),
O(1...n)
(2.17)
where A index takes care of the spinorial indices which already fully appear at untwisted
level. We give for these operators a construction similar to (2.15) where what makes the
52
distinction between them is the value for the momenta. This construction will be further
justified by looking to the OPE of chiral fields.
We will focus further on scalar chiral primaries, namely those fields corresponding
(0,0)
(1,1)
(2,2)
to On (z, z ), On (z, z ), On (z, z ) only. We can characterize the fields as being a
twist operator in 6 bosonic dimensions and having definite momenta on the bosonic fields
coming from the bosonization 81,2 (the dimension and the charge for this operators are
equal):
(0,0)
(z, z ) corresponds to momenta
O(1...n)
n1 n1
n1 n1
,
,
kR = 0, 0, 0, 0,
,
(2.18)
kL = 0, 0, 0, 0,
2
2
2
2
n1
and has dimension ( n1
2 , 2 ),
(1,1)
(z, z ) corresponds, for example, to momenta
O(1...n)
n+1 n1
n+1 n1
,
,
kR = 0, 0, 0, 0,
,
.
kL = 0, 0, 0, 0,
2
2
2
2
(2.19)
3 other combinations of leftright momenta in the case of T 4 are possible. The dimension
is ( n2 , n2 ).
n+1 n+1
kR = 0, 0, 0, 0,
,
2
2
(2.20)
n+1
and has dimension ( n+1
2 , 2 ).
We will define below the precise relation between the On and the corresponding O(1...n)
in a general context. For this let us consider the following basis of forms leaving on the
target space X and spanning its H (1,1)(X); we will denote them as ar a where r counts the
forms (for example, r = 1, . . . , 4 for T 4 , and r = 1, . . . , 20 for K3) and a, a = 1, 2 and
they are X indices. Using this forms and summing over all permutations, it is possible to
describe the scalar chiral primaries up to a normalization constant which will be determined
in the next section. We write here the full expression for this operators:
On(0,0)(z, z ) =
X
1
Oh(1...n)h1 (z, z ),
[N!(N n)!n]1/2
hSN
X a,a
1
Oh(1...n)h1 ar a (z, z ),
Onr (z, z ) =
[N!(N n)!n]1/2
hSN
X ab,a b
1
1
Oh(1...n)h1 ab a b (z, z ).
On(2,2)(z, z ) =
1/2
4 [N!(N n)!n]
(2.21)
hSN
53
From this we are able to write their three-point correlation functions. Using the definition
of the chiral primaries, we see that there are some restrictions coming from charge
conservation, from the fact that except the twist operator, these operators are fermionic
in nature and from SN multiplication law. It is straightforward to observe that we will be
interested from the permutation point of view in two kind of processes:
(1) The joining of two cycles overlapping on only one of their components giving a
longer permutation, which gives correlation functions between fields which have (p, q)
form indices as follows:
(i) (0, 0) + (0, 0) (0, 0),
(ii) (0, 0) + r r,
(iii) (0, 0) + (2, 2) (2, 2),
(iv) s + r (2, 2);
(2) The joining of two cycles overlapping on two components giving a longer
permutation, which gives correlation functions of the form: (0, 0) + (0, 0) (2, 2).
We will first study the processes listed above on the components of the chiral primaries
and then we will sum over all permutation to recover the SN invariance. Using the notation
of [17] we will denote the left-moving twist operator having momenta as O(1...n) (k) where
k is the momenta and we assume that they are normalized in the sense that their two-point
takes the following form:
(2.22)
Oh (k1 ) ()Og (k2 )(0) = h,g 1 k1 ,k2 .
Using the same method, we will derive in Appendix A the OPE of a twist (1 . . . n) having
momenta kn and the twist (nn + 1) having momenta k2 involving only the twist (1 . . . n + 1)
(corresponding to the first kind of process) the result being:
O(nn+1) (k2 )(u)O(1...n) (kn )(0) =
Since we are dealing with operators which are chiral primaries one has the exponent of z
being 0, and the C(2, n|n + 1; k2, kn ) are determined in the appendix. We will write here
the expression for C(2, n|n + 1; k2, kn ) and we will see what they are in our processes
(only left-movers):
1 n + 1 1/2
O(1...n+1) (0) + O(1...n+1n) (0) ,
O(nn+1) (u)O(1...n) (0) =
2
n
1 1
1
1
(0) = O(1...n+1)
(0) + O(1...n+1n)
(0) ,
O(nn+1) (u)O(1...n)
2
1/2
1
n
12
12
12
(0) =
(0) + O(1...n+1n)
(0) .
(2.24)
O(1...n+1)
O(nn+1) (u)O(1...n)
2 n+1
The method also gives us OPE for all the other cases involved in the list above case (1)
for a 2 twist and an n twist overlapping over only one component. Knowing that the total
twist is the product of the left- and right-moving twist and excluding out of the OPE those
cases when the left and right twists do not coincide we see that the result of the OPE is
54
essentially the square of what we have in the previous equations. We can also observe that
an extrapolation to the n = 1 case, when the twist is nothing than the identity operator
results in the following OPE:
a
(0),
O(12)(u) +a (0) = O(12)
1 12
+2
(0) = O(12)
(0).
O(12)(u) +1 (1...n)
2
(2.25)
The equations above show that it is enough to figure out the OPE for the twist fields
and then generalize to the other chiral primary fields. We will use from this point the CFT
rules to compute the OPE for any twist operators overlapping on only one component of
the permutations. What we will do is to compute the OPE for 3 twist of type (1 . . . n 1),
(n 1n) and (n . . . n + k 1) at different locations in the complex plane, and by making
different limits and an induction process we obtain the following results including now the
right moving twist:
n+k1
(O(1...n+k1) (0) + ),
(2.26)
2nk
where the dots are for the field corresponding to the other permutation obtained by
multiplying the two permutations. Using the equations above we derive the OPE for the
processes 1) and we list them here:
(1...n) (0) =
O(n...n+k1) (u, u)O
1
r
(0) + ,
O(1...n+k1)
2k
n
12,1 2
12,1 2
O(1...n+k1)
(1...n) (0) =
(0) + ,
O(n...n+k1) (u, u)O
2k(n + k 1)
2
12,1 2
r
s
r s O(1...n+k1)
(1...n) (0) =
(0) + .
O(n...n+k1) (u, u)O
n+k1
r
(1...n)
(0) =
O(n...n+k1) (u, u)O
(2.27)
Note that if we extrapolate the above expressions for the case n = 1 we have:
1
r
O(1...k)
(0) + ,
2k
1
12,1 2
12,1 2
(1) (0) = 2 O(1...k)
(0) + ,
(2.28)
O(1...k) (u, u)O
2k
where for the definition of untwisted operators we use (2.12).
In addition there is the (2) process which we will focus now on. Using the above rules
there is a single OPE which has to be figured out and this is the one involving only two
twists for the same 2 cycle (overlapping of 2), and then the result can be extended to the
other permutations using the known OPE from the previous list and an induction process.
Consider the OPE of 3 operators: the twists and the one formed from all 4 adjoints of
fermions and no twist. In the limit when we put together the twist and the fermions this
process is nothing else than the one used for normalization of the twist of length 2; in the
limit when one puts together the twists and then the fermions and ask the question on what
kind of operator would give this result one is led to consider the following OPE:
r
(1)
(0) =
O(1...k) (u, u)O
(12)(0) = 1 2 1 2 (0).
O(12)(u, u)O
(2.29)
55
1
12,1 2
O(1...
(0) + . . . , (2.30)
n...n+k2)
nk(n + k 3)
where by n we mean that n is missing in this permutation. We have at this moment the 2
and 3-point functions for all fields which we will use in constructing the chiral fields.
2.2. Correlation functions of chiral primaries
In this section we will put back the sums over permutations with appropriate
normalization factors, which together with the 3-point functions deduced in the previous
section will allow us to write the full 3-point functions for the chiral primaries. For On(0,0)
we have:
X
Oh(1...n)h1 (z, z ).
(2.31)
On(0,0)(z, z ) = const
hSN
The 2-point function for individual twists is given (2.22) with a constant to be determined.
X
(0,0)
(0,0)
(0,0)
()On(0,0)(0) = const2
(2.32)
O
On
1 ()O
1 (0) .
h1,2 SN
h1 (1...n)h1
h2 (1...n)h2
Because the 2-point function for twists is normalized to 1, the sums will be rearranged in
other two sums: a sum over all permutations and a sum on only those permutation which
leave a cycle invariant. The sum over all permutation will give a factor of N! whereas the
second sum will give a factor of (N n)!n leading to the expressions already listed in
Eq. (2.21).
For the three-point functions we will use the normalized chiral primaries operators and
we will also show how one can determine them for only three O (0,0) and then list the
full results for all the other cases. The conservation law for the R-symmetry suggest that
the only possibility is having only the process listed in (1) namely a cycle having length
n + k 1, one having length n and one having k and the individual permutations have to
overlapp on one entry:
(0,0)
On+k1 ()On(0,0)(1)Ok(0,0)(0)
X
()Oh2 (n...n+k1)h1 (1)
O
= const
1 1
h1 (1...n+k1)
h1,2,3 SN
h1
Oh3 (1...n)h1 (0) ,
(2.33)
where the const comes from the normalization of each chiral primary field. The individual
terms which appear in the sum are nonzero only if:
1
1
h1 (1 . . . n + k 1)1 h1
1 h2 (n . . . n + k 1)h2 h3 (1 . . . n)h3 = 1,
and in this case they are all equal to the expressions derived in the previous section. By
rearranging the previous permutation equation it is possible to compute the sums, namely:
1
1
1
(1 . . . n + k 1)1 h1
1 h2 ((n . . . n + k 1)h2 h3 (1 . . . n)h3 h2 )h2 h1 = 1.
56
(0,0)
(N n)!(N k)!(n + k 1)3 1/2
(0,0)
(0,0)
,
On+k1 ()Ok (1)On (0) =
(N (n + k 1))!N!nk
r
(N n)!(N k)!n(n + k 1) 1/2 r s
,
On+k1 ()Ok(0,0)(1)Ons (0) =
(N (n + k 1))!N!k
1/2
(2,2)
(N n)!(N k)!n3
(0,0)
(2,2)
,
On+k1 ()Ok (1)On (0) =
(N (n + k 1))!N!k(n + k 1)
1/2
(2,2)
(N n)!(N k)!nk
r s ,
On+k1 ()Okr (1)Ons (0) =
(N (n + k 1))!N!(n + k 1)
(2,2)
(0,0)
On+k3 ()Ok (1)On(0,0)(0)
1/2
(N n)!(N k)!(N (n + k) + 3)
.
(2.34)
=2
(N (n + k 1))!N!(N (n + k) + 2)nk(n + k 3)
Fixing the charges and taking N :
1 1/2 (n + k 1)3 1/2
,
N
nk
1/2
r
1
n(n + k 1) 1/2 rs
(0,0)
,
On+k1 ()Ok (1)Ons (0) =
N
k
1/2
1/2
(2,2)
1
n3
(0,0)
(2,2)
,
On+k1 ()Ok (1)On (0) =
N
k(n + k 1)
1/2
1/2
(2,2)
1
nk
r s ,
On+k1 ()Okr (1)Ons (0) =
N
(n + k 1)
1/2
1/2
(2,2)
1
1
(0,0)
(0,0)
.
On+k3 ()Ok (1)On (0) = 2
N
nk(n + k 3)
(0,0)
(0,0)
On+k1 ()Ok (1)On(0,0)(0) =
(2.35)
(2.36)
1/2
1 1/2
,
(n + k 1)nk
N
1/2
1/2 rs
r
1
(0,0)
,
(n + k 1)nk
On+k1 ()Ok (1)Ons (0) =
N
1/2
1/2
(2,2)
1
(0,0)
(2,2)
,
(n + k 1)nk
On+k1 ()Ok (1)On (0) =
N
1/2
1/2 r
(2,2)
1
s ,
(n + k 1)nk
On+k1 ()Okr (1)Ons (0) =
N
1/2
1/2
(2,2)
1
(0,0)
(0,0)
.
(n + k 3)nk
On+k3 ()Ok (1)On (0) = 2
N
(0,0)
()Ok(0,0)(1)On(0,0)(0) =
On+k1
57
(2.37)
We notice a certain degree of universality: all the three-point functions exibited above for
the case of AdS3 have the same form factor nk(n + k) as the three-point functions of
chiral primaries in other AdS examples, for example, [1820]. 4
An interesting check we can perform on the form of the correlation functions is to see
how close the infinite N result is to the corresponding result obtained in supergravity [9].
An obvious difference between the two results is the symmetry exhibited in the correlation
functions: in the supergravity description there is an SO(21) symmetry, whereas in this
CFT we observe an SO(20) symmetry only. Aside from the differences, we are interested
in the functional dependence of the correlation functions on the SO(4) indices. We choose
to compare the correlation function with three O (0,0) from (2.35) with the one with three
O0 from gravity [9].
The relation between the twisting indices and the SO(4) indices are l1 = n 1, l2 = k 1
and l3 = n + k 2. In order to compare them, we extend the result obtained in the case
l3 = l1 + l2 only, using the SO(4) symmetry. The result for the correlation function is
expected to be a factor depending on l1 , l2 and l3 (which are related to the Casimirs of the
corresponding representations) multiplied with the corresponding SO(4) ClebschGordon
coefficient. The factor should reduce in the extremal limit, when we are looking to l1 , l2
highest weights and l3 = l1 + l2 lowest weight, to the factor obtained in (2.35). The result
is:
(0,0)I
1 ()O (0,0)I2 (1)O (0,0)I3 (0)
O
(l1 + l2 + l3 + 2)2
C I1 I2 I3 ,
N 1/2 (l1 + 1)1/2(l2 + 1)1/2(l3 + 1)1/2
1
(2.38)
where the means that we drop the constant factors and C I1 I2 I3 is the corresponding
ClebschGordon coefficients. At this point, we can compare with the corresponding result
in supergravity obtained in [9] and notice the similarity in functional dependence on the
harmonic indices for the correlation functions:
O0l1 ()O0l2 (1)O0l3 (0)
4 This form, however, does not hold for the conjugates. We thank Samir Mathur for pointing this out.
(2.39)
58
It is expected that a better understanding of the moduli space of either CFT or AdS gravity
will help in making the correpondence exact.
I I
I I ,
I I I I I I .
I I
(3.1)
The indices , , above run from 1 to 4 and we have 4 operators in the first line, 6 in the
second line, and 4 in the third line. We should add to this an operator E which commutes
(1,0)
with everything, and which appears, for example, in the commutator of a an operator A1
and its conjugate. Note that the procedure of successive commutations of the A1 operators
do not generate terms involving a product of four fermions with their conjugates.
The other generators of the Lie algebra include of course the chiral primaries and
conjugates, along with others generated by the commutations. We also have the following
operators and their conjugates.
I 1 I1 ,
I 1 I 2 I 3 I1 I2 I 3 ,
I
X
I 1 I 2 I1 ,
I
X
I 1 I 2 I 3 I 4 I1 ,
I
X
I 1 I 2 I1 I2 ,
I
X
I 1 I 2 I 3 I 4 I1 I2 ,
I
X
I 1 I 2 I 3 I1 I2 ,
I
X
I 1 I 2 I 3 I 4 I1 I2 I 3 .
59
(3.2)
Some of these operators have already been included in the description of the Cartan
above. The operator of the form ()4 ( )4 does not appear in the the commutators. Let us
call this Lie super-algebra ginv , and the corresponding supergroup Ginv . If we nevertheless
include it we get a super-algebra which has rank 16 and is closely related to the Clifford
algebra which is in turn related to SU(16). 5 This suggests that the Lie super-algebra is
actually U (8|8).
3.1. Coherent state representation
We derive in this section the action governing the dynamics of the chiral primaries for
the untwisted sector and then comment on how it is extended to the full set of chiral
primaries and beyond. Consider then the simplified model obtained by reducing the theory
to a quantum mechanical version with the algebra described above. We deal then with the
following set of operators as annihilation operators:
I ,
= 1, . . . , 4,
I = 1, . . . , N,
(3.3)
where is the SU(2) index, is the T 4 index and we also have the adjoints of these as
creation operators. We also focus on the chiral primaries only meaning that we consider
only highest weight states under the action of the SU(2), namely (+), and for now on we
will also drop this index. The Hamiltonian of the system (the one involving only highest
weight under SU(2)) is considered to be the Hamiltonian of a free system of fermions:
1 X
I I .
(3.4)
L0 + L 0 =
2
I
The chiral primary operators in the untwisted sector are built as:
1 X
I ,
A1 =
N I
1 X
I I ,
A1 =
N I
1 X
I I I ,
A1 =
N I
1 X
I I I I .
A1 =
N I
5 We thank J. Gervais for this remark.
(3.5)
60
The idea of the construction is to build them using only the highest weight operators under
SU(2) and SN invariant combination of operators. The dimension and the charge of the
operators are equal to the number of fermion operators.
Let us study now how we write an action involving only these special set of operators.
For this we can use the technique of [26,27]. It consists in introducing a basis of coherent
states using the SN invariant operators already derived and from the expression for the
partition function we derive the Hamiltonian and the symplectic form which governs the
dynamics of the collective variables. The first step is to derive the measure which will be
use together with the coherent basis (A , A ) as:
B
B
(3.6)
B , B = 0eB A1 eB A1 0 .
Here we have taken the index B to run over all the 15 chiral primaries of the invariant
sector. This can be computed using coherent state techniques to find:
N
(3.7)
A , A = Z(, ) ,
where
1
1
Z , = 1 +
N
2N
1
1
+
3!N
4!N
is the answer for one spieces of fermions. Here the following variables are used:
(3.8)
= ,
1
= ,
N
3
1
= + ,
N
N
1
1
(2,2) = (2,2) + + + ,
3! N
2! N
(3.9)
and the dots stand for terms of lower order in 1/N terms.
For determining the representation of the Hamiltonian in , we evaluate its matrix
elements in the coherent basis. In our case it is suitable to evaluate the following:
B
B
(3.10)
Z , , = 0eB A1 e(L0 +L0 ) eA1 B 0 .
Using the expression for the Hamiltonian (3.4) and coherent state tehniques we compute
the following expression:
Z , ,
h
2
2
= Z 1 + 12 , 1 + 12 , 1 + 12 , 1 + 12 ,
i
3
3
4
4
1 + 12 , 1 + 12 , 1 + 12 , 1 + 12 . (3.11)
We can write then the average for the Hamiltonian as:
H (, )
=
61
h |(L0 + L 0 )| i
( , )
N
1
N Z
1
N
2 ( N + )
.
Z
=
=
=0
ZN
Z =0 (1 N1 + )
(3.12)
In the last line we wrote the first quadratic term, the other being also quadratic in but for
the remaining indices. Noting the expression for the tilde variables (Eq. (3.9)) one has a
sequence of cubic, quartic and higher terms which are explicitely determined by the above
representation.
Using the equations above we can now derive the Lagrangian governing the dynamics
of the collective fields we introduced:
L(, ) = L (, ) H (, ),
(3.13)
+
L , =
t
t
+
b
+
(3.14)
log , ,
t
t
and it is linear in time derivatives. Its expression gives the Poisson structure in the phase
space of , . For the other part of the Lagrangian H , we use the expression we computed
in (3.12).
This coherent state technique can also be used to give a free field realization of
the algebra of observables in a 1/N expansion which reproduces correlation functions
involving a small number of operators compared to N . To convert to free fields we have to
find variables which convert the Lagrangian L(, ) into L(a, a ) = aa /t. This can
be done in a systematic large N expansion and reproduces correlators involving a small
number of insertions compared to N . When the number of insertions becomes comparable
to N , effects like the stringy exclusion principle become important. The properties of the
measure then show properties qualitatively different from free fields. For example
N=1
, = 0.
(3.15)
The coherent state technique offers a complementary insight into the exclusion principle.
It is related to the fact that the symplectic manifold generated by the action of elements
of Ginv on the Fock vacuum has a non-trivial symplectic form which cannot be globally
P
brought to the form = i dpi dq i . This phenomenon was emphasized in simpler
models of large N in [26,27]. We expect along the lines of I, that a transformation valid for
finite N can be done using q-oscillators at roots of unity.
62
well. Some elements were described in I. Here we will discuss further properties of the
algebra, and its representations in terms of free fields. As we go from the zero twist to the
twisted sector the the creationanihilation operators aquire an additional index n, which is
reflected in the commutator algebra. The commutators also become more complex, but are
computable using CFT techniques. The general form of the terms are highly constrained by
conservation of L0 , J0 , and the SN symmetry. The N dependences can be obtained by SN
combinatorics. The precise coefficients are related to computations of correlation functions
of the kind done in Section 2.
0,0
defined at the end of Section 2,
Consider the class of chiral primary operators An
denoted here An . By elementary commutator manipulations, followed by contour
integrals, we derived in I an equation which reduces in the large N limit to
"
#
X
1
n n
O O + O n O n .
(4.1)
An , An = 1 + n/2
N
n Tn
We are keeping only the leading terms in the combinatoric factors, e.g., Nn has been
approximated by N n , to demonstrate the key features of the large N counting. Each
operator is normalized to 1.
We will now consider the form of commutators involving different chiral primaries
Am1 and Am2 . For this we will need to discuss commutators of the chiral building blocks
m
m m
(4.2)
O 1 , O m2 = C3 1 2 O3 .
In the above equation mi Tmi where Tmi denotes the set of permutations in SN which
have one cycle of length mi and remaining cycles of length 1. We are not being specific
about the order of magnitude of C in the large N expansion in the above equation, but we
will cure that in a moment. We will assume that m2 is larger than m1 . We can further
decompose the sum over 3 by specifying the number of non-trivial cycles (of length
greater than 1) in the permutation 3 . When 3 belongs to a permutation with k nontrivial cycles of lengths (n1 , n2 , . . . , nk ) we say 3 Tn1 ,n2 ,...,nk , and we write 3 as
n (n )
(n )
3 1 3 2 3 k . In the following equation we have normalized the operators to have
2-point functions equal to one. The C-factors are of order one. And the form of the leading
N dependences have been made explicit:
X
O m1 , O m2
m1 Tm1 , m2 Tm2
N 1/2 1 + O
3 Tn1
1
N
N 1 1 + O
3 Tn1 ,n2
m m2
(m2 m1 + 1, n1 )Cn11
1
N
m m
(m2 m1 + 2, n1 + n2 )C3 1 2 O n1 O n2
N k/2 1 + O
C3 1
O n1
1
N
(m2 m1 + k, n1 + n2 + + nk )
O n1 O n2 O nk .
(4.3)
63
Note that we could have used a basis where the product operators are just products of the
generating SN invariant generating chiral primaries. For example we could have written
X
O 1 O 2
(4.4)
1 Tn1 ,2 Tn2
rather than the sums associated with conjugacy classes as in (4.3). In the sum in (4.4) the
terms 1 and 2 can involve elements which overlap while they do not involve overlapping
elements in (4.3). The advantage of writing the algebra in the form above is that the leading
coefficient can be read off directly from the 3-point functions that have been computed in
Section 2.
Another noteworthy point in (4.3) is that after correct normalization the terms involving
permutations with higher numbers of non-trivial cycles are sub-leading in the large N
expansion. This means that by restricting attention to leading powers of 1/N we can define
appropriate contractions of the Lie algebra of interest which can be simpler than the exact
Lie algebra and may be investigated in connection with the qualitative properties such as
the exclusion principle and the related properties of correlation functions which we discuss
in Section 5.
While it is easy to prove that the full set of operators appearing in the commutator in
(4.1) is of the form given there, we have not proved that the above form of operators is
the full set of terms that can appear in the commutator in (4.3). We have some evfidence
that the form of the algebra, restricting attention to pure twists is of the above form. For
example we know that the RHS cannot contain a descendant of a chiral primary, which
would not satisfy L0 = J0 as the LHS does. The L0 and J0 conservations alone do allow
other forms of operators which can be ruled out by using SN selection rules. In particular, it
appears that we do not need to include operators of the form (OO )O . The generalization
p,q
of the above ansatz to include the chiral factors which come up in the An is easily done
at the cost of some extra notation.
We will be interested in a class of free field realizations of the algebra obtained by
considering the full non-chiral operators. We note nevertheless that similar free oscillator
realizations can be considered for the chiral half of the algebra and that has some
similarities with the large N counting relevant for the full operators. We can realize the
algebra in terms of free oscillators, with [n , m ] = n,m . In the large N expansion, for
example, by
X
1
m1 m2 (m1 + m2 1, m) +
1 + O N1
Am = m +
N
m1 ,m2
X
1
m1 m2 mk
+ k/2 1 + O N1
N
m ,m ,...,m
1
(m1 + m2 + + mk k + 1, m).
(4.5)
The coefficients are matched with the algebra by using a recursive procedure.
Now we move on to the combination of the left and right operators which give the
full chiral primaries. Using the commutators of the chiral sectors of the form above, we
64
can write down the commutators of the non-chiral operators which contain terms written
below
X
m m
O 1 O 1 , O m1 O m2
m1 Tm1 ,m2 Tn2
N k/2 (m2 m1 + k, n1 + n2 + + nk )
O n1 O n2 O nk O n1 O n2 O nk ,
which have a similar structure to the terms in (4.3).
A free field representation for the algebra can be constructed where
1
1
1 + O N1 ( + ) + + l/2 1 + O
An = n +
N
N
1
N
(4.6)
l+1 .
(4.7)
The second term in the above expression is quadratic in the s. The higher term
that,
weighted by N l/2 is a polynomial in the , of degree l + 1. It is noteworthy
N.
for fixed degree of the
polynomial,
the
expansion
involves
powers
of
1/N
and
not
1/
The odd powers of N can be removed by redefining the operators. This is in agreement
with the idea that the algebra is closely related to the
exclusion principle and to quantum
groups, which both involve the parameter N , and not N .
There are extra operators appearing on the RHS of the equation have the form
On O m (O nm ) = n m + .
(4.8)
The composite operators have also been normalized to have unit two-point function. They
are sub-leading in the 1/N expansion.
4.1. Hamiltonian
The free field realizations of the algebra in the large N limit provide representations of
operators of the theory and of the Hamiltonian. Based on studying the free field reps of
sub-algebras of the full algebra (in particular the global case of the previous section) we
expect that there exists first a free representation where the Hamiltonian is quadratic:
X
p + q p,q p,q
(4.9)
m m .
m+
H (f ) =
2
m,p,q
In this picture the interactions are contained in the non-linear forms of the other (in
particular the creationanihilation) generators. We expect another representation where
the Hamiltonian is nonlinear and exibits interactions in 1/N . For the chiral primaries
represented by (0, 0), (1, 1) and (2, 2) forms this takes the form
X X
p + q p,q p,q
I
m m
m+
H =
2
m,p,q m,p,q
h
X 1
+
v(k, n) nk+1 k n + n k n+k1
N
n,k
(2,2)
r
+ nk+1
kr n + h.c. + nk+1
k(2,2) n + h.c.
(2,2)
+ nk+1 kr nr + h.c.
i
(2,2)
k n + h.c. ,
+ nk+3
65
(4.10)
with the form factor derived from the three-point functions. The above form would
represent the leading term with higher interactions in powers of N . Concerning this
Hamiltonian and the emerging (1 + 1)-dimensional field theory one has the following
comments. It effectively summarizes the dynamics of chiral primaries with their correlation
functions. The dimension conjugate to the twist n corresponds to a coordinate of
AdS obtained after chiral primary reduction. The structure of the (1 + 1)-dimensional
Hamiltonian is similar in form to the collective field theory [11] of 2d strings. The fact that
there is an analogy between the 2d non-critical string and radial dynamics in the AdS/CFT
correspondence has been observed earlier, for example, [2,28].
The above structure (of the algebra) and the Hamiltonian is operational on the space
of chiral primary operators. We should mention the extension to a more general class of
states as follows: using the SUSY algebra would provide couplings to other fields of the full
short multiplet, likewise the lowering operations of both SL(2) SL(2) and SU(2) SU(2)
would specify couplings to the corresponding descendants spanning the full AdS3 S 3
space-time. At leading orther this extensions are direct, at higher order in N they remain a
challenge. The non-linear realizations of some of the basic symmetries [29] might play an
important role in these extensions.
66
1
Bn
B 2 Bnkk |0i
(5.1)
1 n2
P
with the restriction ni ni = N . The relation between the space of chiral primaries defined
as generated by An subject to relations between them which follow from the OPEs is as
follows. Let these relations take the form
Ri (A) = 0.
(5.2)
Consider the ring of polynomials in the An quotiented by the relations above, and let this
space be H /R where R is the ideal generated by the Ri . We have an isomorphism between
H /R and HB .
The relations Ri are closely related to the structure of the Lie algebra we constructed
from the A and A . Setting these elements to zero has to be consistent with the algebra. In
other words
[La , Ri ] = Caij b Lb Rj ,
(5.3)
where La and Lb are elements in the Universal Enveloping algebra of the Lie algebra. An
example of this kind of relation in the case of a simple sl(2) sub-algebra as given in I. In
fact the unitarity conditions in the algebra were used to derive these relations.
5.1. q-deformed super-algebra structure of fusion rules
In I, we argued that a deformed space-time based on quantum groups, namely SU q (2)
SU q (1, 1), could account for some properties of the spectrum and interactions of the chiral
primaries. A natural symmetry algebra governing the properties of such a space-time,
given that it has to reduce to the super-algebra SU(2|1, 1) in the limit of large N , may
be guessed to involve SU q (2|1, 1). In this section we show that some further properties of
the correlation functions are natural from the point of view of such a symmetry. A complete
accounting of the finite N properties of the correlation functions from the point of view of
a quantum group symmetry remains to be found, but the discussion here and in the next
sub-section will hopefully provide useful steps in the right direction.
For simplicity we restrict the discussion here to couplings involving only the pure-twist
(0,0)
operators An . For n 6 N these are generating elements of the chiral ring. For small
values of n we have fusion of the form
(0,0)
(0,0)
(0,0)
An Am Anm+1 .
(5.4)
There can also appear on the RHS products of As but we can restrict attention to the
terms on the RHS which are relevant for studying the 3-point functions of generators of
the chiral ring. On the left we have operators of SU(2)L spins 2j1 = n 1 and 2j2 =
m 1. On the right we have 2j = 2j1 + 2j2 . This is consistent with SU(2) fusion rules but
not all SU (2) reps. allowed by SU(2) tensor products appear on the RHS. Rather a good
model for the fusion is given by the tensor products of short reps. of SU L (2|1, 1) (since
the SU R (2|1, 1) quantum numbers are identical to the left-moving ones it suffices to focus
on the left-moving symmetry). The spin j then labels the largest SU(2) spin present in the
decomposition of the short SU(2|1, 1) rep. into reps. of its SU(2) sub-algebra.
67
A qualitatively new feature appears when n comes close to N , and n + k > N . Then
(0,0)
(0,0)
An Ak 0.
(5.5)
We can see this explicitly from the formulae for correlation functions we wrote down in
(2.34). We rewrite the equation relevant for this feature here:
(0,0)
(N n)!(N k)!(n + k 1)3 1/2
(0,0)
(0,0)
.
(5.6)
On+k1 ()Ok (1)On (0) =
(N (n + k 1))!N!nk
We see that when n + k 1 exceeds N the denominator contains a 0 function with a
negative integer argument which causes the expression to vanish. This feature can be
i
explained using fusion rules of SU q (2|1, 1) where q is, as in I, given by q = e N+1 . The
q-deformed super-algebra will have a family of unitary short (atypical) reps. which will
include unitary irreps. of the SU q (2). Since all the reps. of the SU q (2) appearing in the
decomposition from SU q (2|1, 1) to SU q (2) have to be unitary, there will be a bound on
these short reps. 2j 6 N 1.
5.2. SU q (2|1, 1) and multiparticle states
We have explained the qualitative features of the couplings between this family of
generators of the chiral ring, using SU q (2|1, 1). It will be very interesting to find a
more detailed comparison between the fusions of the chiral ring and the q-deformed
super-algebra. Some novel features, unfamiliar from ordinary rational CFT, but that
have appeared in studies of WZW on super-groups [31] have to be taken into account.
For example it has been found that one needs, in general, to take into account the
indecomposable reps. of the super-algebra as well. It is very plausible, that in this case, we
also need to include reps. of SU q (2|1, 1) which contain indecomposables of the SU q (2)
sub-algebra. While the SU q (2|1, 1) with the q given above, nicely describes the cutoffs
(p,q)
on generating An [4], it also seems to have enough structure to account for some other
cutoffs in this theory. For example, SU q (2) has, in addition to the standard irreps., with
p
a cutoff 2j 6 N 1, a family of irreps. Iz in the notation of [32]. The detailed form of
these reps. is given in [33]. Usually one drops these reps. in studying standard connections
between SU q (2) and the and SU(2) WZW, but in this model the chiral primaries transform
in more complicated reps. of the SU(2) current algebra than the standard integrable reps.
[34,35] usually considered in SU(2) WZW of level k = N . We may expect a larger class
of SU q (2) reps. to appear in the corresponding quantum group model.
We mention a very suggestive numerical observation in favour of the above line of
p
argument. The family I0 has a cutoff at 2j 6 2N . The An operators coming from twisted
sectors can be raised to powers which allow them to exceed the bound at N 1. It turns
out, however that that there is a cutoff on the highest weight 2j 6 2N which works for
any chiral primary, as explained in the previous section. This means that if we associate
reps. containing these indecomposables to some of the chiral primaries which are products
p,q
of An s, we can explain both the cutoff 2j 6 N 1 for the generators, and the cutoff
2j 6 2N for arbitrary chiral primaries.
68
A2 = 2 + ,
(0,0)
= 3 + .
A3
(5.7)
(5.8)
This wouldcontradict the fact that the correlation function hA3 A2 A3 i is non-zero
2 is the twist operator
at order 1/ N . Rather the object which behaves more like 2
associated with the conjugacy class with two cycles of length 2, which is one of the terms
in (4.6), for example. While Al2 is not, for small l, a free oscillator raised to the lth power,
its behaviour in correlation functions should not be qualitatively different from a power
of a free field because it is a linear combination including an operator which behaves like
l . When l hits N/2 the corresponding operator ceases to exist because we cannot have
2
a permutation with more than N/2 cycles of length 2. We observe that this happens at
L0 = N/4. If we try the same thing with Al3 we get a threshold of l = N/3 where we
may expect a qualitative change in behaviour. If we consider a general operator of the form
(A2 )n2 (A3 )n3 Aknk
(5.9)
the corresponding free operator ceases to exist when 2n2 + 3n3 + + knk = N . It has
1
L0 = (n2 + 2n3 + + knk )
2
1
= N (n2 + n3 + + nk )
2
1
2
(k 1)
1 N
+ n3 + n3 + +
nk .
=
2 2
2
2
2
(5.10)
p,q
It is now clear that the lowest threshold we get is N/4. Considering operators of type An
with p, q 6= 0 only increases the threshold. Precisely this value L0 = N/4 was obtained by
[36,37] as the threshold where black holes start to be relevant. We have argued that the same
threshold appears if we ask for the lowest value of L0 where operators in the chiral ring
69
start to display behaviour qualitatively different from free fields. It will be very interesting
to characterize the corresponding change of behaviour in the correlation functions and
compare with expectations from black hole physics.
6. Conclusions
We have studied in this paper a Lie algebra associated with the chiral primaries of
S N (T 4 ) CFT and their CFT conjugates. The structure constants of this Lie algebra are
simply related to the correlation functions after an appropriate choice of basis. We obtained
by CFT computations some of these structure constants, those which are determined by
the 3-point functions of bosonic chiral primaries. This leads to a dynamics described by an
effective (1 + 1)-dimensional field theory which corresponds to the simplest representation
of the algebra.
Connections between the Lie algebra and the stringy exclusion principle were studied in
I, where the relation with q-algebras and non-commutative space-times was emphasized.
These aspects were elaborated in Sections 5.15.3. In Section 5.4 we focused on a
characterization of the exclusion principle as a deviation of the chiral primaries from free
field behaviour in correlation functions, and we were lead to look for the lowest threshold
where such deviation is expected. We found that the lowest threshold is at L0 = N/4 which
has been argued to be relevant to black holes in AdS3 . Developing further the relation
between these correlation functions and black holes will be very interesting.
The Lie algebra is a much richer structure than the truncated Fock space of the chiral
primaries themselves. The latter is only one representation of the Lie algebra. This
simplest representation was studied in connection with actions associated to the correlation
functions of chiral primaries. This was done in detail in Section 3 for the untwisted
sector. A coherent state basis was useful there, and we explained how the parameters in
the coherent states are related to the multi-oscillator Heisenberg algebra at large N . In
Section 4, we pursued the study of the large N map between the Lie algebra and the multioscillator Heisenberg algebra (free fields).
Our discussion naturally raises the question of other representations of the algebra which
are not directly related to the Fock space of the chiral primaries. These will be important
in studying the stringy states, which are certainly important since the free orbifold CFT
is expected to be dual to a gravitational background where the graviton as well as other
stringy degrees of freedom are important, as emphasized in [1214]. Studying the stringy
states as representations of this algebra derived from chiral primaries and their conjugates is
particularly interesting, since the chiral primaries and their truncations show clear evidence
of a non-commutative space-time. This point of view on the stringy states should clarify
the relevance of the non-commutative space-time to the dynamics of the full set of stringy
degrees of freedom.
The precise role of the non-commutative space-time in determining the dynamics of this
gravitational theory remains to be further studied, but the simplest possibility is that, in
analogy to the case of non-commutative YangMills [38] there is simple action on a
70
Acknowledgements
We are happy to acknowledge enjoyable and instructive discussions with Miriam Cvetic,
Sumit Das, Pei Ming Ho, Robert de Mello Koch, Vipul Periwal, Radu Tatar, T. Yoneya,
E. Witten. M.M. would like to thank the organizers of TASI99 for hospitality while
part of the work was being done. This research was supported by DOE grant DEFG02/19ER40688-(Task A).
Appendix A
Let us consider first the OPE we used which can be read from [17] where it is detemined
by a geometric construction. We will also neglect the fact that in our case the theory is a
T 4 orbifold, whereas in [17], it was a non-compact orbifold. The result for the constant is
taken from Eq. (5.24) in [17] with the following redefinitions:
N = n + 1,
p1 = kn ,
n0 = n,
p2 = 0,
D = 6,
p = k2 ,
(7.1)
(7.2)
9 k2
a= + 2,
8
4
1
1 2 1 2
n1 2
3
n kn k2 kn
k ,
b=
8 n+1 4
2
4 2
k2
1
k2
1 1
n+
2 n.
c= +
8 4
n
4
2n
(7.3)
a b
where
We outline here the derivation for the equations used in the text only for the twist field
In this case, in Eqs. (7.2) we take
which does not involve any T 4 indices, O(1...n) (u, u).
n1
,
)
which
give a = 1, b = 1/2, c = 1/2
k2 = (0, 0, 0, 0, 12 , 12 ), kn = (0, 0, 0, 0, n1
2
2
and we have the first equation in (2.24). Combining this result with an identical one for the
right-movers we obtain:
1n+1
O(1...n+1) (0) + O(1...n+1n) (0) .
(1...n) (0) =
(7.4)
O(nn+1) (u, u)O
4 n
This result has a clear interpretation for the case when the lenght of the cycle at 0 is set to
1, namely n = 1 case. We used this extrapolation for the other cases to arrive at Eqs. (2.25).
71
Starting from Eq. (7.4) we can use now the conformal field theory rules in order to
determine more involved correlators. We look to an OPE involving three twists:
(n...n+k1) (z, z )O(1...n) (0)
O(n+k1n+k) (u, u)O
= const O(1...n+k) (0) + ,
(7.5)
where stand for the other 4 possible terms, and the right-hand side is in the limit u 0;
z 0. For consistency, the limit should not depend on the order, and from this we derive
the following recurrence equation for the constant which appear in the OPE of a twist of
lenght n and of one of lenght k, denoted in this appendix as C(n, k):
C(n, k + 1) = C(n, k)
(n + k)k
,
(n + k 1)(k + 1)
(n + 1)
.
4n
These equations are solved by:
C(n, 2) =
(7.6)
n+k1
.
(7.7)
2nk
It is interesting to observe that in this derivation of the constants appearing in the OPE
of the twist operators it was helpful to use both the geometric construction [17] and the
associativity of the OPE in conformal field theory.
Using a similar derivation, we are able to obtain also the OPE for the twist operators
overlapping over a two cycle.
C(n, k) =
References
[1] J. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor.
Math. Phys. 2 (1998) 231; hep-th/9711200.
[2] S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory corellators from non-critical string
theory, hep-th/9802109.
[3] E. Witten, Anti-de-Sitter space and holography, hep-th/9802150.
[4] A. Jevicki, S. Ramgoolam, Non-commutative gravity from the AdS/CFT correspondence,
JHEP 9904 (1999) 032; hep-th/9902059.
[5] J. Maldacena, A. Strominger, AdS3 black holes and a stringy exclusion principle, JHEP 9812
(1998) 005; hep-th/9804085.
[6] J. Madore, Non-commutative geometry for Pedestrians, grqc/9906059.
[7] M. Berkooz, H. Verlinde, Matrix theory, AdS/CFT and HiggsCoulomb equivalence, hepth/9907100.
[8] A. Jevicki, A. van Tonder, Finite (q-oscillator) description of 2D string theory, Mod. Phys. Lett.
A 11 (1996) 1397; hep-th/9601058.
[9] M. Mihailescu, Correlation functions for chiral primaries for D = 6 supergravity on AdS3 S 3 ,
hep-th/9910111; JHEP 02 (2000) 007.
[10] A. Jevicki, B. Sakita, Nucl. Phys. B 165 (1980) 511; Nucl. Phys. B 185 (1981) 89.
[11] S.R. Das, A. Jevicki, String field theory and physical interpretation of D = 1 strings, Mod. Phys.
Lett. A 5 (1990) 16391650.
[12] T. Banks, M. Douglas, G. Horowitz, E. Martinec, AdS dynamics from conformal field theory,
hep-th/9808016.
72
[13] F. Larsen, E. Martinec, U(1) Charges and moduli in the D1D5 system, JHEP 9906 (1999) 019;
hep-th/9905064.
[14] N. Seiberg, E. Witten, The D1/D5 system and singular CFT, hep-th/9903224.
[15] L. Dixon, D. Friedan, E. Martinec, S. Shenker, The conformal field theory of orbifolds, Nucl.
Phys. B 282 (1987) 13.
[16] R. Dijkgraaf, G. Moore, E. Verlinde, H. Verlinde, Elliptic genera of symmetric products and
second quantized strings, Commun. Math. Phys. 185 (1997) 197209; hep-th/9608096.
[17] G.E. Arutyunov, S.A. Frolov, Four graviton scattering amplitude from S N R 8 supersymmetric
orbifold sigma model, Nucl. Phys. B 524 (1998) 159206; hep-th/9712061.
[18] D. Freedman, S. Mathur, A. Mathusis, L. Rastelli, hep-th/9804058.
[19] S. Lee, S. Minwalla, M. Rangamani, N. Seiberg, Three point functions of chiral operators in
D = 4, N = 4 SYM at lareg N , hep-th/9806074.
[20] F. Bastianelli, R. Zucchini, hep-th/9907047.
[21] J. Avan, A. Jevicki, Phys. Lett. B 266 (1991) 35; Phys. Lett. B 272 (1991) 17.
[22] D. Minic, J. Polchinski, Z. Yang, Translation invariant backgrounds in (1 + 1)-dimensional
string theory, Nucl. Phys. B 369 (1992) 324.
[23] J.H. Harvey, G. Moore, On the algebras of BPS states, hep-th/9609017.
[24] C. Vafa, Puzzles at large N , hep-th/9804172.
[25] P. Berglund, E. Gimon, D. Minic, The AdS/CFT correspondence and spectrum generating
algebras, hep-th/9905097.
[26] F.A. Berezin, Models of GrossNeveu type are quantization of a classical mechanics with nonlinear phase space, Commun. Math. Phys. 63 (1978) 131.
[27] A. Jevicki, N. Papanicolaou, Classical dynamics at large N , Nucl. Phys. B 171 (1980) 362.
[28] V. Periwal, String field theory Hamiltonians from YangMills theories, hep-th/9906052.
[29] A. Jevicki, Y. Kazama, T. Yoneya, Generalized conformal symmetry in D-brane matrix models,
Phys. Rev. D 59 (1999); hep-th/9810146.
[30] W. Lerche, C. Vafa, N.P. Warner, Chiral rings in N = 2 superconformal theories, Nucl. Phys.
B 324 (1989) 427.
[31] L. Rozansky, H. Saleur, Nucl. Phys. B 389 (1993) 365.
[32] G. Keller, Fusion rules of Uq SL(2, C), q m = 1, Lett. Math. Phys. 21 (1991) 273.
[33] N. Reshetikhin, V.G. Turaev, Invariants of three manifolds via link polynomials and quantum
groups, Invent. Math. 103 (1991) 547597.
[34] D. Gepner, E. Witten, String theory on group manifolds, Nucl. Phys. B 278 (1986) 493.
[35] P. Goddard, D. Olive, KacMoody and Virasoro algebras in relation to quantum physics,
IJMPA 1 (2) (1986) 303.
[36] J. de Boer, Large N elliptic genus and AdS/CFT correspondence, JHEP 9905 (1999) 017; hepth/9812240.
[37] J. Maldacena, G. Moore, A. Strominger, Counting BPS black holes in toroidal type II string
theory, hep-th/9903163.
[38] A. Connes, M. Douglas, A. Schwarz, Non-commutative geometry and matrix theory: compactification on tori, JHEP 02 (1998) 003.
[39] J. Teschner, The deformed two-dimensional black hole, hep-th/9902189.
[40] C.S. Chu, P.M. Ho, Y.C. Kao, World-volume uncertainty relations for D-branes, hepth/9904133.
Abstract
We discuss the consistency conditions of a novel orientifold projection of type IIB string theory
on C2 /ZN singularities, in which one mods out by the combined action of world-sheet parity and a
geometric operation which exchanges the two complex planes. The field theory on the world-volume
of D5-brane probes defines a family of six-dimensional RG fixed points, which had been previously
constructed using type IIA configurations of NS-branes and D6-branes in the presence of O6-planes.
Both constructions are related by a T-duality transforming the set of NS-branes into the C2 /ZN
singularity. We also construct additional models, where both the standard and the novel orientifold
projections are imposed. They have an interesting relation with orientifolds of DK singularities, and
provide the T-duals of certain type IIA configurations containing both O6- and O8-planes. 2000
Elsevier Science B.V. All rights reserved.
PACS: 11.25
Keywords: Type IIB orientifold; D-branes; Six-dimensional fixed points
1. Introduction
One of the most interesting quantum field theory lessons that we have learned from
string theory is the existence of six-dimensional supersymmetric field theories with nontrivial infrared dynamics [1,2]. We are now familiar with the existence of large families
of interacting superconformal field theories with (0, 2) and (0, 1) supersymmetry. There
are two approaches which have been extensively used to construct these theories. The first
is the study of type IIB D5-branes at A-D-E singularities [3,4] and orientifolds thereof
[57]. 1 The second is the construction of type IIA brane configurations (in the spirit of
[17]) of NS-branes, D6-branes and possibly D8-branes and orientifold planes [1820]. 2
angel.uranga@cern.ch
1 Strings in orientifold backgrounds have been studied for instance in [816].
2 A third approach, the study of F-theory on elliptically fibered singular CalabiYau threefolds [21,22] will not
74
There is a close relation between both constructions. In fact, type IIA brane configurations where one of the directions (along which the D6-branes have finite extent) is compact
can be T-dualized to a system of D5-branes probing orbifold and orientifold singularities
(see, e.g., [23,24]). This observation has led to a rich interplay between both approaches.
For example, the construction of superconformal field theories in the type IIB picture in
[3,57] was a source of information in the study of the T-dual type IIA configurations
in [19,20]. On the other hand, some configurations in [19,20] (those containing oppositely
charged O8-planes) produced new field theories which required the existence of new orientifolds of C2 /ZN . These were constructed in [25] making use of the information available
from the IIA construction.
In this paper we continue this program by constructing the type IIB orientifolds
associated to yet another set of field theories constructed in [19,20]. As we discuss below,
the new orientifold of C2 /ZN has some unusual and amusing properties. 3
Before continuing with our introduction, it will be useful to know more about the
relevant IIA brane configurations [19,20]. We consider a set of N NS-branes with worldvolume along 012345, several stacks of D6-branes (along 0123456) stretched between
them, and an O6-plane parallel to the D6-branes. The direction x 6 , along which the D6branes have finite extent, is taken compactified on a circle. The field theory on the noncompact part of the D6-brane world-volume has D = 6, N = 1 supersymmetry. This is the
six-dimensional version of the four-dimensional models studied in [26].
As determined in [27] (see also [28] for a world-sheet derivation of this fact) the RR
charge of the O6-plane changes sign whenever it crosses a NS-brane, and the projection
it imposes on the D6-branes changes accordingly. 4 Therefore, a consistent configuration
is obtained only for an even number of NS-branes. If we place ni D6-branes in the ith
interval between NS-branes, the gauge group is of the form
SO(n0 ) USp(n1 ) SO(nN2 ) USp(nN1 ).
(1.1)
The matter content arises from strings stretching between neighbouring D6-branes, and is
of the form
N1
X
i=0
1
(i , i+1 ),
2
(1.2)
where the index i is defined mod N , and the 12 means the matter arises in halfhypermultiplets, due to the O6-plane projection. In brane configurations realising sixdimensional field theories, the NS-branes have the same number of non-compact
dimensions as the D6-branes, so their world-volume fields are dynamical. In our model,
they give rise to N 1 tensor multiplets; an additional tensor multiplet is decoupled and
hence irrelevant.
3 Several group theoretical features of the corresponding projection were pointed out in [7].
4 When k D6-branes sit on top of an O6 -plane (which carries 4 units of D6-brane charge, as counted in the
double cover) their gauge group is projected down to SO(k). When they sit on top of an O6+ (with charge +4)
75
n2i+1 = N.
(1.3)
2. Orientifold construction
Here we propose a type IIB T-dual realization of the field theory in the previous section,
in terms of D5-branes probing a new kind of orientifold of C2 /ZN . As discussed below,
5 Orientifold projections involving exchange of complex planes have also appeared in [32,33] in a different
kind of models.
76
we propose that the orientation reversing element has a geometric action on the C2 /ZN ,
xy = v N , given by x y, v v. This action is a symmetry only when N is even,
and involves the exchange of the two complex planes. This choice can be heuristically
motivated by directly T-dualizing the IIA model described above. Following [31], upon
T-duality the set of N NS-branes becomes a N -centered TaubNUT space, which can be
described as xy = v N in suitable complex coordinates, while the D6-branes transform into
D5-brane probes. The directions 89 can be identified with v. The directions 7 and 60 (the
T-dual of 6, with asymptotic radius R60 ) are related to x and y in a complicated manner.
7
60
Roughly speaking, for large x and fixed y one has x ei(x +ix )/R60 and for large y at
7
60
fixed x one has y e(x +ix )/R60 . The T-dual of the O6-plane action is expected to flip
the signs of the coordinates 60 789, which is certainly the case for the action proposed
above.
This argument should be considered a heuristic motivation. Stronger checks will arise
from the detailed discussion of the model. We therefore turn to constructing the new
orientifold of C2 /ZN and to showing that the world-volume field theory on D5-brane
probes is the one described in Section 1. Let us consider type IIB theory on C2 /ZN , modded
out by 5, where is world-sheet parity and 5 acts as
5: z1 z2 ,
z2 z1
(2.1)
C2 /ZN .
(i)
(j )
NSNS: k L Nk R
(j )
(i)
RR:
k
Nk R
4(1, 1), i, j = 1, 2,
(1, 1) + (1, 3), k = 1, . . . , N 1.
77
(2.3)
The second column gives the representation under the spacetime little group SU (2)
SU (2). We obtain one hyper and one tensor multiplet of D = 6 N = 1 supersymmetry per
twisted sector.
In the usual orientifolds studied in the literature [15,16], and for generic twisted
sectors (k 6= N/2), the states surviving the projection are
(j )
(j )
(i)
(i)
NSNS: k L Nk R + Nk L k R 4(1, 1),
(2.4)
(j )
(j )
(i)
(i)
(1, 1) + (1, 3).
RR:
k
Nk R
Nk L
Notice that invariant combinations are a mixture of states appearing in oppositely twisted
sectors of the type IIB theory. The orientifolded model produces one hyper and one tensor
multiplet per such pair of oppositely twisted sectors.
For the k = N/2 twisted sector, which does not mix with any other, the action of
may be defined with an additional (1) sign [34]. This Z2 choice determines the type of
multiplet surviving the projection. One gets a tensor multiplet or a hypermultiplet if the
additional sign is present or not. As discussed in [34], consistent coupling between closed
and open strings implies the following constraint on the D-brane ChanPaton matrices
N/2 = TN/2 1 ,
(2.5)
where the positive sign is taken when includes the additional (1) sign, and the negative
sign is taken otherwise. The two possibilities correspond to D-brane gauge bundles with or
without vector structure [35].
Let us now turn to the 5 orientifold. It is easy to realize that 5 invariant states do not
require mixing different twisted sectors of the type IIB theory. Because of that, for every
sector there is a priori a Z2 ambiguity in defining the orientifold action. Actually, below
we will show that the only consistent possibility is to include the additional (1) sign in
the action of 5. Assuming momentarily this claim, the resulting invariant states are
(2)
(1)
k(2) L Nk
(1, 1),
NSNS: k(1) L Nk
R
R
(1)
(1)
Nk ,
RR:
k
(2.6)
(2) L (2) R
k L Nk R (1, 3),
(2)
(2)
(1)
(1)
+
.
k
Nk R
Nk R
We obtain one tensor multiplet per twisted sector. We now show that this choice is the
only consistent one, by using the constraints coming from consistent coupling of open and
closed strings. Consider a set of D5-branes at the C2 /ZN singularity, before the orientifold
projection is imposed. 7 Their ChanPaton matrix ,5 has N different eigenvalues si with
multiplicities ni . It can be shown that 5,5 is block-diagonal (with ni ni blocks) in
7 The 5 orientifold does not allow for the introduction of D9-branes. In Section 3 D9-branes may appear
in some models where both the 5 and the usual projections are imposed. The consistency conditions for
D9-branes in those cases can derived by simple modifications in our arguments below.
78
this basis, and therefore k ,5 and 5,5 commute. The fact that 5 acts diagonally on
the index i can be derived, e.g., by looking at the ChernSimons couplings of open string
states and closed string RR modes in the ZN orbifold [3]
Z
(2.7)
tr( k ,5 i )Ck eFi ,
where Ck is a formal sum of the twisted RR forms, and i selects the entries of the Chan
Paton matrix corresponding to the ith set of D-branes. Since 5 is a symmetry of the
orbifold theory, the coupling must be invariant. Recalling that 5 acts diagonally in k, it
follows that its action on the open string sector is diagonal in i. 8
The argument in [34] now allows us to use this information on the open string sector to
constrain the closed string sector. Since 5 commutes with k , we have
1
.
k ,5 = +5,5 Tk ,5 5,5
(2.8)
Comparing with (2.5), we learn that the projection 5 in the closed string sector must
include the additional (1) sign, as claimed above. This fixes the Z2 ambiguity in all
twisted sectors.
The bottomline of this argument is that the closed string spectrum of the 5 orientifold
of C2 /ZN produces N 1 tensor multiplets.
2.2. Open string spectrum
We now turn to the open string spectrum and further consistency conditions on 5,5 .
It is convenient to introduce a concrete expression for ,5 , which we take 9
1
N1
,5 = diag 1n0 , e2i N 1n1 , . . . , e2i N 1nN1 .
(2.9)
An interesting constraint on 5,5 can be derived by requiring (5)2 = N/2 in the
open string sector. The action of 5 on a state |, ij i, corresponding to an open string
stretching between the ith and j th D-brane, is
1
.
(2.10)
5: |, ij i (5 )ii 0 |5 , j 0 i 0 i 5
j 0j
We are interested in comparing the actions of (5)2 and N/2 , given by
1
1
T
T
|5 , i 0 j 0 i 5,5
5,5
,
(5)2 : |, ij i 5,5 5,5
j 0j
0
ii
N/2 : |, ij i ( N/2 ,5 )ii 0 N/2 , i 0 j 0 1
N/2 ,5 j 0 j .
(2.11)
Let us split the set of D-branes into two types, denoted even and odd, according to
whether their N/2 eigenvalue is +1 or 1. In the eveneven and oddodd sectors, N/2
T , with independent choices of sign for even
acts as +1, and (2.11) imposes 5 = 5
8 In other words, i labels the different kinds of fractional branes [36], which can be understood as higherdimensional branes wrapped on the collapsed two-cycles of the orbifold singularity. Since the orientifold acts
diagonally on the cycles, so it does on the wrapped branes.
9 In the orientifolds, a different model would be obtained for
N ,5 = 1. It is easy to show that they are
identical in the 5 orientifold.
79
and odd D-branes. In the evenodd and oddeven sectors, N/2 acts as 1 and (2.11)
implies the symmetry of 5 must be opposite for even and odd branes. Without loss
of generality we get the conditions
T
5,5 = +5,5
T
5.5 = 5,5
(2.12)
(2.13)
Notice that this type of constraint is satisfied only for N even. We also would like to point
out that the different symmetry of 5,5 for even and odd branes is related by T-duality
to the change of sign of the O6-plane whenever it crosses a NS-brane. The 5 orientifold
hence provides a geometrical description of such process in a IIB T-dual realisation. It
would be interesting to compare it with the geometries proposed in [37].
This completes the discussion on consistency conditions. Let us turn to computing the
massless open string spectrum. The projection on the ChanPaton factors for gauge bosons
is
1
,
= ,5 ,5
1
= 5,5 T 5,5
(2.14)
(2.15)
1
Z2 = e2i/N ,5 Z2 ,5
,
1
Z1 = 5,5 Z2T 5,5
.
(2.16)
1
(i , i+1 ).
2
(2.17)
Notice that the action of 5 as exchange of the two complex planes is essential in obtaining
half (rather than full) hypermultiplets.
Thus, the spectrum in the closed and open string sectors agrees with the field theory
constructed in Section 1 from type IIA brane configurations of NS- and D6-branes in the
presence of an O6-plane.
This field theory is potentially anomalous. We now show that, in analogy with the
models in [6,7,25], the irreducible gauge anomaly vanishes once tadpole cancellation
conditions are imposed. The reader not interested in these details in encouraged to skip
the computation. The tadpoles can be obtained using the general techniques in [1416],
and we only stress the differences between the and 5 projections. Since the cylinder
diagrams do not involve crosscaps, their contribution to the tadpoles is the familiar one
N1
X
2 k
(Tr k ,5 )2 ,
4 sin
(2.18)
C=
N
k=1
80
where the untwisted tadpole k = 0 vanishes in the non-compact limit and is therefore
ignored. The tadpoles from the Mbius strip diagrams are quite similar to those in [15,
16]. The only difference is that the eigenvalues of the twists k 5 that act along with are
e2i/4 (and independent of k) rather than e2ik/N , and this modifies the trigonometric
coefficient of the tadpole. Concretely we have
M = 16
N1
X
4 sin2
k=0
Tr Tk 5 1
k 5 = 32N Tr N/2 ,
4
(2.19)
N1
X
T (1, k ) +
k=0
= 64
N1
X
N1
X N1
X
T ( n , k )
n=1 k=0
4 sin2
2
4
4 sin2 4 4 sin2 3
4
k=0
+ 64
N1
X N1
X
1 = 64N 2 .
(2.20)
n=1 k=0
2
2 k
Tr k ,5 16Nk,N/2 = 0
4 sin
N
4 sin2 k
N
1
(2.21)
k
Tr k ,5 16Nk,N/2 = 0.
N
(2.22)
The tadpole cancellation conditions can be expressed in terms of the integers nr , yielding
the condition
2nr + 16(1)r + nr1 + nr+1 = 0.
(2.23)
This is the condition of cancellation of the irreducible anomaly in the field theory (2.15),
(2.17). Their solution is n2i = N + 8, n2i+1 = N . This solution reproduces the result
based on RR charge conservation in the type IIA brane configuration discussed in the
introduction. The factorization of the remaining anomalies and their cancellation by a
GreenSchwarz mechanism involving the N 1 tensor multiplets [30] has been discussed
in [7], and we will not repeat it here.
10 Recall that in the orientifolds only the pieces with n = 0, N/2 give a net contribution. For other twists, the
contributions from symmetric and antisymmetric combinations of n and n twisted sectors cancel in the trace.
81
52 = K ,
5 n = n 5.
(3.1)
These relations define the non-Abelian discrete group DK . Hence we are dealing with
orientifolds of C2 /DK . How does this arise in the IIA brane configuration?
3.1. The IIA brane configurations
The answer goes as follows. In a type IIA brane configuration with O6- and O8planes, the orientifolds impose the projections O6 (1)FL R7 R8 R9 and O8 R6 ,
respectively, where Ri acts as x i x i . By closure, the configuration is also modded out
by the orbifold element R = (1)FL R6 R7 R8 R9 .
It is illustrative to consider momentarily the configuration modded out only by R,
without any orientifold projection. Such brane configurations were studied in [39] and
shown to reproduce field theories with gauge groups and matter contents defined by DK
quiver diagramas [4]. Furthermore, these configurations have been argued to be T-dual to
systems of D5-branes at C2 /DK singularities [38]. The type IIA brane configurations we
are actually interested in contain an additional orientifold projection, say, by O6 . Hence
their type IIB T-duals should correspond to orientifolds of C2 /DK singularities, as found
above.
Let us consider the IIA brane configurations is some more detail. Since they contain
an O6-plane which flips charge whenever it crosses a NS-brane, N is constrained to be
even, N = 2K. Another consistency condition is that no NS-brane can intersect the O8planes, since the O6-plane crossing them would not respect the Z2 symmetry imposed by
the O8-plane. For a fixed K, in order to specify the model completely one has to make
the following choices. First, the charge of the two O8-planes 11 , which can be positive for
one and negative for the other in which case their RR charge cancels and no D8-branes
11 Our convention is that an O8+ -plane has +16 units of D8-brane charge, and projects the gauge group of k
coincident D6-branes down to SO(k), and that an O8 -plane, with 16 units of D8-brane charge, produces a
gauge group USp(k) on k D6-branes.
82
are required or negative for both in which case RR charge cancellation requires
32 D8-branes, as counted in the covering space; in this case one also has to choose
the distribution of the D8-branes in the different intervals, in a way consistent with the
orientifold symmetries. Second, the charge assignment for the different pieces of O6-plane
in the intervals. Finally, the number of D6-branes ni at each interval i is determined by the
conditions or RR charge conservation [19,20]. The solution to these conditions is unique
up to an overall addition of an equal number of D6-branes in all intervals.
To make the discussion a bit more concrete, consider the case of even K. There are two
possible patterns for the gauge group
G0 USp(n1 ) SO(n2 ) USp(nK1 ) GK ,
G0 SO(n1 ) USp(n2 ) SO(nK1 ) GK .
(3.2)
The first (respectively, second) possibility corresponds to the case when the O8-planes are
intersected by O6 - (respectively, O6+ )- planes. The nature of the factors G0 and GK is
more model-dependent and will be discussed below. When K is odd, the general structure
of the gauge group is
G0 USp(n1 ) SO(n2 ) USp(NK2 ) SO(nK1 ) GK .
(3.3)
(3.4)
i=1
where R0 and RK are discussed below. If D8-branes are present, suitable flavours in
fundamental representations must be added [38].
Let us now specify the structure of the end sectors in the above spectra, where a certain
number n of D6-branes suffers the projection by an O8-plane and an O6-plane. There are
four different cases to be considered, depending on the charges of the orientifold planes,
and they lead to different group factors and matter contents [38].
(i) When the projection is imposed by an O8 -plane and an O6 -plane, the
corresponding gauge factor is SU (n/2) and the representation R0 or RK in (3.4) is
+ .
(ii) When there is an O8+ -plane and an O6+ -plane, we obtain the same answer: the
gauge factor is SU (n/2) and the representation R0 or RK is + .
(iii) For an O8 -plane and an O6+ plane, the gauge factor is USp(n) USp(n0 ) and the
representation R0 or RK is (, 1) + (1, ).
(iv) For an O8+ -plane and an O6 plane, the gauge factor is SO(n) SO(n0 ) and the
representation R0 or RK is (, 1) + (1, ).
Concerning the NS-brane world-volume fields, we obtain K tensor multiplets from the
K NS-branes in the configuration. There are also twisted fields living at the fixed points of
(1)FL R6 R7 R8 R9 . For end sectors of the type (i) or (ii) each orbifold plane gives rise
to a hypermultiplet. For end sectors of the type (iii) or (iv), each produces one tensor
multiplet.
83
In the resulting field theory, the ranks of the gauge groups are determined by imposing
the charge cancellation conditions in the IIA brane configuration. These include the the
charges with respect to the twisted fields of the orbifold plane. In the cases without D8branes, on which we center henceforth for the sake of clarity, this last condition implies
n = n0 in the cases (iii) and (iv).
In the following subsection we show how these rules arise in the construction of the type
IIB orientifolds.
3.2. The orientifold construction
Even though as explained above we are dealing with an orientifold of a DK singularity,
it will be convenient to continue discussing in terms of a ZN singularity modded out by
the two orientifold projections 5 and . This is useful since it allows to benefit from
the known results about models with either of the projections, taken from Section 2 for the
5 and from [6,25] for the projections. There are nevertheless some important instances
where the additional elements k 5 in the orbifold group play a role.
One of these situations is the computation of the closed string spectrum. The model
contains a twisted sector for each one of the K + 2 non-trivial conjugacy classes of the
orbifold group DK . These classes are
Cn = { n , n },
n = 1, . . . , K 1,
CK = { K },
(3.5)
We see that the new elements in the orbifold group generate new twisted sectors, but
also introduce identifications in the twisted sectors of ZN . In type IIB theory, each sector
produces a hyper and a tensor multiplet. In the orientifolded theory, the action of and
5 on the classes Cn , n = 1, . . . , K selects the tensor multiplet, while the action on the
remaining classes may select the tensor or hypermultiplet. The relevant cases are discussed
in the examples below. Notice that the action of on the N/2 -twisted sector corresponds
to a projection with vector structure.
Another consequence of the new elements in the orbifold group is that the D5- and
D9-branes in the model generate new disk tadpoles twisted by these elements k 5. These
do not have a corresponding crosscap diagram, so cancellation of these tadpoles requires
conditions of the type
Tr k 5,9 2 Tr k 5,5 = 0,
(3.6)
where the factor of two arises form momentum zero modes, and the first term is required
only in cases with D9-branes. This condition is the IIB counterpart of the cancellation of
charge under the twisted fields of the orbifold in the IIA brane configuration. We will make
sure this condition is satisfied for the matrices in our models.
Let us turn to the explicit construction. Instead of being completely general, it will be
illustrative to consider two examples, which include all the different end sectors discussed
in Section 3.1. Let us consider C2 /ZN with N = 2K and K odd, and mod it by 5 and
, with 2 = . As explained in [25], the presence of in the projection implies the
84
model contains no D9-branes, a conveniently simple case. Let us consider the ChanPaton
matrices
1
K1
N
=
n0
1 1n1
(P 1) 1nP 1
P nP
P 1 1nP 1
1n1
(3.7)
Notice that the matrices k 5 = k 5 are traceless, so the additional disk tadpoles
mentioned above vanish. The open string spectrum one obtains from the projection using
the matrices above is
U (n0 /2) SO(n1 ) USp(nK1 ) U (nK /2),
K2
X1
1
1
[(0, 1 ) + (0 , 1 )] +
(i , i ) + [(K1, K ) + (K1 , K )]. (3.8)
2
2
2
i=1
The closed string spectrum contains K tensor multiplets and two hypermultiplets.
This model has a clear type IIA T-dual configuration. It contains N NS branes, two
O8-planes (not intersected by any NS-brane) and one O6-plane. The projection above
corresponds to the case where the O8-planes are oppositely charged [25]. The choice of
ChanPaton matrices 5 , above specifies that the O8+ is intersected by an O6+ , and
the O8 by an O6 . Using the rules of Section 3.1 we obtain a gauge group and matter
content in agreement with (3.8).
Let us turn to the discussion of cancellation of tadpoles in the type IIB side. Instead
of performing it in the usual way, by factorizing Klein bottle, Mbius strip and cylinder
diagrams, one can take advantage of knowing directly the contributions of the [25] and
5 crosscaps (Section 2) 12 . The tadpole condition reads
k
Tr k ,5 32k,1 mod 2 + 16Nk,N/2 = 0.
(3.9)
4 sin2
N
This expression agrees with that obtained performing the standard computation. This
condition can be recast in terms of the integers nr , giving
2nr + nr1 + nr+1 + 16r,0 16r,K 16(1)r = 0.
(3.10)
85
brane configuration. The residual gauge anomaly is factorized and cancelled as discussed
in [20,38] by exchange of the closed string tensor multiplets [30]. Also, the U (1) anomalies
are cancelled by hypermultiplet exchange.
The second example we consider is a slight modification of the previous model. Let us
again consider C2 /ZN , with N = 2K and K odd, modded out by and 5. Let us
choose the ChanPaton embedding of ,5 and ,5 to be exactly as in (3.7), but let us
take
5,5 = diag(1 1n0 /2 , n1 , . . . , 1nK1 , i2 1nK /2 , 1nK1 , . . . , n1 ).
(3.11)
Notice that 5,5 by itself is equivalent to (2.13), despite the modification of the blocks at
the positions 0 and K. The modification is required so that the matrices k 5 are traceless,
as required for consistency. The final spectrum of this model is
SO(n0 /2) SO(n0 /2)0 SO(n1 ) USp(nK1 )
1
1
USp(nK /2) USp(nK /2)0 (0 , 1 ) + (00 , 1 )
2
2
K2
X1
1
1
(i , i ) + (K1 , K ) + (K1 , 0K ).
(3.12)
+
2
2
2
i=1
Acknowledgements
I am pleased to thank J. Park for useful discussions, and the Institute for Advanced
Study, Princeton, for hospitality at the beginning of this project. I am also grateful to
86
M. Gonzlez for her patience and support. This work has been financially supported by
the Ramn Areces Foundation (Spain).
References
[1] N. Seiberg, E. Witten, Comments on string dynamics in six dimensions, Nucl. Phys. B 471
(1996) 121, hep-th/9603003.
[2] N. Seiberg, Nontrivial fixed points of the renormalization group in six dimensions, Phys. Lett.
B 390 (1997) 169, hep-th/9609161.
[3] M.R. Douglas, G. Moore, D-branes, quivers, and ALE instantons, hep-th/9603167.
[4] C.V. Johnson, R.C. Myers, Aspects of type IIB theory on ALE spaces, Phys. Rev. D 55 (1997)
6382, hep-th/9610140.
[5] K. Intriligator, RG fixed points in six dimensions via branes at orbifold singularities, Nucl. Phys.
B 496 (1997) 177, hep-th/9702038.
[6] J.D. Blum, K. Intriligator, Consistency conditions for branes at orbifold singularities, Nucl.
Phys. B 506 (1997) 223, hep-th/9705030.
[7] J.D. Blum, K. Intriligator, New phases of string theory and 6-D RG fixed points via branes at
orbifold singularities, Nucl. Phys. B 506 (1997) 199.
[8] A. Sagnotti, in: G. Mack et al. (Eds.), Cargese 87, Non-perturbative Quantum Field Theory,
Pergamon Press, 1988, p. 521, Some properties of open string theories, hep-th/9509080.
[9] J. Dai, R.G. Leigh, J. Polchinski, New connections between string theories, Mod. Phys. Lett. 4
(1989) 2073.
[10] R.G. Leigh, DiracBornInfeld action from Dirichlet sigma model, Mod. Phys. Lett. 4 (1989)
2767.
[11] P. Horava, Strings on world-sheet orbifolds, Nucl. Phys. B 327 (1989) 461; Background duality
of open string models, Phys. Lett. B 231 (1989) 251; Two-dimensional stringy black holes with
one asymptotically flat domain, Phys. Lett. B 289 (1992) 293; Equivariant topological sigma
models, Nucl. Phys. B 418 (1994) 571.
[12] G. Pradisi, A. Sagnotti, Open strings orbifolds, Phys. Lett. B 216 (1989) 59.
[13] M. Bianchi, A. Sagnotti, On the systematics of open string theories, Phys. Lett. B 247 (1990)
517; Twist symmetry and open string Wilson lines, Nucl. Phys. B 361 (1991) 519.
[14] E. Gimon, J. Polchinski, Consistency conditions for orientifolds and D manifolds, Phys. Rev.
D 54 (1996) 1667, hep-th/9601038.
[15] E.G. Gimon, C.V. Johnson, K3 orientifolds, Nucl. Phys. B 477 (1996) 715, hep-th/9604129.
[16] A. Dabholkar, J. Park, Strings on orientifolds, Nucl. Phys. B 477 (1996) 701, hep-th/9604178.
[17] A. Hanany, E. Witten, Type IIB superstrings, BPS monopoles, and three-dimensional gauge
dynamics, Nucl. Phys. B 492 (1997) 152, hep-th/9611230.
[18] I. Brunner, A. Karch, Branes and six-dimensional fixed points, Phys. Lett. B 409 (1997) 109,
hep-th/9705022.
[19] I. Brunner, A. Karch, Branes at orbifolds versus HananyWitten in six dimensions, JHEP 9803
(1998) 003, hep-th/9712143.
[20] A. Hanany, A. Zaffaroni, Branes and six-dimensional supersymmetric theories, Nucl. Phys.
B 529 (1998) 180, hep-th/9712145.
[21] P.S. Aspinwall, Pointlike instantons and the spin(32)/Z(2) heterotic string, Nucl. Phys. B 496
(1997) 149, hep-th/9612108.
[22] P.S. Aspinwall, D.R. Morrison, Pointlike instantons on K3 orbifolds, Nucl. Phys. B 503 (1997)
533, hep-th/9705104.
[23] A. Karch, D. Lust, D. Smith, Equivalence of geometric engineering and HananyWitten via
fractional branes, Nucl. Phys. B 533 (1998) 348, hep-th/9803232.
87
[24] A. Karch, Field theory dynamics from branes in string theory, hep-th/9812072, PhD thesis.
[25] J. Park, A.M. Uranga, A note on superconformal N = 2 theories and orientifolds, Nucl. Phys.
B 542 (1999) 139, hep-th/9808161.
[26] K. Landsteiner, E. Lpez, D.A. Lowe, N = 2 supersymmetric gauge theories, branes and
orientifolds, Nucl. Phys. B 507 (1997) 197, hep-th/9705199
[27] N. Evans, C.V. Johnson, A.D. Shapere, Orientifolds, branes, and duality of 4-D gauge theories,
Nucl. Phys. B 505 (1997) 251, hep-th/9703210.
[28] S. Elitzur, A. Giveon, D. Kutasov, D. Tsabar, Branes, orientifolds and chiral gauge theories,
Nucl. Phys. B 524 (1998) 251, hep-th/9801020.
[29] J. Erler, Anomaly cancellation in six dimensions, J. Math. Phys. 35 (1994) 1819, hepth/9304104.
[30] A. Sagnotti, A note on the GreenSchwarz mechanism in open string theories, Phys. Lett. B 294
(1992) 196, hep-th/9210127.
[31] H. Ooguri, C. Vafa, Two-dimensional black hole and singularities of CY manifolds, Nucl. Phys.
B 463 (1996) 55, hep-th/9511164.
[32] R. Blumenhagen, L. Grlich, Orientifolds of non-supersymmetric asymmetric orbifolds, Nucl.
Phys. B 551 (1999) 601.
[33] R. Blumenhagen, L. Grlich, B. Krs, Supersymmetric orinetifolds in 6D with D-branes at
angles, hep-th/9908130.
[34] J. Polchinski, Tensors from K3 orientifolds, Phys. Rev. D 55 (1997) 6423, hep-th/9606165.
[35] M. Berkooz, R.G. Leigh, J. Polchinski, J.H. Schwarz, N. Seiberg, E. Witten, Anomalies,
dualities, and topology of D = 6 N = 1 superstring vacua, Nucl. Phys. B 475 (1996) 115,
hep-th/9605184.
[36] M.R. Douglas, Enhanced gauge symmetry in M(atrix) theory, JHEP 07 (1997) 4, hepth/9612126
[37] J. Park, R. Rabadan, A.M. Uranga, N = 1 type IIA brane configurations, chirality and T-duality,
hep-th/9907074.
[38] A. Hanany, A. Zaffaroni, Issues on orientifolds: on the brane construction of gauge theories
with SO(2n) global symmetry, JHEP 9907 (1999) 009, hep-th/9903242.
[39] A. Kapustin, D(n) quivers from branes, JHEP 9812 (1998) 015, hep-th/9806238.
Abstract
In the framework of a 2HDM effective Lagrangian for the MSSM, we analyse important
phenomenological aspects associated with quantum soft SUSY-breaking effects that modify the
relation between the bottom mass and the bottom Yukawa coupling. We derive a resummation of the
dominant supersymmetric corrections for large values of tan to all orders in perturbation theory.
With the help of the operator product expansion we also perform the resummation of the leading and
next-to-leading logarithms of the standard QCD corrections. We use these resummation procedures
to compute the radiative corrections to the t b H + , H + t b decay rates. In the large tan
regime, we derive simple formulae embodying all the dominant contributions to these decay rates
and we compute the corresponding branching ratios. We show, as an example, the effect of these new
results on determining the region of the MH + tan plane excluded by the Tevatron searches for
a supersymmetric charged Higgs boson in top quark decays, as a function of the MSSM parameter
space. 2000 Elsevier Science B.V. All rights reserved.
PACS: 11.10.Gh; 12.38.Cy; 12.60.Jv; 14.80.Cp
Keywords: Supersymmetry; Charged Higgs phenomenology; Higher-order radiative corrections
1. Introduction
In minimal supersymmetric extensions of the Standard Model (SM), soft Supersymmetry (SUSY) breaking terms [14] are introduced to break SUSY without spoiling the
cancellation of quadratic divergences in the process of renormalization. These terms must
have dimensionful couplings, whose values determine the scale MSUSY , lower than a few
1 On leave from the Theoretical Physics Department, Fermilab, Batavia, IL 60510-0500, USA.
2 On leave from the High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
and the Enrico Fermi Institute, Univ. of Chicago, 5640 Ellis, Chicago, IL 60637, USA.
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 4 6 - 2
89
TeV, above which SUSY is restored; they are also responsible for the mass splittings inside
the supersymmetric multiplets. Little is known for sure about the origin of these SUSYbreaking terms. Upcoming accelerators will test the energy range where we hope that the
first supersymmetric particles will be found. From their masses and couplings we could
learn about the pattern of SUSY-breaking at low energies, which translates, through the
renormalization group equations, into the pattern of breaking at the scale at which SUSYbreaking is transmitted to the observable sector. Meanwhile, one can obtain some information on the soft terms by looking at any low-energy observables sensitive to their values,
and in particular to the Yukawa sector of the theory.
In this work we consider the simplest supersymmetric version of the SM, the Minimal
Supersymmetric Standard Model (MSSM) [57]. We analyse the limit of a large ratio
v2 /v1 = tan of the vacuum expectation values v1 , v2 of the Higgs doublets. We show
that in this limit a large class of physical observables involving the Yukawa coupling of the
physical charged Higgs boson can be described in terms of a two-Higgs-doublets model
(2HDM) [8] effective Lagrangian, with specific constraints from the underlying MSSM
dynamics.
The finding of a charged Higgs boson would be instant evidence for physics beyond the
SM. It would also be consistent with low-energy SUSY, as all supersymmetric extensions
of the SM contain at least a charged Higgs boson, H . Current experiments, looking at
the kinematical region MH + < mt mb , have been able to place an absolute bound of
MH + > 71.0 GeV at the 95% confidence level [9,10] and/or to exclude regions of the
MH + BR(t b H + ) plane [1114]. 3 If the charged Higgs mass happens to be greater
than the top mass, future e+ e , pp and even e p accelerators will have a chance to find
it [1631].
Present bounds from LEP on a SM light Higgs boson, MhSM > 105.6 GeV ([32], see
also Refs. [9,10]), are beginning to put strong constraints on values of tan lower than
a few, a region that can only be consistent with low-energy SUSY if the third-generation
squark masses are large, of the order of a TeV and, in addition, if the mixing parameters
in the stop sector are of the order of, or larger than, the stop masses. Therefore, the LEP
limits give a strong motivation for the study of the large tan region. The region of large
values of tan is also theoretically appealing, since it is consistent with the approximate
unification of the top and bottom Yukawa couplings at high energies, as happens in minimal
SO(10) models [3339]. The aim of this work is to compare, for large values of the tan
parameter, the effective potential results truncated at one loop with the diagrammatic oneloop computation for the supersymmetric QCD (SUSY-QCD) and electroweak (SUSYEW) corrections in the coupling of tbH + [4044]. We then use the effective potential
approach to include a resummation of the SUSY-QCD and SUSY-EW effects and we show
how relevant these higher-order effects are to the final evaluation of the H + t b and
t b H + partial decay rates.
3 See also the study in Ref. [15], where it is shown how these bounds are affected by some usually overlooked
> 1 region.
decay modes in the intermediate tan
90
2. Effective Lagrangian
2.1. Supersymmetric corrections
The effective 2HDM Lagrangian contains the following couplings of the bottom quark
to the CP-even neutral Higgs bosons [55]:
(1)
4 For the QCD corrections to the neutral Higgs decay rate the reader is referred to [5254] and references
therein.
91
Fig. 1. One-loop SUSY-QCD diagram contributing to the effective coupling 1hb . The solid lines
inside the loop denote the gluino propagator, the dashed lines correspond to sbottom propagators.
The cross represents the Mg insertion coming from the gluino propagator.
The H20 b b tree-level coupling is forbidden in the MSSM. Yet a non-vanishing 1hb is
dynamically generated at the one-loop level by the diagram of Fig. 1. 5
0 acquire their vacuum
Although 1hb is loop-suppressed, once the Higgs fields H1,2
expectation values v1,2 , the small 1hb shift induces a potentially large modification of
the tree-level relation between the bottom mass and its Yukawa coupling, because it is
enhanced by tan = v2 /v1 :
mb = hb v1 mb = v1 (hb + 1hb tan) = hb v1 (1 + 1mb ).
(2)
Since the numerical value of mb is fixed from experiment, Eq. (2) induces a change in
the effective Yukawa coupling. This affects not only the CP-even neutral Higgs field, but
the whole Higgs multiplet, with phenomenological consequences for the charged Higgs
particle. In particular, Eq. (2) modifies the Yukawa coupling of the charged Higgs to top
and bottom quarks as follows:
hb sin =
mb
m
1
tan hb = b
tan,
v
v 1 + 1mb
(3)
q
where v = v12 + v22 ' 174 GeV. In the last equation we have assumed a large tan regime.
It turns out that, in the MSSM with large tan, the dominant supersymmetric radiative
corrections to the Yukawa interactions of the Higgs doublet H1 = (H1+ , H10 ) stem from the
relation (3). Explicit loop corrections to the H1 ff 0 Yukawa coupling are suppressed by at
least one power of tan. This remarkable feature has far-reaching consequences: first in
observables involving the coupling hb of H1 to bottom quarks the MSSM behaves like a
two-Higgs-doublets model. The main effect of a heavy SUSY spectrum is to modify the
coupling strength via 1mb in Eq. (2), which depends on the masses of the supersymmetric
particles. In certain regions of the parameter space a sizeable enhancement of hb occurs.
Secondly these dominant corrections encoded in 1mb are universal. They are not only
equal for the neutral and the charged Higgs bosons, on which we will focus in the
following, but they are also independent of the kinematical configuration. This means that
they affect the decay rate of a charged Higgs into a top and bottom (anti-) quark in the
same way as the tbH + vertex in a rare b-decay amplitude or, after replacing the top by a
charm quark, as Higgs-mediated b c decays. Further the universality property of these
5 There are similar diagrams involving supersymmetric electroweak quantum corrections, see Section 3.2.
92
tan-enhanced radiative corrections allows for a simple inclusion into the Higgs search
analysis.
The proper tool to describe such universal effects is an effective Lagrangian. Expanding (1) to include the charged Higgs sector one finds that the relevant terms in the large
tan limit are:
L = hb bL bR H10 + hb Vt b sin tL bR H + 1hb bL bR H20 + h.c.
(4)
1hb is the loop-induced Yukawa coupling associated with the supersymmetric QCD
corrections in Fig. 1 and similar electroweak contributions. H + is the physical charged
Higgs boson. The Higgs mechanism defines the relation between the bottom mass mb
0 and one-loop bbH
0
and the couplings hb and 1hb in L: calculating the tree-level bbH
1
2
vertices with zero Higgs momentum, and replacing the Higgs fields by their vacuum
expectation values v1,2 , yields the desired relation in Eqs. (2) and (3):
1mb =
1hb
SQCD
tan = 1mb
+ 1mSEW
,
b
hb
(5)
b1,2
I (a, b, c) =
1
(a 2 b2 )(b2 c2 )(a 2 c2 )
a2
b2
c2
2 2
2 2
2 2
a b log 2 + b c log 2 + c a log 2 .
b
c
a
(7)
An interesting limit of Eq. (6) applies when all mass parameters are of equal size. One has,
depending on the sign of
s (Q = MSUSY )
tan,
(8)
3
clearly showing that the effect does not vanish for a heavy SUSY spectrum and can be of
O(1) for large tan values.
For sizeable values of the trilinear soft SUSY-breaking parameter At , the supersymmetric electroweak corrections are dominated by the charged higgsino-stop contribution,
which is proportional to the square of the top Yukawa coupling, ht = mt /v2 . Wino-sbottom
contributions are generally smaller, being proportional to the square of the SU(2)L gauge
coupling, g, and to the soft SUSY breaking mass parameter M2 . Neglecting the bino effects, which we found to be numerically irrelevant, these corrections read [56]
SQCD
1mb
=
1mSEW
b
h2t
At tan I (mt , mt , )
1
2
16 2
g2
M2 tan cos2 t I (mt , M2 , ) + sin2 t I (mt , M2 , )
2
1
2
16
+ 12 cos2 b I (mb , M2 , ) + 12 sin2 b I (mb , M2 , ) .
93
(9)
When including radiative corrections, one has to specify the definition of the quark mass
mb appearing in the leading order: mb denotes the pole mass corresponding to the on-shell
renormalization scheme, in which the on-shell self-energy is exactly cancelled by the mass
counterterm.
Note that the supersymmetric corrections contained in 1mb enter hb in Eq. (3) as a
factor 1/(1 + 1mb ). To order s one is entitled to expand this factor as (1 1mb ). In
the phenomenologically most interesting case of a large |1mb | of O(1), this leads to
disturbingly large numerical ambiguities. Their resolution seems to require painful higherorder loop calculations, and a large |1mb | may even put perturbation theory into doubt.
Yet these tan-enhanced contributions have the surprising feature that they are absent in
higher orders:
There are no contributions to 1mb of order
n
tan
for n > 2.
(10)
s
MSUSY
Here MSUSY represents a generic mass of the supersymmetric particles. An analogous
result applies to the electroweak corrections. In other words, to the considered order, 1mb
is a one-loop exact quantity, and the factor 1/(1 + 1mb ) contains the corrections to hb of
the form in (10) to all orders in s .
To prove our theorem, consider possible n-loop SUSY-QCD contributions to 1mb
proportional to tann : the only possible source of additional factors of tan is the offdiagonal element of the bottom squark mass matrix, mb tan, which can enter the
result via the squark masses as m 2 m 2 ' 2mb tan or through counterterms to the
b2
b1
squark masses. It is easier to track the factors of mb tan by working with chiral squark
eigenstates and assigning these factors to chirality flipping two-squark vertices. Thus
any extra factor of tan is necessarily accompanied by a factor of mb . This dimensionful
factor is multiplied with some power of inverse masses stemming from the loop integrals.
The next step in our reasoning is to show that the loop integrals always give powers of
1/MSUSY and can never produce a factor of 1/mnb . The appearance of any inverse power
of mb in a loop integral would imply a power-like infrared singularity in the limit mb 0
with gluino and squark masses held fixed. But the KLN theorem [5759] guarantees the
absence of any infrared divergence in all bare diagrams except for those in which gluons
couple to the b-quark lines. A two-loop example of the latter set is shown in Fig. 2. The
infrared behaviour of these diagrams can be studied with the help of the operator product
expansion (OPE). The result of the OPE is nothing but the effective Lagrangian in (4). To
apply the OPE to our problem we first have to contract the lines with heavy supersymmetric
particles to a point, i.e., we replace the MSSM by an effective theory in which the heavy
SUSY particles are integrated out. For the case of the diagram in Fig. 2 this yields the
diagram in Fig. 3, in which the loop-induced interaction is represented by the dimension-4
0 . The information on the heavy SUSY masses is contained in the Wilson
operator bbH
2
94
coefficient 1hb in Eq. (4). The key feature of the OPE exploited in our proof is the fact
that the effective diagram in Fig. 3 and the original diagram in Fig. 2 have the same infrared
behaviour. Power counting shows that the diagram of Fig. 3 has dimension zero. It depends
only on mb and the renormalization scale Q. Since Q enters the result logarithmically,
the diagram of Fig. 3 depends on mb as log mb /Q, no power-like dependence on mb is
possible. This argument essentially power counting immediately extends to higher
orders. Terms from diagrams in which gluons are connected with the b-quark line and
one of the SUSY-particle lines in the heavy loop, are either infrared-finite or suppressed
by even one more power of mb /MSUSY , because they are represented in the OPE by
operators with dimension higher than 4. Finally there are diagrams with counterterms.
In mass-independent renormalization schemes the counterterms are polynomial in mb . In
the on-shell scheme the diagrams with counterterms can be infrared-divergent for mb 0,
but only logarithmically. In conclusion the loop integrals cannot give factors of 1/mnb .
Therefore any correction to 1mb of order sn tann comes with a suppression factor of
n
. Higher-order loop corrections to 1mb are therefore either suppressed by
mnb /MSUSY
mb /MSUSY or lack the enhancement factor of tan, which proves our theorem.
So far we have discussed 1mb from the one-loop vertex function of Fig. 1 as in [38,39].
A different viewpoint has been taken, e.g., in [40,41]: the renormalization of the Yukawa
coupling (mb /v) tan is performed by adding the mass counterterm to mb . In the large
tan limit and to one-loop order, this amounts to the replacement
m
mb
tan hb = b (1 1mb ) tan
(11)
v
v
instead of (3). This procedure gives the correct renormalization of the Yukawa coupling
in regularization schemes respecting gauge symmetry [6062], such as dimensional
regularization. The relation to the Yukawa renormalization using the vertex function
in Fig. 1 leading to (2) is provided by a SlavovTaylor identity [6062]. In general a
correction factor related to the anomalous dimension of the quark mass occurs in (2), but
95
the large tan-enhanced contributions considered by us are finite and do not contribute to
the anomalous mass dimension. To one-loop order, Eqs. (11) and (3) are equivalent. Yet the
crucial difference here is the point that 1mb in Eq. (11) stems from the supersymmetric
contribution to the quark self-energy diagram in Fig. 4. While the vertex diagram has
dimension zero, the self-energy diagram has dimension one and the above proof does not
apply. Indeed, higher-order corrections to Fig. 4 do contain corrections of the type in (10).
In Appendix A these corrections are identified and it is shown that they sum to
1
,
(12)
1 + 1mb
so that both approaches lead to the same result (3) to all orders in (/MSUSY ) s tan.
2.2. Renormalization group improvement
The tan-enhanced supersymmetric corrections discussed so far are not the only
universal corrections. It is well known that standard QCD corrections to transitions
involving Yukawa couplings contain logarithms log(Q/mb ), where Q is the characteristic
energy scale of the process. For the decays discussed in Sections 35 one has Q = mH + or
Q = mt and s log Q/mb is of O(1) thereby spoiling ordinary perturbation theory. The
summation of the leading logarithms
Q
, n = 0, 1, 2 . . .
(13)
sn logn
mb
to all orders in perturbation theory has been performed in [52,53] for the standard QCD
corrections to the tL bR H + Yukawa interaction. This summation is effectively performed
by evaluating the running Yukawa coupling hb at the renormalization scale Q. This
amounts to the use of the running mass at the scale Q, mb (Q), after expressing hb sin in
terms of (mb /v) tan. Hence these large logarithms are likewise universal, depending only
on the energy scale Q at which the Yukawa coupling is probed, and can also be absorbed
into the effective Lagrangian.
The full one-loop QCD corrections to neutral [52,53] and charged [45,46] Higgs decay
and top decay [51] also contain non-logarithmic terms of the order s . A consistent use
of these one-loop corrected expressions therefore requires the summation of the next-toleading logarithms
Q
, n = 0, 1, 2 . . .
(14)
sn+1 logn
mb
to all orders, because all these terms have the same size as the one-loop finite terms. Since
squarks and gluinos are heavy, leading logarithms of the type in (13) are absent in the
96
(15)
Here and in the following, MS quantities are overlined. The renormalization scale Q is
explicitly displayed in (15). Note that 1hb depends on Q through s , Mg and the squark
masses. The relation (3) between hb and mb is defined at the low scale Q = mb . Hence
as in the
we must evolve (15) down to Q = mb . Since we encounter the same operator bb
leading order, the renormalization group evolution down to Q = mb is also identical to
the leading-order evolution and just amounts to the use of the running Yukawa coupling
h b (Q = mb ) in the desired relation:
h b (Q = mb ) =
mb (Q = mb )
1
tan.
v
1 + 1mb (Q = MSUSY )
(16)
Notice that 1mb is evaluated at the high scale Q = MSUSY : the heavy particles freeze
out at the heavy scale Q = MSUSY and the strong coupling s in 1mb likewise enters
the result at this scale. This can be intuitively understood, as the loop momenta in Fig. 1
probe the strong coupling at typical scales of order MSUSY . Further any renormalization
group running below Q = MSUSY is done with the standard model result for -functions
and anomalous dimensions. Since the QCD contributions to the anomalous dimensions of
h b and mb are the same, h b at an arbitrary scale Q is given by
1
mb (Q)
tan.
h b (Q) =
v
1 + 1mb (MSUSY )
(17)
97
If one expands h b (MSUSY ) around h b (mb ) to order s2 , one reproduces the large logarithm
of the form log(MSUSY /mb ) contained in the diagram of Fig. 2. The running mass must be
evaluated with the next-to-leading order formula:
mb (Q) = U6 (Q, mt ) U5 ( mt , mb ) mb ( mb ),
(18)
where we have assumed that there are no other coloured particles with masses between Q
and mt . The evolution factor Uf reads
s (Q2 )
Uf (Q2 , Q1 ) =
s (Q1 )
d (f ) =
d (f )
s (Q1 ) s (Q2 ) (f )
J
1+
,
4
12
,
33 2 f
J (f ) =
(19)
Here f is the number of active quark flavours. For Q 6 mt one must replace U6 (Q, mt )
U5 ( mt , mb ) by U5 (Q, mb ) in Eq. (18). J (f ) depends on the renormalization scheme,
the result in Eq. (19) is specific to the MS scheme. The b-quark mass in this scheme
is accurately known from (1S) spectroscopy and momenta of the bb production cross
section [64,65]:
mb ( mb ) = (4.25 0.08) GeV.
(20)
Physical observables such as the H + and top decay rates discussed in Sections 35 are
scheme independent to the calculated order. Passing to a different renormalization scheme
would change J (f ) , but in Eq. (18) the change in s ( mb ) J (5) is compensated by a
corresponding change in the numerical value of mb ( mb ). Likewise the scheme dependence
in s (Q) J (6) is compensated by the one-loop standard QCD corrections [45,46,51] to the
decay rates. This concludes the discussion of the universal renormalization group effects.
A discussion of additional aspects specific to the decay rates (t b H + ) and (H +
can be found in Appendix B.
t b)
Finally we arrive at the desired effective Lagrangian for large tan:
L=
mb (Q)
g
2MW 1 + 1mb
1mb
+ 1mb
h bb(Q)
H bb(Q)
, (21)
+
cos
sin
cos
sin
where the renormalization scale Q entering mb and the renormalization constants of the
quark bilinears are explicitly shown. In Eq. (21) we have expressed L in terms of the
physical Higgs fields H, h, A and H + and traded v for the W mass and the SU(2) gauge
coupling g. We have used the standard convention [8,55] for these fields and the hH
mixing angle . For completeness also the coupling of the CP-odd Higgs boson A has
98
been included. The phenomenology of the neutral Higgs bosons in the large tan regime
has been studied in detail in [55].
5 b and H + tL bR interactions
The effective Lagrangian in Eq. (21) describes the A b
correctly for large tan, irrespective of the mass hierarchy between MSUSY and MH + .
Even if MSUSY MH + , the supersymmetric loop form factors of these interactions are
suppressed by one power of tan with respect to the terms described by L. On the contrary,
this is no longer true for the H bL bR and h b L bR form factors [66]. For these couplings L
2
MA2 .
is only correct in the limit MSUSY
3. Quantum corrections to (t b H + ), (H + t b)
The tree-level partial widths read
tree (t b H + ) =
g2
|Vt b |2 m3t 1/2 (1, qH + , qb )
2
64MW
(1 qH + + qb ) cot2 + qb tan2 + 4qb ,
=
tree (H + t b)
g 2 Nc
2
32MW
(22)
3
1/2
|Vt b |2 MH
(1, rt , rb )
+
(1 rt rb ) rt cot2 + rb tan2 4rt rb ,
(23)
(1, x, y) = 1 + x 2 + y 2 2 (x + y + x y).
From now on, we shall assume |Vt b | ' 1 and neglect light fermion generations. For
p
> 15 (the inflexion point being given by tan > m /m
values of the parameter tan
t
b
7) virtual quantum effects are largely dominated by the corrections to the right-handed
bottom Yukawa coupling. In that limit the tree-level widths reduce to
tree (t b H + ) =
g 2 m3t
(1 qH + )2 qb tan2 ,
2
64MW
2
= g Nc M 3 + (1 rt )2 rb tan2 ,
tree (H + t b)
H
2
32MW
(24)
(25)
99
The one-loop QCD-corrected expressions for the t (H + ) decay rates [4551] can be
greatly simplified after expanding them in a series in powers of rb (qb ) and retaining
only the first-order term. As we are mainly interested in the region of large tan, we will
provide formulae valid for those values of tan, for which Eqs. (24), (25) apply. An explicit
evaluation of the departure from this approximation for the one-loop result will be done in
Section 4.
In the H + case we perform a simultaneous expansion in powers of rb and rt . Retaining
terms up to rt3 and considering the logarithmic factors to be of O(1), the resulting
approximation to the one-loop formula is
QCD (H + t b)
app
s
16 3
g 2 Nc
3
2
2
3 + 6 rt + rt2
r + 2 log(rb )
MH + (1 rt ) rb tan 1 +
=
2
27 t
32MW
10 2 40 3
r
r log(rt ) .
(26)
+ 4rt
3 t
9 t
As can be seen from the above equation, there is no need to do the resummation of the
log rt logarithms, as they are either small when rt is close to 1 or suppressed by at least a
power of rt when it is small.
For rt < 0.03, the correction induced by the rt -terms in (26) is smaller than a 2%. Then,
Eq. (26) reduces to
= 1 + s (3 + 2 log rb ) (0) (H + t b),
(27)
QCD (H + t b)
where we have introduced the quantity (0) , which is formally identical to tree but has as
input parameters the on-shell renormalized ones. The finite part in Eq. (27), 3s / , stands
for a correction of about +10% (for s ' 0.1), whereas the full correction is large and
negative, due to the much bigger logarithmic term.
For the t b H + decay, the expansion in qb reads
QCD (t b H + )
app
'
g 2 m3t
(1 qH + )2 qb tan2
2
64MW
qH +
4s 9 2 2 3
+ log qb
log qH +
1+
3 4
3
2
1 qH +
2 5qH +
+
log(1 qH + ) + log qH + log(1 qH + ) + 2 Re Li2 (1 qH + ) .
2qH +
(28)
In the limit qH + 1, the ratio QCD / (0) becomes infinite and perturbation theory
breaks down, as the b-quark moves too slowly in the top rest frame. Nevertheless, the
correction goes to zero due to the presence of the kinematic suppression factor.
At this point we are ready to incorporate the resummation of the leading and next-toleading qb , rb logarithms, as explained in Section 2.2, which amounts to replacing mb in
100
Eqs. (26) and (28) by the running bottom mass at the proper scale. 6 The one-loop QCDcorrected widths are then, in the large tan limit and including renormalization group
effects up to next-to-leading order, given by the following improved (imp) formulae
QCD (t b H + )
imp
g2
mt (1 qH + )2 m2b (m2t ) tan2
2
64MW
8 2
s (m2t )
2 log(1 qH + ) + 2(1 qH + )
7
1+
9
4 2
2
+ log(1 qH + ) (1 qH + )
+
,
9 3
(29)
QCD (H + t b)
imp
g 2 Nc
2
2
MH + (1 rt )2 m2b (MH
+ ) tan
2
32MW
2 )
s (MH
17
16 3
+
+ 6rt + rt2
r
1+
3
27 t
10 2 40 3
rt
rt log(rt ) ,
+ 4rt
3
9
(30)
where s (Q2 ) is the MS-scheme running coupling constant and mb (Q2 ) the MS running
mass expressed in terms of the bottom pole mass.
Finite parts in imp and app differ (see, e.g., the 17/3 in Eq. (30) and the 3 in Eq. (26)).
There is an implicit scheme conversion in going from Eqs. (26), (28) to Eqs. (29), (30): the
bottom pole mass has been replaced for the running MS mass in the prefactor and the
app
log(rb ) has been absorbed into mb . Notice that the non-logarithmic terms of QCD have
imp
been explicitly included in QCD , as they are not accounted for by the renormalization
group resummation techniques.
3.2. Supersymmetric corrections
The effective Lagrangian prediction for the SUSY-QCD and SUSY-EW corrected decay
rates can be read from Eq. (21). No tan-enhanced vertex corrections contribute to the
matching and the result is obtained by simply inserting the effective coupling, Eq. (3), into
the zeroth-order width
1
eff
=
(0).
(31)
SUSY
(1 + 1mb )2
We want to compare Eq. (31) with the diagrammatic on-shell expressions for the oneloop SUSY-QCD and SUSY-EW corrected t b H + , H + t b partial widths [40,41,43,
6 We refer the reader to Appendix B for a proof of that statement.
101
1-loop
44], which we will denote by SUSY . For large tan values, the only sizeable diagrams
are those that contribute to the scalar part of the bottom quark self-energy, entering the
computation through the mass counterterm. For the SUSY-QCD corrections, the diagram
that matters is shown in Fig. 4. By simple power counting one can realize that it is finite.
2
) contributions, its value is essentially given by that of
Moreover, neglecting O(m2b /MSUSY
.
the three-point diagram in Eq. (6): 1mSQCD
b
Similarly, the diagram relevant to the SUSY-EW corrections is a two-point one with a
chargino (neutralino) and a stop (sbottom) inside the loop. As for the SUSY-QCD case, it is
finite, and its value can be approximated by the corresponding three-point diagram where
in
an extra H20 leg is attached to the scalar line. Its contribution is thus given by 1mSEW
b
Eq. (9).
Collecting the results from Eqs. (6) and (9) via Eq. (5), the one-loop SUSY corrected
decay rates can be cast into the formula
1-loop
(32)
SUSY = (1 21mb ) (0) + 1SUSY .
The term 1SUSY , which contains non-universal and tan-suppressed contributions to the
decay, is very small provided tan is large, as we have numerically checked.
Both prescriptions, Eqs. (32) and (31), are equivalent at first order in perturbation theory
(PT) and consequently do not differ significantly when the corrections are small. In general,
though, 1mb can be a quantity of O(1) for large enough tan values, in which case
Eq. (31) is preferred as it correctly encodes all higher-order 1mb effects (see the discussion
in Section 2.1 and in Appendix B).
3.3. Full MSSM renormalization group improved correction
In Section 2.2, we saw how the effective Lagrangian (21) accounts for the higher-order
tan-enhanced SUSY quantum corrections and also for the leading and next-to-leading
QCD logarithms, including those in diagrams like Fig. 2. We define the improved values
for the decay rates of the two processes under study in the MSSM as
imp
imp
MSSM = QCD
1
+ 1SUSY,
(1 + 1mb )2
which also incorporates the one-loop finite QCD effects. Neglecting the small tansuppressed 1SUSY effect, one has
MSSM (H + t b)
imp
2 )
m2b (MH
g 2 Nc
+
2
M
tan2
+ (1 rt )
H
2
2
(1
+
1m
32MW
b)
2 )
s (MH
17
16 3
+
+ 6rt + rt2
r
1+
3
27 t
10 2 40 3
r
r log(rt ) ,
+ 4rt
3 t
9 t
(33)
102
MSSM (t b H + )
imp
2
2
g2
2 mb (mt )
m
(1
q
)
tan2
+
t
H
2
(1 + 1mb )2
64MW
8 2
s (m2t )
2 log(1 qH + ) + 2 (1 qH + )
7
1+
9
4 2
2
+ log(1 qH + ) (1 qH + )
.
+
9 3
(34)
The above formulae contained all the improvements discussed in this article. In order to
1-loop
compare them to the diagrammatic one-loop MSSM results, we introduce MSSM
1-loop
1-loop
imp SUSY
MSSM = QCD
,
(35)
(0)
imp
which only differs from MSSM in that no resummation of the SUSY-QCD, SUSY-EW
1-loop
imp
corrections is performed. Comparing MSSM , MSSM one can assess the size of the higherorder tan-enhanced effects.
x tree
.
tree
(36)
103
sake of simplicity, we omitted this term in Eq. (28), but we have included it when drawing
app
the QCD curve in Fig. 5. The extra correction is almost negligible for the tan = 30
curve. In the H + t b decay rate, Fig. 6, and for tan > 10, Eq. (26) is always extremely
app
close to the one-loop result, and the QCD curves are not shown.
104
app
1-loop
As can be seen in Fig. 5, a discrepancy appears between QCD and QCD close to the
threshold, which can be traced back to the fact that we dropped the mb kinetic terms in the
approximated formula. Similar problems should be present in the H + t b case, Fig. 6,
when approaching the threshold, but our plot starts at a conservative MH + = 250 GeV
value for which the truncated series, Eq. (26), with rt = 0.5, is still valid. Although the
inclusion of higher-order corrections in rt allows to get closer to the threshold, at some
point the perturbative expansion will no longer be reliable: the decay products move
slowly in the decay particles rest frame, and long-distance non-perturbative effects can
significantly modify the perturbative prediction. Moreover, in this region the branching
ratio is very small and therefore the corresponding decay channel loses its relevance for
the charged Higgs phenomenology.
As was justified in Section 2 using the operator product expansion, the replacement
of the renormalized bottom mass and strong coupling by their running two-loop MS
values correctly resums leading and next-to-leading rb , qb logarithms. In Eqs. (29), (30)
the substitution was explicitly done. In Fig. 5 the numerical effect of the improvement
app
imp
corresponds to the difference between the dashed (QCD ) and solid (QCD ) curves. For
the H + t b decay the improvement is essentially given by the difference between the
1-loop
imp
dotted (QCD ) and solid (QCD ) curves.
Even for moderate tan values around 10, the QCD corrections are larger than 50%,
driven by the big qb , rb logarithms. The resummation of the leading logarithms is
mandatory, specially for the H + decay where log rb is unbounded as MH + increases. The
effect of the LO and NLO resummation diminishes the top partial decay rate in about 5%
and the charged Higgs decay rate in about 15%.
105
Fig. 7. The SUSY contributions to the H + t b partial decay width, as a function of tan, for
MH + = 350 GeV and two values of . The dashed lines denote the approximation 1SUSY = 0
of Eq. (32), whereas the solid lines correspond to the effective width, Eq. (31). For the heavy
spectrum one has Mg = mb = mt1 = 1 TeV, b1 , t1 being the lightest sbottom and stop, respectively.
1
At = 500 GeV, the values are shown in the plot. For the light spectrum we have set Mg = 500 GeV,
mb = 250 GeV and mt1 = 180 GeV. In this case, we also show a dotted curve corresponding to the
1
one-loop result of Ref. [42].
estimate of the one-loop correction. This illustrates the fact that our effective Lagrangian
L in Eq. (21) describes the charged Higgs interaction correctly even if MSUSY < MH + .
It shows that 1mb accounts for most of the effects and we can trust the validity of the
improved result.
Typical values we found for the SUSY correction are 15%30% with the heavy spectrum
and 40% with the light one. In both cases, the results depend heavily on the and
tan parameters, the size of the correction growing almost linearly with their absolute
values. Although not shown in the plots, the main contribution to SUSY comes from the
SUSY-QCD diagrams. Only for a very large At values can the electroweak corrections be
comparable.
eff
curves correspond to the relative correction to the widths as evaluated using
The SUSY
1-loop
Eq. (31), an expression derived from the effective Lagrangian in Section 2. While SUSY ,
app
SUSY do not include higher-order 1mnb effects (which can be potentially of O(1)) these
tan-dominant effects are correctly resummed to all orders in PT in the expression for
eff .
SUSY
app
eff
and SUSY first appears at order (1mb )2 , and is always
The difference between SUSY
positive, opposite to the negative standard QCD corrections, for 1mb > 1.5. 7 Therefore,
7 The comparison is between 1/(1 + 1m )2 and 1 21m , Eq. (32), the approximated one-loop result as
b
b
defined in this paper and in [42,44].
106
1-loop
for negative (positive) values of 1mb , that is, positive (negative) corrections SUSY , the
higher-order terms tend to reinforce (suppress) the correction. As 1mb is mainly given by
the SUSY-QCD contribution, Eq. (6), this correlation is seen in association with the sign
of .
1-loop
Just to give some examples, for a negative SUSY = 30% correction, which correeff
increase the
sponds to 1mb = 0.15, the extra higher-order terms contained in SUSY
partial width by 8%. For 1mb = 0.2, a number that can be obtained from Eq. (8) by set1-loop
eff
and SUSY is of order +16%.
ting tan = 20, s = 0.1, the difference between SUSY
The only restriction to the potential size of SUSY is set by the renormalized bottom
Yukawa coupling, which is required to remain perturbative from the GUT scale to the scale
of the corresponding decay. This is guaranteed in our calculations by demanding hb < 1.2
at low energies (see, e.g., [38,39]), implying the following combined bound on tan and
1mb :
1mb >
1 g mb (m2t )
(37)
In the above example, with tan = 20, the minimum allowed value for 1mb is 0.72.
If Eq. (8) holds for negative , and using s (MSUSY ) 0.1, it is found that a maximum
>
eff
allowed correction, SUSY
+200%, is obtained around tan = 40.
4.3. Full MSSM correction
We shall now show the combined effects of the QCD, SUSY-QCD and SUSY-EW
corrections in the partial decay widths under study, starting from three different sets of
imp
curves: QCD , i.e., the QCD correction including the renormalization group resummation
1-loop
of the bottom mass logarithms up to NLO; MSSM , the full one-loop MSSM contribution
imp
as defined in Eq. (35); and the MSSM-improved contribution, MSSM , defined in Eq. (34).
Fig. 8 shows the dependence of the relative corrections to the width (t b H + ) on
the mass scale MSUSY , defined as a common value for the gaugino mass, M2 , the gluino
mass and the masses of the lightest stop and sbottom. As we keep the value of fixed,
the SUSY contribution smoothly goes to zero like /MSUSY when MSUSY increases.
Contrarily, if all mass parameters are sent to infinity together, the SUSY correction
tends towards a constant value, determined by 1mb ' (s /3) tan, Eq. (8). A similar
with a different renormalized value for
behaviour occurs for (H + t b)
QCD .
imp
imp
The difference between QCD and MSSM is due to the SUSY corrections, which were
1-loop
imp
already considered in the above section. The mismatch between MSSM and MSSM is
produced by the tan-enhanced higher-order effects that are resummed in the latter.
Fig. 9 shows how the full MSSM correction evolves with tan. While QCD has a mild
dependence on tan that is almost saturated around tan = 20, the SUSY part gets more
and more important as tan increases. One can see that for the chosen parameters SUSY
becomes of O(10%) around tan = 30. For negative values of , of O(MSUSY ), and for
107
Fig. 8. Evolution of the corrections to the t b H + width, for MH + = 125 GeV and tan = 30, as
a function of a common SUSY mass, MSUSY = M2 = Mg = mb = mt1 , and At = 500 GeV. The
1
dashed line corresponds to the QCD-improved width, Eq. (29), the dotted line denotes the one-loop
MSSM result, Eq. (35), and the solid line denotes the MSSM-improved one, Eq. (34).
Fig. 9. The corrections to the t b H + width for MH + = 125 GeV as a function of tan. The
rest of the parameters are those of the heavy spectrum in Fig. 7. The dashed line corresponds to the
QCD-improved width, Eq. (29), the dotted line denotes the one-loop MSSM result, Eq. (35), and the
solid line denotes the MSSM-improved one, Eq. (34).
sufficiently large tan values, the total correction can be considerably reduced with respect
108
Fig. 10. Curves of constant branching ratio for the t b H + channel. The figure shows the
QCD-improved, Eq. (29), result. The transition between the solid and dashed styles occurs when
the bottom Yukawa coupling crosses the bound hb (mt ) < 1.2. As explained in the text, this bound
guarantees the perturbativity of the Yukawa up to the GUT scale. Finally, the shaded area defines the
95% C.L. exclusion boundary in the tanMH + plane for mt = 175 GeV and (t t) = 5.5 pb that
can be derived from the D0 frequentist analysis in Ref. [11,12].
109
Fig. 11. As Fig. 10, but plotting the MSSM-improved result, Eq. (34), for = 500 GeV (left
plot) and = 500 GeV (right plot). The rest of relevant SUSY parameters are given by
Mg = M2 = m = m = 1 TeV, At = 500 GeV. In the = 500 GeV plot, the shaded area is
t1
b1
with Mg = mt = m = 1 TeV. As in Fig. 10, the dark area on the bottom-right corner
b1
1
corresponds to the experimentally excluded region.
For positive values of 1mb (left plot in Fig. 11), both QCD and SUSY-QCD corrections
reduce the tree-level partial width of the t b H + decay channel, and the bound on the
BR moves to higher tan values. In our example plot, with = 500 GeV, the excluded
region starts at tan > 100 and it is not shown. Conversely, for negative 1mb values, the
supersymmetric corrections partly compensate for the QCD reduction of the width, and
the bound is found for lower tan values. This fact can be checked in the plot on the
right of Fig. 11, corresponding to = 500 GeV. Values larger than 0.4 for BR(t
>
55. The experimental bound starts around tan = 65,
b H + ) are obtained when tan
in a region where hb (mt ) > 1.2, which implies that the bottom Yukawa coupling becomes
non-perturbative below the GUT scale [38,39]. This fact is denoted in the plots by changing
from solid to dashed line style. The same remark applies for Fig. 10.
The H + t b branching ratio, which is expected to be tested at the LHC and at the
NLC, is depicted in Fig. 12. On the left plot, contour lines of constant BR are drawn
using the QCD improved width, Eq. (30). Similarly, the right plot shows curves of constant
for the MSSM-improved result, Eq. (33), with = 500 GeV, the rest of
BR(H + t b)
SUSY parameters being equal to those of Fig. 11. It has been assumed that no decays
of H + into pairs of R-odd SUSY particles [15] were possible. This is guaranteed by the
choice of the soft SUSY-breaking masses and by cutting the plots at MH + = 500 GeV.
6. Conclusions
Using an effective Lagrangian description of the MSSM, we have investigated the virtual
supersymmetric effects that modify the tree-level relation between the bottom Yukawa
coupling and the bottom mass, which are dominant in the large tan regime. Motivated
by the fact that these effects do not vanish for large values of the SUSY masses and are
110
Fig. 12. Curves of constant branching ratio for the H + t b channel. On the left, the QCD-improved
values, Eq. (30), on the right, the MSSM-improved result, Eq. (33). The parameters chosen for these
plots are = 500 GeV, Mg = M2 = mt = m = 1 TeV, At = 500 GeV.
1
b1
potentially of O(1), we have derived the expressions for the bottom Yukawa couplings that
resum all higher-order tan-enhanced quantum effects. These expressions have a natural
interpretation and are easily deduced in the context of the effective Lagrangian formulation.
We have also shown that they can be equivalently deduced in the framework of the full
MSSM.
As an interesting application of our results, we have computed the partial decay
rates for the t b H + and H + t b decay channels, relevant to supersymmetric
charged Higgs searches at present and future colliders. First we have considered the
QCD quantum corrections to these processes and, applying the OPE, we have performed
the resummation of the leading and next-to-leading logarithms of the form log Q/mb .
Concerning the supersymmetric corrections, we have compared our results with those
of previous diagrammatic one-loop analyses in the literature and we have shown the
numerical relevance of the resummation of the tan-enhanced effects derived in this
work. Collecting the above improvements, we have finally computed the corresponding
As an example, we have shown,
branching ratios, BR(t b H + ) and BR(H + t b).
for different sets of the MSSM parameters, the effect of the quantum corrections in
determining the region of the MH + tan plane excluded by the D0 indirect searches for a
supersymmetric charged Higgs boson in the decay of the top quark.
Acknowledgements
We would like to thank D. Chakraborty for providing us with the frequentist analysis
data on the indirect charged Higgs search at D0. We are also grateful to L. Groer for
sending us the corresponding data for the direct charged Higgs search at CDF. U.N. thanks
M. Spira and P.M. Zerwas for a clarifying discussion on Yukawa coupling renormalization
and gauge symmetry. The work of D.G. was supported by the European Commission TMR
programme under the grant ERBFMBICT 983539.
111
(38)
hb being the counterterm of hb . The l.h.s. of the previous equation is graphically depicted
in Fig. 13.
We are not displaying the wave function renormalization to avoid an unnecessary
complication of the argument. Note that v1 receives no one-loop QCD corrections and thus
its renormalization only adds effects suppressed by EW /s , which allows us to identify
R
hb v1 with mb . Besides, in any renormalization scheme one has mR
b = hb v1 , with mb and
hb denoting renormalized quantities. Therefore,
(hb + hb )v1 = mR
b + mb ,
(39)
112
Fig. 13. Feynman diagrams contributing to the bottom pole mass up to first order in PT. From left to
right, the renormalized bottom mass, the bottom mass counterterm and the finite one-loop Feynman
graph contributions are shown. The dashed line in the last diagram denotes a sbottom and the solid
line a gluino. The cross represents the insertion of the bottom mass counterterm.
(40)
The l.h.s. of Eq. (40) is just the bare bottom mass, m0b .
When evaluated beyond first order, scheme differences appear in the r.h.s. of Eq. (40). In
the on-shell scheme, the renormalization condition being given by mb = hb v1 = mR
b , one
would obtain that the bare bottom mass is equal to mb (1 1mb ), while in the MS-scheme,
for which mb is zero as 1mb is finite, one would have mb /(1 + 1mb ). Both results are
equivalent at first order in 1mb , as they should.
To proceed with the resummation, we come back to the relation between the Yukawa
and the pole mass. Although no n-loop diagrams produce sn tann corrections for n > 2,
there is one and only one genuine nth-order diagram left (see Fig. 14), which contains the
insertion of a (n 1)-loop counterterm into a one-loop diagram. Then, all dominant terms
in the large tan limit, at all orders in PT, are contained in the equation
hb v1 + hb v1 + h b v1 1mb + h b v1 1mb = mb .
(41)
Fig. 14. Full set of SUSY-QCD dominant diagrams, in the large tan limit, contributing to the bottom
pole mass at all orders in PT. The first three diagrams are those of Fig. 13. In the fourth one, the
cross denotes the insertion of the h b counterterm, and the solid and dashed lines denote gluino and
sbottom propagators, respectively (see Fig. 16).
113
(42)
which can be regarded as the identity of the bare quark and squark Yukawa couplings,
which is guaranteed by the underlying supersymmetry governing the relations between
the bare Lagrangian parameters in the ultraviolet. 9 No soft SUSY-breaking dimensionful
couplings can induce modifications to Eq. (42), allowing for the extraction of a common
hb v1 + hb v1 factor in (41). At the level of bare couplings one does not need to make
reference to any particular renormalization scheme. Thus, one has
mb
,
(43)
(hb + hb )v1 = mR
b + mb =
1 + 1mb
where the r.h.s. is expressed in terms of physical quantities, 1mb being independent of mb .
For the rest of this appendix we will derive expressions valid to all orders in PT in the
respectively,
large tan limit for the H + and H, h, A dressed couplings to t b and bb,
recovering the effective Lagrangian results one can find in [55]. The calculation involves
contributions from three-point loop diagrams with one external on-shell Higgs leg whose
momentum we have neglected. In Section 4, the departure from this assumption for the
H + and t decay rates has been shown to be small, as the extra contribution inducing the
momentum dependence does not include any tan enhancement factor. More complete
formulae including the momentum dependence for the decay rates of the neutral Higgs
bosons can be found in Ref. [66].
Let us start with the simplest case, that of the charged Higgs H + and of the pseudoscalar A, for which there are no vertex loop diagrams tan-enhanced with respect to the
tree-level coupling. The relevant Feynman diagrams are just the tree-level Yukawa and the
counterterm. From Eq. (43), the renormalized decay amplitudes are given by
i(hb + hb ) sin H + tPR b = i
mb tan
H + tPR b,
(1 + 1mb )v
mb tan
sin
5 b.
Ab
(hb + hb ) Ab
5b =
2
2(1 + 1mb )v
(44)
Therefore, in this case, the result of the resummation is to effectively modify the tree-level
Yukawa coupling by the universal 1/(1 + 1mb ) factor.
The case of the CP-even neutral Higgs bosons is a little bit more involved. Depending
on the relation between and the mixing angle, , the one-loop correction to the vertex
diagrams can be importantly enhanced. The full set of potentially relevant graphs is shown
in Fig. 15.
renormalized amplitude
One obtains, for the H bb
cos
sin 1mb
H bb
i(h b + h b )
i(hb + hb ) H bb
2
2 tan
9 This is true if a regularization method preserving SUSY is used, such as dimensional reduction. Deviations
from Eq. (42) in the MS-scheme will be loop-suppressed, not affecting the conclusions of this appendix.
114
Fig. 15. Vertex diagrams contributing to the renormalization of the Higgs-fermion Yukawa
interaction. From left to right, the renormalized Yukawa coupling, the Yukawa counterterm, the
one-loop contribution and the higher-order diagram containing the insertion of the h b counterterm.
The solid and dashed lines inside the loops denote gluino and sbottom propagators, respectively. The
cross in the fourth diagram denotes the h b counterterm.
mb cos
tan
= i
H bb.
1 + 1mb
tan
2(1 + 1mb )v1
(45)
Again, the resummation amounts to the inclusion of the universal 1/(1 + 1mb ) factor.
However, there is an additional 1mb term inside the parenthesis, which constitutes the non
tan-suppressed contribution coming from the SUSY-QCD vertex diagrams. Similarly, for
one has
the hbb
sin
cos 1mb
hbb
i(h b + h b )
i(hb + hb ) hbb
2
2 tan
mb sin
1mb
hbb.
(46)
=i
1
tan tan
2(1 + 1mb )v1
It can easily be checked that for large MH + values, the limit that corresponds to the
coupling
effective decoupling of one of the Higgs doublets, one recovers the SM hbb
m
i b hbb,
2v
coupling, H being heavy, still feels the decoupled sector
whereas the H bb
i
mb tan
2(1 + 1mb )v
H bb.
Fig. 16. The fourth self-energy diagram in Fig. 14, shown in greater detail. A gluino propagator is
denoted by the solid line inside the loop. The dashed lines denote sbottom propagators and the cross
the insertion of a h b counterterm.
h b v1
(8s ) CF Mg tan
2
115
(47)
bj
where CF = 4/3 is a colour factor and the two-dimensional rotation matrices Z transform
the weak eigenstate sbottom basis into the mass eigenstate basis. Expressed in terms of the
mixing angle b , the components of Z read: Z11 = Z22 = cosb , Z12 = Z21 = sinb .
The term between parentheses in the numerator of (47) and the combination h b v2 =
b , after
h b v1 tan come from the counterterm to the tan-dominant interaction H20 bR
L
the Higgs field develops its vacuum expectation value v2 .
Splitting the implicit i, j sum into the i = j part and the rest of the terms we obtain
Z
1
1
1
d nk
2
Lb
sin 2b
+
bP
2
(2)n k 2 Mg2 (k 2 m2 )2 (k 2 m2 )2
b1
b2
Z
nk
1
d
L b,
(48)
bP
+ cos2 2b
(2)n (k 2 Mg2 )(k 2 m2 )(k 2 m2 )
b1
b2
the constant being a short-hand for the constant prefactor of the integral in (47).
The second term in (48) is of the same form as 1mb . Adding and removing this term
times tan2 2b and rearranging terms one arrives at
Z
1
d nk
Lb
bP
n
2
2
2
(2) (k Mg )(k m2 )(k 2 m2 )
+
sin2 2b
2
b1
b2
(m2 m2 )2
1
d nk
b2
b1
L b.
bP
(2)n k 2 Mg2 (k 2 m2 )2 (k 2 m2 )2
b1
b2
(49)
(50)
b2
(51)
b2
The second integral in (51) has two extra propagators and thus in the limit of heavy
6
2
), whereas the first one is of O(1/MSUSY
). One can
SUSY masses it is of O(1/MSUSY
conclude that the two- and three-point loop diagrams in Fig. 14 are just related by h b /h b ,
2
.
apart from contributions that are suppressed by powers of either tan or m2b tan2 /MSUSY
The amplitude for the diagram in Fig. 16 reduces to
2 2
mb tan
2
1
L b.
O
(52)
bP
+
O
i h b v1 1mb 1 + 2
2
tan
MSUSY
MSUSY
116
Here
mb (Q)
g
Vt b tan H + tL bR (x, Q)
J (x) =
2MW 1 + 1mb
is the scalar current stemming from the Yukawa interaction in Eq. (21). All currents and
couplings in this appendix are considered to be renormalized using a mass-independent
renormalization scheme such us the MS scheme [63]. For the moment we also assume this
for the quark masses and discuss the use of the pole mass definition, which is commonly
used for the top mass, later. The decay rate involves highly separated mass scales mb
MH + , mt . First we assume that MH + and mt are of similar size so that log MH + /mt
is not dangerously large. We return to the case mb mt MH + later. To prepare the
resummation of the large logarithm log mb /MH + , we first perform an operator product
expansion of the bilocal forward scattering operator in Eq. (53):
Z
i d 4 x eiqx h H + |T J (x)J (0)| H + i
X
Cn (q 2 , mt , Q)h H + |On | H + i(mb , Q).
(54)
=
n
2
Here all dependence on the heavy mass scales mt and q 2 = MH
+ is contained in the
Wilson coefficient Cn , while the dependence on the light scale mb resides in the matrix
element of the local operator On . Both depend on the renormalization scale Q at which the
OPE is carried out (so that Q is sometimes called factorization scale). The OPE provides
in terms of (m /M + )2 . Increasing powers of m /M +
an expansion of (H + t b)
b
b
H
H
correspond to increasing twists of the local operator On . Here the twist is defined as the
dimension of the operator On minus the number of derivatives acting on the Higgs fields
in On .
The OPE in Eq. (54) is depicted in Fig. 17 where also the leading twist operator O1 =
m2b (Q)H + H is shown. At leading twist the OPE, depicted in Fig. 17, is trivial: the matrix
element h H + |O1 | H + i simply equals m2b (Q) and the Wilson coefficient C1 can be read
off from Eq. (33). In the leading order (LO) of QCD it reads
117
Fig. 17. The OPE in (54) to leading order in mb /MH + and s . The self-energy diagram on the left
represents the left-hand side of Eq. (54). The right diagram depicts h H + |O1 | H + i.
Im C1 =
g 2 Nc
1
2
2
MH
tan2 .
+ (1 rt )
2
(1 + 1mb )2
32MW
(55)
The QCD radiative corrections in contain powers of the large logarithm s log mb /MH + .
The OPE in Eq. (54) splits this logarithm into s log Q/MH + + s log mb /Q: the former
term resides in the coefficient function C1 while the latter is contained in the matrix
element h H + |O1 | H + i. If we choose Q = O(mt , MH + ), then the logarithms in the
Wilson coefficient are small and perturbative, but log mb /Q in the matrix element is big
and needs to be resummed to all orders. One could likewise choose Q ' mb and resum
the large logarithm in the Wilson coefficient, but the former way is much easier here. In
order to sum log mb /Q we have to solve the renormalization group (RG) equation for O1 .
Since the Higgs fields in O1 have no QCD interaction, the solution of the RG equation
simply amounts to the use of the well-known result for the running quark mass mb (Q)
(see Eq. (18)) at the scale Q = O(mt , MH + ) in O1 . In the next-to-leading order (NLO)
one has to include the O(s ) corrections to in Eq. (33). First there are no explicit oneloop corrections to h H + |O1 | H + i, so that in the NLO we obtain Im C1 (Q) by simply
multiplying the result in Eq. (55) with the curly bracket in (33). Secondly in the NLO we
have to use the two-loop formula for mb (Q) in the matrix element. Since one is equally
entitled to use Q = MH + (as chosen in (33)) or Q = mt or any other scale of order
mt , MH + , there is a residual scale uncertainty. This feature is familiar from all other RGimproved observables. To the calculated order s this uncertainty cancels, because there is
an explicit term s log Q/MH + in the one-loop correction, so that the scale uncertainty is
always of the order of the next uncalculated term. In our case this is O(s2 ) and numerically
tiny. In conclusion, our OPE analysis shows that at leading order in mb /MH + all large
can indeed be absorbed into the running quark mass in our
logarithms in (H + t b)
effective Lagrangian in Eq. (21). Some clarifying points are in order:
1. The summation of large logarithms in the NLO does not require the calculation of the
two-loop diagrams obtained by dressing the diagram in Fig. 17 with an extra gluon,
as performed in [67]. This calculation only gives redundant information, already
contained in the known two-loop formula for the running quark mass.
2. At the next-to-leading order the result depends on the chosen renormalization
scheme. Changing the scheme modifies the constant term 17/3 in Eq. (33). After
inserting the NLO (two-loop) solution (18) for the running mass, this scheme
dependence cancels between this term and J (f ) in Eq. (19). In the literature,
sometimes, the one-loop result for is incorrectly combined with the one-loop
running bottom mass resulting in a scheme-dependent expression.
118
No running top-quark mass is needed for the case mt ' MH + , and one can adopt the
pole mass definition for mt as we did.
3. The OPE also shows that the correct scale to be used in the running s in Eq. (33) is
the high scale Q = O(mt , MH + ) and not the low scale mb .
4. The absorption of the large logarithms into the running mass does not work for
terms that are suppressed by higher powers (mb /MH + )n with respect to the leading
contribution considered by us. Higher-twist operators contain explicit b-quark fields.
MH +
and to verify that the overall factor (1 rt (MH + ))2 indeed reproduces the rt log rt term in
Eq. (33). The terms of order rt2 log rt are not correctly reproduced by the running top mass
as anticipated by the occurrence of non-trivial twist-8 operators. The important result of
our consideration of the case mb mt MH + is the absence of terms of the form rt0 log rt
to all orders in s . In this case the additional large logarithm log rt is always suppressed by
powers of rt and therefore these terms are negligible for rt 1 and need not be resummed.
For the decay t b H + the above discussion can be repeated with the appropriate
changes in the OPE: the leading-twist operator is now O1 (Q) = m2b (Q) tt (Q) and the
external state in Eq. (54) is a top quark instead of a charged Higgs boson. We have mb
mt , MH + and the factorization scale Q is again of order mt , MH + . While O1 now involves
119
strongly interacting fields, its matrix element h t |O1 | t i(Q) still does not contain large
logarithms log mb /Q other than those contained in the running mass mb (Q). Hence the
applies likewise for (t b H + ).
proof above for (H + t b)
After exchanging Vt b tL bR for Vcb cL bR in Eq. (21), we can likewise apply our effective
Lagrangian L to semileptonic B-meson decays corresponding to b c ` ` by using
the appropriate scale Q ' mb in L. The QCD radiative corrections involve no large
logarithm, because the gluons couple only to the b and c quarks. Hence the effective fourfermion operator cL bR `R L obtained after integrating out the heavy H + renormalizes in
the same way as the quark current cL bR in L. The corresponding loop integrals do not
depend on MH + at all and this feature is correctly reproduced by using mb (mb ) in L.
The situation is different in physical processes in which the charged Higgs connects two
quark lines, as for example in the loop-induced decay b s . Here effective four-quark
operators, which involve a non-trivial renormalization group evolution, occur. The largetan supersymmetric QCD corrections associated with 1mb and the H + tL bR Yukawa
coupling, however, are still correctly reproduced by applying L to b s or other
loop-induced rare b-decays. Yet it must be clear that these corrections are part of the
mixed electroweak-QCD two-loop contributions and that there are already supersymmetric
electroweak contributions at the one-loop level, which are process-specific and of course
not contained in L.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
120
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
[66]
[67]
Abstract
We obtain general analytical forms for the solutions of the one-loop renormalization group
equations in the top/bottom/ sector of the MSSM. These solutions are valid for any value of tan
as well as any non-universal initial conditions for the soft SUSY breaking parameters and nonunification of the Yukawa couplings. We establish analytically a generic screening effect of nonuniversality, in the vicinity of the infrared quasi fixed point, which allows to determine sector-wise a
hierarchy of sensitivity to initial conditions. We give also various numerical illustrations of this effect
away from the quasi fixed point and assess the sensitivity of the Higgs and sfermion spectra to the
non-universality of the various soft breaking sectors. As a by-product, a typical anomaly-mediated
non-universality of the gaugino sector would have marginal influence on the scalar spectrum. 2000
Elsevier Science B.V. All rights reserved.
PACS: 11.10.Hi; 12.60.Jv; 12.60.-i
Keywords: MSSM; Renormalization group; Non-universality; Fixed point; tan
1. Introduction
The Minimal Supersymmetric Standard Model (MSSM) [16] has been intensively
studied recently [7] with the emphasis on prediction of particle spectrum. It crucially
depends on the mechanism of SUSY breaking and the one which is commonly accepted
introduces the so-called soft terms at a high energy scale [1518]. These soft terms are
running then down to low energy according to the well known RG equations starting from
some initial values. In the minimal version the soft terms obey the universality hypothesis
which leaves one with a set of 5 independent parameters (before radiative electroweak
symmetry breaking): m0 , m1/2 , A0 , 0 and tan . However, recently there appeared some
1 E-mail: moultaka@lpm.univ-montp2.fr; phone: (33) 4.67.14.35.53; fax: (33) 4.67.54.48.50
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 0 7 - 3
122
(1)
(2)
2
/Q2 and
where d/dt, t = log MGUT
bi = {33/5, 1, 3},
ct i = {13/15, 3, 16/3},
at l = {6, 1, 0},
a l = {0, 3, 4}.
c i = {9/5, 3, 0},
i =
Yk =
i0
1 + bi i0 t
Yk0 uk
1 + akk Yk0
123
(3)
Rt
0
uk
(4)
Et
R
0 t
(1 + 6Yb
ub =
,
ub )1/6
Eb
,
Rt
Rt
(1 + 6Yt0 0 ut )1/6 (1 + 4Y0 0 u )1/4
E
,
u =
Rt
(1 + 6Yb0 0 ub )1/2
(5)
3
Y
(1 + bi i0 t)cki /bi .
(6)
i=1
Let us stress that Eqs. (3,4) give the exact solution to Eqs. (1,2), while the uk s in
Eqs. (5), although solved formally in terms of the Ek s and Yk0 s as continued integrated
fractions, should in practice be solved iteratively. Yet the important gain here is twofold:
(i) As shown in [23], the convergence of the successive iterations to the exact solution
can be fully controlled analytically in terms of the initial values of the Yukawas, allowing
in practice to obtain approximations at the level of the percent or less after one or two
iterations, and
(ii) The structure of the solutions is explicit enough to allow for exact statements about
some regimes of the initial conditions, as we will see later on. Furthermore, these nice
features will be naturally passed on to the solutions for the soft SUSY breaking parameters
since the latter will be obtained from (3)(5) through the method of Ref. [24].
124
i = i (1 + mi + m
i + 2mi m
i ),
Yk Yk = Yk 1 Ak Ak + (k + Ak A k ) ,
(7)
(8)
where mi are the gaugino masses, Ak are the scalar triple couplings and k are certain
combinations of the soft masses
e2Q3 + m
e2U 3 + m2H 2 ,
t = m
e2Q3 + m
e2D3 + m2H 1 ,
b = m
e2L3 + m
e2E3 + m2H 1 .
= m
Here = 2 and = 2 are the spurion fields depending on Grassmannian parameters
, ( = 1, 2).
It has been proven in Ref. [26] that the singular part of effective action, which determines
the renormalization properties of any softly broken SUSY theory, is equal to that of an
unbroken one in presence of external spurion superfields. This means that in order to
calculate it in a softly broken case one just has to take the unbroken one, replace the
couplings according to Eqs. (7,8) and expand over the Grassmannian parameters and .
Moreover, as it has been demonstrated in [24], the same replacement can be done directly
in RG equations in order to get the corresponding equations for the soft terms, or even in the
solutions to these equations. In the last case one obtains the solutions to the RG equations
for the soft terms. Below we demonstrate how this procedure works in case of the MSSM,
when combined with the solutions (3), (4).
m0i
(9)
R
A0k /Yk0 + akk uk ek
R
,
Ak = ek +
1/Yk0 + akk uk
(10)
k = k + A2k
(11)
1 + bi i0 t
R
(A0k )2 /Yk0 k0 /Yk0 + akk uk k
+ 2ek Ak
R
,
1/Yk0 + akk uk
where the new functions ek and k have been introduced which obey the iteration equations
R
R
et
A0b ub ub eb
1 dE
+
R
,
et =
Et d
1/Yb0 + 6 ub
125
R
R
R
R
eb A0t ut ut et
A0 u u e
1 dE
R
+
R
+
,
Eb d
1/Y0 + 4 u
1/Yt0 + 6 ut
R
R
e
A0b ub ub eb
1 dE
+3
R
,
e =
E d
1/Yb0 + 6 ub
!2
R
R
R
R
et A0b ub ub eb
et
A0b ub ub eb
1 dE
1 d2 E
+2
R
R
+7
t =
Et d d
Et d 1/Yb0 + 6 ub
1/Yb0 + 6 ub
Z
Z
Z
Z
1
0
0 2
0
b + (Ab )
+ 6 ub ,
ub 2Ab ub eb + ub b
Yb0
" R
#
R
R
R
eb A0t ut ut et
eb
A0 u u e
1 dE
1 d2 E
R
+2
R
+
b =
Eb d d
Eb d
1/Y0 + 4 u
1/Yt0 + 6 ut
!2
!2
R
R
R
R
A0 u u e
A0t ut ut et
R
R
+5
+7
1/Y0 + 4 u
1/Yt0 + 6 ut
!
!
R
R
R
R
A0t ut ut et
A0 u u e
R
R
+2
1/Y0 + 4 u
1/Yt0 + 6 ut
Z
Z
Z
Z
1
+
6
ut
ut 2A0t ut et + ut t
t0 + (A0t )2
Yt0
Z
Z
Z
Z
1
+
4
u
u 2A0 u e + u
0 + (A0 )2
,
Y0
!2
R
R
R
R
e A0b ub ub eb
e
A0b ub ub eb
1 dE
1 d2 E
+6
R
R
+ 27
=
E d d
E d 1/Yb0 + 6 ub
1/Yb0 + 6 ub
Z
Z
Z
Z
1
3 b0 + (A0b )2
+
6
u
ub 2A0b ub eb + ub b
b , (12)
Yb0
eb =
(13)
i=1
!2
3
3
X
X
ek
1 d2 E
2
0
=
t
c
m
+
2t
cki i (m0i )2
ki
i
i
Ek d d ,=0
i=1
t2
3
X
i=1
cki bi i2 (m0i )2 .
(14)
i=1
When solving Eqs. (5) and (12) in the nth iteration one has to substitute in the r.h.s. the
(n 1)-th iterative solution for all the corresponding functions.
In the particular case where Yb = Y = 0 Eqs. (5)(12) give an exact and well known
solutions already in the first iteration.
126
Let us finally note that upon inspection of the solutions (9)(12), the Ai s and i s
depend respectively linearly and quadratically on the initial conditions, as expected, and
thus can be generally cast in the form:
X
X
aj (t)A0j +
bk (t)m0k ,
(15)
At,b, (t) =
j =t,b,
k=1,2,3
t,b, (t) =
i,j =t,b,
i,j =1,2,3
i=t,b,, j =1,2,3
(16)
where the various running coefficients aj , bk , cij , dij , eij and ki are fully determined by
our solutions and can be seen to depend exclusively on the initial conditions of the gauge
and Yukawa couplings. In Section 6 we will evaluate these coefficients at the E.W. scale,
using a truncation of the general solutions.
uFP
Rk
akk uFP
k
(17)
with
Et
uFP
t = R FP 1/6 ,
( ub )
Eb
uFP
b = R FP 1/6 R FP 1/4 ,
( ut ) ( u )
E
uFP
= R FP 1/2
( ub )
(18)
extending the IRQFP [29,30] to three Yukawa couplings. What is worth stressing here is
that both the dependence on the initial condition for each Yukawa as well as the effect of
Yukawa non-unification, r1 , r2 have completely dropped out of the runnings. (Note that in
practice this regime is already obtained if Yk0 > 0GUT , assuming here for simplicity the
unification of the gauge couplings 10 = 20 = 30 = 0GUT .) The fact that the ratios r1 , r2
drop out implies the validity of the described properties in any tan regime.
This in turn leads to the IRQFPs for the soft terms. Disappearance of Yk0 in the FP
solution naturally leads to the disappearance of A0k and k0 in the soft term fixed points. To
127
see this one can either go to the limit of large Yk0 in Eqs. (9)(12) or directly perform the
Grassmannian expansion of the FP solutions (17), (18). One gets
R
AFP
k
= ekFP
kFP
= kFP
uFP eFP
+ R k FPk ,
uk
2
+ (AFP
k )
R
= kFP
uFP FP
R k FPk
uk
R FP FP !2
u e
R k FPk
(ekFP )2 +
uk
+ 2ekFP AFP
k
uFP FP
R k FPk
uk
(19)
(20)
with
R
et
eFP
1 uFP
1 dE
R b FPb ,
=
Et d
6
u
R FPb FP
R
FP
e
1 d E b 1 ut e t
1 uFP
e
FP
R
R
,
eb =
Eb d
6
4
uFP
uFP
t
R FP FP
e
1 ub e b
1 dE
R FP ,
eFP =
E d
2
ub
R FP FP !2
R
R
FP
et uFP
et
ub e b
FP
7
1 d2 E
1 1 dE
1 uFP
b eb
FP
R FP
R FP
R b FPb ,
+
t =
Et d d 36
3 Et d
6
ub
ub
ub
R
R
R
R
2
2
FP uFP e FP
eb
uFP eFP
uFP eFP
uFP
7
1 d2 E
5
1
t et
R
R
R t FPt
R
+
+
+
bFP =
FP
Eb d d 36
16
12
uFP
uFP
ut
u
t
R FP FP
R FP FP
R FP FP
R FP FP
eb 1 ut et
1 u e
1 u
1 ut t
1 dE
R
R
R FP +
R FP
,
Eb d 3
2
6
4
uFP
uFP
ut
ut
R FP FP !2
R
R
FP
2E
e uFP
e
ub e b
FP
3
1
1 dE
1 uFP
d
b eb
FP
R FP
R FP
R b FPb .
+
(21)
=
E d d 4
E d
2
ub
ub
ub
etFP
One can see from Eqs. (19), (20) that the constant terms in ek and k do not contribute
to Ak and k and can be dropped from Eqs. (21). Thus, all the dependence on the initial
conditions Yk0 , A0k and k0 disappears from the fixed point solutions. The only dependence
left is on the gaugino masses. This is a general screening property valid for the exact
solution as well as for any of its approximate (truncated) forms.
In view of the above screening properties of the initial conditions at the quasi-fixed point,
one should recall the existing connection between the true IR attractive fixed point of the
Yukawa couplings and that of the soft parameters [24,31,32]. What we have established
can be seen as an extension of such connections to the transient regime of quasi-fixed point
at the one-loop level. It is also worth stressing that the above properties are valid for any
renormalization scale and are thus operative despite the uncertainty in the choice of the
physical scales.
128
ek =
ek
1 dE
,
Ek d
k =
ek
1 d2 E
,
Ek d d
(22)
(23)
(24)
2
/MZ2 66.
where the numbers are calculated for t = log MGUT
One can see from Eqs. (23), (24) that the prevailing term is that of m03 . A0t and t0
decouple due to the fixed point behaviour as explained above and the contribution of m01
and m02 is small due to smallness of 1 and 2 compared to 3 at t = 66.
The second case looks similar. We first consider the triple couplings Ak . In this case one
has a set of initial values {A0k , m0i }. In Fig. 1 we show the variation of the coefficients ai , bk
of Eq. (15) for At , Ab and A as functions of Yk0 in the interval Yk0 = 0 100 .
One can clearly see that the coefficients of A0k are small and have a fast decrease with
increasing Yk0 . The coefficients of m0i quickly saturate and approach their asymptotic values
129
Fig. 1. Initial value contributions of the various soft SUSY breaking parameters to the running At , Ab
and A at the EW scale, as a function of a common initial value for the three Yukawa couplings.
130
with the hierarchy 1 : 10 : 100 for m01 , m02 and m03 . The effect is less pronounced for A due
to the absence of the SU(3) coupling in the lepton sector.
Now we come to k . We have chosen the intermediate value of Yk0 = 50 where the
effective fixed point is practically already reached and calculate the coefficients at t = 66
as in Eqs. (23), (24). One has
t = 0.0497 A2t 0 + 0.0076 A2b0 + 0.0142 At 0Ab0 + 0.0057 At 0m10
0.0018 Ab0m10 + 0.0333 At 0m20 0.0114 Ab0m20 + 0.1509 At 0m30
0.0516 Ab0m30 + 0.0198 m210 + 0.2509 m220 + 6.3299 m230
2
0.0057 m10m20 0.0336 m10m30 0.2669 m20m30 0.0252 m
e0D3
2
2
2
2
0.0252 m0H1 + 0.0525 m0H2 + 0.0273 m
e0Q3 + 0.0525 m
e0U3 ,
(25)
2
0.0133 m10m20 + 0.0116 m10m30 0.0687 m20m30 0.2177 m
e0D3
2
2
2
2
+ 0.2649 m
e0E3 + 0.0473 m0H1 + 0.2649 m
e0L3 0.2177 m
e0Q3 . (27)
One can again see how the coefficients of the initial values of A0k and k0 almost vanish
and the prevailing one is that of (m03 )2 . The next-to-leading ones are those of (m02 )2 and
m02 m03 being however almost 30 times smaller. This is true for both t0 and b0 but
is less manifest for 0 . We note here that a soft gaugino mass hierarchy like the one
predicted by anomaly-mediated susy breaking, m3 : m2 : m1 = 3 : 0.3 : 1 [19], enforces
even more the insensitivity of the running Ai s and i s to the non-universality of the
gaugino sector.
6.2. The next iterations
To demonstrate the validity of the iterative procedure and reliability of the the first
iteration we consider the effect of the next ones on the above mentioned coefficients. We
have performed the numerical integration up to the 6th iteration and have observed fast
131
convergence of the coefficients to their exact values. To show the numbers we have chosen
the leading coefficients of m03 in Ak and m203 in k . In case Yt0 = Yb0 = Y0 = 50 and
0 = 0.00329 the results are the following:
Iteration
1st
2nd
3rd
4th
5th
6th
At
Ab
1.6127
1.6161
1.6125
1.6133
1.6131
1.6131
1.7584
1.7330
1.7372
1.7375
1.7373
1.7374
0.6871
0.5037
0.5526
0.5440
0.5456
0.5454
6.3299
6.3270
6.3192
6.3213
6.3206
6.3207
6.8166
6.6822
6.7069
6.7054
6.7053
6.7054
2.8162
2.0989
2.2937
2.2588
2.2660
2.2649
One can see explicitly the fast convergence of the iterations. As expected it is worse
for A and , so in this case one has to take few more iterations. We present the
general arguments for the convergence of iterations for the soft terms in Appendix B. The
advantage of this solution is that one can improve the precision taking further iterations
and in principle can achieve any desirable accuracy. Typically one has an accuracy of a few
percent after 23 iterations. This is in contrast with the approximate solutions presented in
Ref. [33] which give simple explicit expressions but without improvement.
Taking the sixth iteration in Eqs. (5), (12) expressions for the soft terms now look like
At = 0.0558 At 0 0.0294 Ab0 + 0.0080 A 0
0.0186 m10 0.1586 m20 1.6131 m30,
Ab = 0.0341 At 0 + 0.0984 Ab0 0.0450 A 0
0.0014 m10 0.1583 m20 1.7374 m30,
A = 0.0394 At 0 0.2221 Ab0 + 0.2871 A 0
0.0825 m10 0.2344 m20 + 0.5454 m30,
t = 0.0487 A2t 0 + 0.0068 A2b0 + 0.0014 A2 0 + 0.0130 At 0Ab0
0.0017 At 0A 0 0.0017 Ab0A 0 + 0.0058 At 0m10 0.0015 Ab0m10
0.0001 A 0m10 + 0.0335 At 0m20 0.0097 Ab0m20 + 0.0005 A 0m20
+ 0.1514 At 0m30 0.0460 Ab0m30 + 0.0070 A 0m30 + 0.0211 m210
+ 0.2547 m220 + 6.3207 m230 0.0057 m10m20 0.0340 m10m30
2
2
2
0.2720 m20m30 0.0294 m
e0D3 0.0214 m0H1 + 0.0558 m0H2
2
2
2
2
+ 0.0264 m
e0Q3 + 0.0558 m
e0U3 + 0.0080 m
e0L3 + 0.0080 m
e0E3 ,
b = +0.0067 A2t 0 0.0736 A2b0 0.0035 A2 0 + 0.0200 At 0Ab0
0.0064 At 0A 0 + 0.0302 Ab0A 0 0.0003 At 0m10 0.0000 Ab0m10
+ 0.0021 A 0m10 0.0110 At 0m20 + 0.0396 Ab0m20 0.0076 A 0m20
132
133
(28)
At this level one can already make rough qualitative statements about the physical scalar
masses. We note first that, as can be seen from the above equations, the sensitivity to the
initial conditions reappears partly in the running of the soft scalar masses, even in the
vicinity of the IRQFP. However, this dependence remains confined in the initial values
of the soft masses themselves in a universal (scale independent) form, and in the initial
conditions of the gaugino soft masses through the fi s and the s. [The dependence on
the initial values of Yukawa couplings as well as on the A0 s and the 0 s, that could come
from the running, remain completely screened.] The ratios giving the universal sensitivity
of the running soft scalar masses to the soft scalar masses initial conditions is as follows:
(e
mQ3 )2
(e
mU 3 )2
(e
mD3 )2
(mH 1 )2
(mH 2 )2
(e
mL3 )2
(e
mE3 )2
(e
m0Q3 )2
(e
m0U 3 )2
17
17
5
5
17
5
5
3.4
40
1
1
21
1
1
(e
m0D3 )2
(m0H 1 )2
(m0H 2 )2
(e
m0L3 )2
(e
m0E3 )2
4
4
9.25
6
4
6
6
3
3
4.5
5.67
3
5.67
5.67
3.4
21
1
1
19.67
1
1
1
1
1.5
1.89
1
29
11.67
1
1
1.5
1.89
1
11.67
8.67
These numbers are renormalization scale independent and give the trend of the relative
sensitivity in the vicinity of the IRQFP.
On the other hand, the dependence on the initial soft gaugino masses is renormalization
scale dependent. At the electroweak scale (t ' 66), one finds that in the soft masses of the
third squark generation and of the Higgs doublets the sensitivity to (m03 )2 remains leading
134
(by a factor of 15 to 25) as compared to (m02 )2 . In contrast, a large cancellation occurs for
eE3 , and even a
the sleptons, leading to comparable sensitivities to (m03 )2 and (m02 )2 in m
bigger sensitivity to (m02 )2 (by a factor of 4) in m
eL3 .
To go further to the physical scalar masses, one has to consider the behaviour of the
parameter which enters the mixing of the Rleft and right states. The running of this
t
parameter has the simple form (t) 0 exp[ 0 ( Y )] where , Y are generic gauge
and Yukawa couplings. Thus, here too, the initial conditions for the Yukawas are screened
near the IRQFP in the evolution of , the Ai s and i s being absent anyway. However,
when the electroweak symmetry breaking (EWSB) is required to take place radiatively,
the parameter becomes, as usual, correlated to the other parameters of the MSSM at
the electroweak scale. To be specific, in the leading one-loop top/stop-bottom/sbottom
approximation to the EWSB conditions, the sensitivity of to the initial conditions will
come basically from the soft scalar masses of the Higgs doublets and scalar partners of
the third quark generation [34]. As stated before, the latter dependence is dominated,
on one hand by the initial conditions of the soft scalar masses, in a well determined
scale independent way, and on the other hand by the (scale-dependent) m03 contributions.
The same dependence pattern is then taken over to the physical scalar masses. A further
inclusion of the scalar contributions to the EWSB conditions will basically not affect this
e2E3 have comparable sensitivity between
dependence pattern. Indeed, although m
e2L3 and m
0
0
m2 and m3 at the electroweak scale, they are less sensitive to this sector altogether than
to the squark soft masses. All in all our analytical results allow to draw at this stage a
qualitative sensitivity hierarchy for the physical scalar masses:
basically no sensitivity to Yukawa couplings initial conditions (whether unified or
not), or A0i initial conditions (whether universal or not),
important sensitivity to initial conditions of the soft gaugino masses, however
basically only through m03 , i.e., weak sensitivity to non-universality of this sector,
important sensitivity to initial conditions of the soft scalar masses, however through
a universal scale independent pattern.
8. Conclusion
In the present paper we have obtained general analytical forms for the solutions of the
one-loop renormalization group equations in the top/bottom/ sector of the MSSM. These
solutions are valid for any value of tan as well as any non-universal initial conditions
for the soft SUSY breaking parameters and non-unification of the Yukawa couplings. They
allow a general study of the evolution of the various parameters of the MSSM and to trace
back, sector-wise, the sensitivity to initial conditions of the Yukawa couplings and the soft
susy breaking parameters. We have established analytically a generic screening of nonuniversality, in the vicinity of the infrared quasi fixed points. In practice, this property gives
the general trend of the behaviour, despite the large number of free parameters, and even
when one is not very close to such a quasi fixed point. This shows that non-universality
of the A parameters and gaugino soft masses, as well as Yukawa unification conditions,
135
would basically have no influence on the squark and Higgs spectra. The main input from
the gaugino sector comes from the soft gluino mass contribution (which dominates by
far the other two), i.e., insensitive to non-universality conditions of this sector. The only
substantial sensitivity to non-universality is associated to the initial conditions of the scalar
soft masses, but is renormalization scale independent and well defined. A similar pattern
holds for the sleptons, apart from the fact that now the contribution of the wino soft mass
becomes comparable to that of the gluino, yet the overall sensitivity to the gaugino sector
is much smaller than in the case of the squarks.
Detailed illustrations of the physical spectrum, including the lightest Higgs, will be given
in a subsequent study.
Acknowledgements
We are grateful to members of GDR-supersymtrie for useful discussions and to
S. Codoban for suggesting to us useful tricks for the iterative numerical integration. D.K.
would like to thank the University of Montpellier II for hospitality and CNRS for financial
support. We also thank Y. Mambrini and M. Rausch de Traubenberg for various comments
and for spotting a missing term in Eq. (21) in an earlier version.
et
E
,
R
(n1) 1/6
(1 + 6Yb0 u b
)
eb
E
,
u (n)
R
R (n1)
b =
(n1)
(1 + 6Yb0 u t
)1/6 (1 + 4Yb0 u
)1/4
e
E
,
u (n)
R
=
(n1) 1/2
(1 + 6Yb0 u b
)
(n)
u t
(A.1)
with
e
u (0)
k Ek
(k = t, b, ),
where the twiddled quantities are obtained from the non twiddled ones by proper rescaling
with r1 or r2 , and we indicate explicitly the order of iteration. It is now easy to show
inductively that if at the nth iteration
(n)
u (n)
k
(u k )FP
(Yb0 )pk,n
(A.2)
136
0
FP
where (u (n)
k ) is Yb independent but r1 , r2 dependent, then the same is true at the (n + 1)th iteration. Furthermore, since (A.2) is obviously true for n = 1 as can be easily seen from
(A.1) we conclude that the exact uk s behave also like
u k
(u k )FP
(Yb0 )
pk
with
0 < pk < 1.
This means that the 1s can be legitimately dropped both in Eq. (4) and Eq. (5). The
complete cancellation of Yb0 , r1 and r2 in the final result is then obvious, leading to
Eqs. (17), (18).
0
et (t) 0
(B.1)
E(t) = eb (t)
0
0,
0
e (t) 0
0
Ub (t1 ; t)
0
(B.2)
0
U (t1 ; t) ,
U(t1 ; t) = Ut (t1 ; t)
0
0
3Ub (t1 ; t)
where
Ui (t1 ; t)
ui (t1 )
Rt ,
+ a i 0 ui
i = t, b, ,
1/Yi0
(B.3)
and at = ab = 6, a = 4,
1 dEet
Et d
Rt
+ A0b Ub (t1 ; t)
0
C(t) =
0
eb
1 dE
Eb d
k=t,
Rt
A0k Uk (t1 ; t)
0
0
e
1 dE
E d
+ 3A0b
Rt
(B.4)
Ub (t1 ; t)
with C(0) = 0. The system of integral equations for the ei s can then be written in the
matrix form
Zt
(B.5)
E(t) = C(t) + U(t1 ; t)E(t1 ) dt1 .
0
137
Zt
U(t1 ; t)E(t1 ) dt1
(B.6)
for any matrix M in a given evolution interval [0, T ]. One then has the inequality
Zt
!
t
XZ
|Uik | kEk
(UE)ij 6
0
(B.7)
(B.8)
k 0
valid for any i, j . On the other hand, one has from Eqs. (B.2), (B.3)
R
ub
1
i = 1,
R
6 ,
1/YbR+ 6 ub
t
R
X
u
ut
5
R
R
+
6 , i = 2,
|Uik | =
0+6 u
0+4 u
12
1/Y
1/Y
t R
k 0
u
1
b
i = 3.
R
6 ,
3
1/Yb0 + 6 ub 2
(B.9)
Combining the above inequalities (B.8), (B.9) with Eq. (B.6) one obtains
1
(B.10)
kE10 E20 k 6 kE1 E2 k,
2
that is, the mapping (B.6) is a contraction, the solution to Eq. (B.5) is unique and
approximated at worse with an error of 1/2n , after n iterations. Actually, the situation
is much better than given by this upper bound error, as one can see from the numerical
illustrations of Section 6.2. Finally, we note that the rational is exactly the same for the
convergence of the s. Indeed, apart from a different definition for C(t), the s satisfy a
matrix equation similar to (B.5) with the same U as the one given in (B.2).
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
138
[12] D.M. Pierce, J.A. Bagger, K. Matchev, R. Zhang, Nucl. Phys. B 491 (1997) 3.
[13] M. Carena, P. Chankowski, M. Olechowski, S. Pokorski, C.E.M. Wagner, Nucl. Phys. B 491
(1997) 103.
[14] J. Casas, J. Espinosa, H. Haber, Nucl. Phys. B 526 (1998) 3.
[15] A.H. Chamseddine, R. Arnowitt, P. Nath, Phys. Rev. Lett. 49 (1982) 970.
[16] R. Barbieri, S. Ferrara, C.A. Savoy, Phys. Lett. B 119 (1982) 343.
[17] L. Hall, J. Lykken, S. Weinberg, Phys. Rev. D 27 (1983) 2359.
[18] M. Dine, A.E. Nelson, Phys. Rev. D 48 (1993) 1277.
[19] L. Randall, R. Sundrum, hep-th/9810155.
[20] G.F. Giudice, M.A. Luty, H. Murayama, R. Rattazzi, JHEP 9812 (1998) 27.
[21] A. Pomarol, R. Rattazzi, hep-ph/9903448.
[22] T. Gherghetta, G.F. Giudice, J.D. Wells, hep-ph/9904378.
[23] G. Auberson, G. Moultaka, Eur. Phys. J. C 12 (2000) 331; hep-ph/9907204.
[24] D.I. Kazakov, Phys. Lett. B 449 (1999) 201; hep-ph/9812513.
[25] L. Girardello, M. Grisaru, Nucl. Phys. B 194 (1982) 65.
[26] L.V. Avdeev, D.I. Kazakov, I.N. Kondrashuk, Nucl. Phys. B 510 (1998) 289.
[27] I. Jack, D.R.T. Jones, Phys. Lett. B 415 (1997) 383.
[28] G. Giudice, R. Rattazzi, Nucl. Phys. B 511 (1998) 25.
[29] C.T. Hill, Phys. Rev. D 24 (1981) 691.
[30] C.T. Hill, C.N. Leung, S. Rao, Nucl. Phys. B 262 (1985) 517.
[31] I. Jack, D.R.T. Jones, Phys. Lett. B 443 (1998) 177.
[32] M. Lanzagorta, G.G. Ross, Phys. Lett. B 349 (1995) 319.
[33] S. Codoban, D.I. Kazakov, hep-ph/9906256.
[34] G.K. Yeghiyan, M. Jurciin, D.I. Kazakov, Mod. Phys. Lett. A 14 (1999) 601; hep-ph/9807411.
[35] M. Jurciin, D.I. Kazakov, Mod. Phys. Lett. A 14 (1999) 671; hep-ph/9902290.
Abstract
The U (1) gauge theory on a D3-brane with non-commutative worldvolume is shown to admit
BIon-like solutions which saturate a BPS bound on the energy. The mapping of these solutions to
ordinary fields is found exactly, namely non-perturbatively in the non-commutativity parameters.
The result is precisely an ordinary supersymmetric BIon in the presence of a background B-field. We
argue that the result provides evidence in favour of the exact equivalence of the non-commutative
and the ordinary descriptions of D-branes. 2000 Elsevier Science B.V. All rights reserved.
PACS: 11.25.-w; 11.15.-q
Keywords: Solitons and monopoles; D-branes; Gauge theories
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 6 0 - 7
140
action are the closed string metric g, the closed string coupling constant gs and the constant
background B-field itself. Furthermore, the whole dependence of the effective action on B
is accounted for by writing it in terms of the modified field strength F F + B ? , where
F = dA and the superscript ? denotes the pull-back to the brane worldvolume. This is
the combination which is invariant under the two U (1) gauge symmetries which the open
string -model enjoys at the classical level: a first one under which
B = 0,
A = d,
(1.1)
A = ? .
(1.2)
(1.3)
(1.6)
where A is the non-commutative gauge field. With this regularization, the gauge
symmetry under which the effective action is invariant is
A],
A = d + i[,
b = i[,
b].
F
F
(1.7)
As explained in [3], since both the ordinary and the non-commutative descriptions
arise from different regularizations of the same worldsheet theory, there should be a field
redefinition which maps gauge orbits in one description into gauge orbits in the other. This
1 We will set 2 0 = 1.
141
requirement, plus locality to any finite order in , enabled the authors in [3] to establish the
following system of differential equations: 2
1
A
b ,
A = = A , A + F
4
i
b
1
X
bi + D X
bi ,
bi =
= A , X
(1.8)
X
4
b These equations, which we
b i[A , X].
where {f, g} f g + g f and D X X
will call -evolution equations, determine how the fields should change when is varied,
in order to describe equivalent physics. Their integration provides the desired map between
two descriptions with different values of , to which we will refer as the SeibergWitten
map.
The fact that two apparently so different descriptions can be equivalent is certainly
remarkable. The authors in [3] provided a direct check that this is indeed the case by
showing that the effective action in one description is mapped to the effective action
in the other by the SeibergWitten map. However, they worked in the approximation of
slowly-varying fields, which consists of neglecting all terms of order F (or 2 X). This
approximation was used at two different stages. First, when the effective action for the
massless fields on the brane was taken to be the DiracBornInfeld (DBI) action. Indeed,
this action can be derived from string theory precisely by neglecting such terms. Second,
this approximation was also used to simplify the SeibergWitten map considerably.
Although this check provides direct evidence in favour of the equivalence of the two
descriptions, it would be desirable to have an exact proof. This would consist of three
steps. First, one would have to determine the effective action in each description exactly,
namely to all orders in 0 . Second, one would have to integrate (1.8) also exactly, namely
to all orders in . Third, one would have to substitute the change of variables in one action
and see that the other one is recovered. Of course, this procedure is impossible to put into
practice, but we will see that it is still possible to provide some evidence that the Seiberg
Witten map works non-perturbatively in .
The idea is as follows. If one exact effective action is mapped to the other by the
SeibergWitten map, then a classical solution of one action should also be mapped to
a classical solution of the other. Of course, since we do not know the exact effective
actions, we do not know any non-trivial exact solutions either, except for one case: the BIon
[46]. This is a 1/2-supersymmetric solution of the ordinary DBI theory of a D-brane. In
general, supersymmetric solutions of the worldvolume theories of branes have a natural
interpretation as intersections of branes. The BIon is the prototype of this fact: it is the
worldvolume realization of a fundamental string ending on a D-brane. What makes the
BIon solution special is that, although originally discovered as a solution of the DBI action
[4,5], it has been shown to be a solution of the exact effective action to all orders in 0 [7].
Perhaps this should not be surprising since, after all, the fact that a fundamental string can
end on a D-brane is the defining property of D-branes [8].
2 The two Eqs. (1.8) are simply the dimensional reduction of that for a ten-dimensional gauge field A (M =
M
142
When no background B-field is present, the string ends orthogonally on the brane (see
Fig. 1(a)). When a constant electric 3 B-field is turned on, supersymmetry requires the
string to tilt a certain angle determined by B (see Fig. 1(b)). This can be intuitively
understood, because the background B-field induces a constant electric field on the brane. 4
Since the endpoint of the string is electrically charged, the string is now forced to tilt in
order for its tension to compensate the electric force on its endpoint.
Our strategy will be to identify a BIon-like solution of the effective action in the noncommutative description with the value of determined by B as in (1.5). Since the exact
effective action is not known, we will work with the lowest order approximation in 0
(see (3.5)). We will then integrate the -evolution equations exactly to find what this noncommutative BIon is mapped to in the ordinary description. The result is that it is mapped
to a tilted ordinary BIon, as described above, and that the tilting angle agrees precisely
with the value determined by B. 5
Our result can be interpreted in two ways. On one hand, if one accepts that the
equivalence between the ordinary and the non-commutative descriptions, as determined
by the SeibergWitten map, is valid non-perturbatively in , then the result proves that the
non-commutative BIon is a solution of the exact non-commutative effective action, namely
it is a solution to all orders in 0 . On the other hand, motivated by the defining property of
D-branes mentioned above, one could directly conjecture that the non-commutative BIon
is a solution of the exact non-commutative effective action. In this case, one could regard
our result as evidence in favour of the exact equivalence of both descriptions beyond the
perturbative level in . Whichever point of view one adopts, the result sheds some light on
another related question raised in [3]: the convergence of the series in generated by the
(a)
(b)
Fig. 1. The worldvolume realization of a fundamental string ending on a D-brane: (a) in the absence
of B-field, and (b) in the presence of an electric B-field.
0, . . . , 9), which was given in [3].
3 An electric B-field leads, through (1.5), to an electric in the non-commutative theory. There is some
controversy about whether or not such a theory makes sense. We will postpone the discussion of this issue until
the last section.
4 Fundamental strings ending on D-branes with constant electric fields on their worldvolumes have also been
studied in [9]. We will clarify the relationship of the results in [9] with ours at the end of Section 2.
5 The fact that tilted brane configurations are related to monopoles and dyons in non-commutative gauge
theories was first pointed out in [10] by working to first order in .
143
Eq. (1.8). As far as we are aware, this series has only been shown to converge for the case
of constant field strength [3]. Our example shows that it converges in a less simple case.
In this paper we will concentrate on the case of D3-branes, but the analysis for BIons
applies to D-branes of arbitrary dimension. The reason for this restriction is that we
will briefly comment on solutions more general than BIons, namely on dyons. These are
solutions of a D3-brane worldvolume theory carrying both electric and magnetic charges
which constitute the worldvolume realization of (p, q)-strings ending on a D3-brane.
They fill out orbits of the SL(2, Z) duality group of type IIB string theory which, on
the worldvolume of a D3-brane, becomes an electromagnetic duality group. In the last
section we will briefly comment on the possible role of SL(2, Z) in the non-commutative
description of D3-branes. Section 2 is mainly a review. Sections 3 and 4 contain original
results.
B = cos X,
(2.1)
where E and B are the electric and magnetic fields on the brane, tan = p/q, and X is the
only scalar field involved in the solution. The bound on the energy E which the solutions
of (2.1) saturate is [6]
q
2 + Z2 ,
(2.2)
E > Zel
mag
where the electric and magnetic charges above are
144
Z
Zel =
Z
d E X,
Zmag =
d 3 B X,
(2.3)
and is the D3-brane worldspace. Since both E and B are divergence free,
E = 0,
B = 0,
(2.4)
(the former as a consequence of the Gauss law and the latter because of the Bianchi
identity), these charges can be rewritten as surface integrals over the boundary of the brane
worldspace:
Z
Z
Zmag = dS XB.
(2.5)
Zel = dS XE,
This ensures that the charges are topological, in the sense that they only depend on the
boundary conditions imposed on the fields. It is this topological nature which guarantees
that the saturation of the bound automatically implies the equations of motion.
We see from (2.1) and (2.4) that X must be harmonic, that is, 2 X = 0. Given a
harmonic function X, the electric and magnetic fields are determined by (2.1). A dyon
is then associated with an isolated singularity of X.
We will concentrate for the rest of this section on the BIon. It corresponds to sin = 1
in (2.1), and therefore satisfies
E = X,
(2.6)
where we have chosen the minus sign for convenience. The most general SO(3)-symmetric
solution is then given (up to a gauge transformation) by
e
,
A = 0,
(2.7)
X = A0 =
4| |
where = ( a ), a = 1, 2, 3. It corresponds to a fundamental string ending orthogonally
on the brane at = 0, as depicted in Fig. 1(a). As mentioned above, (2.7) is a solution
of (classical) string theory to all orders in 0 [7], that is, when all corrections to the DBI
action involving higher derivatives of the fields are taken into account.
Since the BIon (2.7) saturates the BPS bound (2.6), its energy equals its charge.
Furthermore, the latter is easily calculated with the help of (2.5). The boundary of the
BIon worldspace consists of a two-sphere at | | and another at | | 0. The surface
integral over the former vanishes. Over the latter, it diverges. We can regularize it by
integrating over a small two-sphere S of radius . Since X is constant on this sphere
we are left with
Z
(2.8)
E = |Zel | = lim X() dS E = e lim X().
0
0
S
This shows that the energy of the BIon is precisely the energy of an infinite string of
constant tension [4,6]. To compare with the BIon in the presence of a B-field and with the
non-commutative BIon, it will be convenient for us to rewrite (2.8) as
E = |Zel | = zel lim I (),
0
(2.9)
(a)
145
(b)
Fig. 2. (a) Decomposition of the electric B-field, and (b) projection on the 1 X plane of the tilted
BIon (2.12).
where
zel = e2 ,
I () =
1
.
4
(2.10)
Let us now see how the BIon solution is modified when a constant electric background
B-field is turned on. We assume that the target-space ten-vector B0M no longer vanishes
(but we still impose the restriction that B has no magnetic components). In general,
any non-vanishing component of B along directions transverse to the brane can be
gauged away. In our case, however, we have to be careful, because we are looking for
a configuration in which one scalar field is excited. In other words, the worldspace of the
brane is not a flat three-plane extending along the directions 1, 2 and 3. Therefore, we need
to consider the component of B along the direction labelled by X (see Fig. 2(a)), to which
we will refer as BX . Without loss of generality, we can take the component of B along
the 123-space to point along the 1-direction. We will denote this component by B . The
remaining components of B0M can be gauged away and we will therefore set them to zero.
In the presence of the B-field, the BIon BPS equation which guarantees the preservation
of some fraction of supersymmetry must be modified: the field strength F = dA is replaced
by F = F + B ? . This can be easily understood, since the supersymmetry condition has
to be gauge-invariant under the two U (1) gauge symmetries (1.1) and (1.2). Thus, (2.6)
becomes
F0a = a X,
(2.11)
e
,
4| |
A = 0,
X=
1
B
e
1.
1 + BX 4| | 1 + BX
(2.12)
Note the appearance of a term linear in the worldspace coordinate 1 in the expression for
the scalar field. This is the term responsible for the tilt of the string (see Fig. 2(b)). Indeed,
although the spike coming out of the brane at | | = 0 still points along the X-axis and
the brane worldspace is still asymptotically flat, the latter no longer asymptotically extends
along the 1-direction.
146
The energy of the solution (2.12) is computed analogously as we did with (2.7). The
surface integral at infinity still vanishes (the term in the scalar field which is linear in 1
does not contribute because it changes sign under ). Thus, we obtain again (2.9),
but with zel now given by
zel =
e2
.
1 + BX
(2.13)
We would like to close this section with some clarifications. The first one is that it might
appear that (2.12) is not the most general solution of (2.11): we could have included terms
linear in the worldspace coordinate both in X and in A0 , with appropriate coefficients
such that (2.11) be satisfied. However, this apparently more general solution is related to
(2.12) by a gauge transformation of the type (1.2); they are therefore physically equivalent.
(Admittedly, such a transformation would shift the value of B, but this is allowed since B
is generic in our analysis.) In particular, by choosing the target-space one-form as
(2.14)
= B 1 + BX X d 0 ,
the solution (2.12) in the presence of a non-vanishing B-field is mapped to a configuration
with A0 = X (where X is still given by (2.12)) and vanishing B-field. This configuration
satisfies (2.6), and is precisely of the form considered in [9]. These considerations therefore
clarify the relation between the solution studied in [9] and the one we have presented: they
are related by a gauge symmetry of the theory; whether the constant electric field on the
brane is induced from the background or whether it arises from the worldvolume gauge
field itself is a matter of gauge choice. We have chosen to work in a gauge in which B 6= 0
since it is in this case that a non-commutative alternative description exists.
The second remark is that the proof in [7] that the BIon (2.7) is an exact solution to all
orders in 0 assumed the absence of a background B-field. Therefore, one might question
whether the result also holds for our BIon (2.12). The answer is affirmative, because, as we
have just explained, (2.12) is related by a gauge symmetry of the theory to the configuration
studied in [9] in which B = 0. As pointed out in [9], this latter configuration is indeed a
solution to all orders in 0 , because it satisfies (2.6), which was the only assumption in [7]
(as opposed to any assumption concerning the specific form of the solution, such as (2.7)).
147
on the brane). The system in which G takes this form is obtained as follows (see Fig. 3).
Let us take the first worldspace coordinate 1 along the direction of B, and the scalar field
Y to be orthogonal to it. In this system we have
B = b d 0 d 1 ,
(3.1)
where b is a positive constant, but G does not yet take the desired form. However, the
simple rescaling
p
p
1 = 1 b2 1
(3.2)
0 = 1 b2 0 ,
brings G into such a form:
2
= d02 + d12 + d22 + d32 + dY 2 .
ds(G)
(3.3)
In this coordinate system also takes a very simple form: its only non-vanishing
components are
01 = 10 = b.
(3.4)
After these preliminaries, we are ready to write down the non-commutative D3-brane
worldvolume action. We will take it to be the lowest order approximation in 0 , namely
Z
b 1 D Y
b F
b D Y
b .
(3.5)
S = d 1+3 14 F
2
All indices above are contracted with G.
One might think that a Hamiltonian analysis of the non-commutative action is required
in order to obtain its energy functional, which we certainly need to derive a BPS bound on
the energy. This seems difficult in view of the non-local nature of the action. (Note that this
non-locality is not only in space but also in time, since we will not impose the restriction
0a = 0.) The problem can be circumvented by noting that, although the presence of
breaks Lorentz invariance, the theory is still translation-invariant. Therefore we can obtain
the Lagrangian energy E as the conserved quantity under time translations. The result is
Fig. 3. The convenient coordinate system in the analysis of the non-commutative BIon, and the
projection to the 1 Y plane of the solution (4.12).
148
Z
E=
d 3
1 b2
2E
b)2 ,
b2 + 1 (D0 Y
b)2 + 1 (D Y
+ 12 B
2
2
(3.6)
b and B
b are the non-commutative electric and magnetic fields, that is,
where E
b0a ,
ba = F
E
ba = 1 abc F
bbc .
B
2
(3.7)
i
b DY
b + cos B
b DY
b ,
+ sin E
(3.8)
(3.9)
The electric and magnetic charges above are the natural non-commutative generalizations
of (2.3):
Z
Z
3 b
b
b DY
b.
Zmag = d 3 B
(3.10)
Zel = d E D Y ,
b = cos D Y
b
B
(3.11)
hold. 6 They are also the natural generalizations of their commutative counterparts (2.1).
As well as the BPS equations, the Gauss law and the Bianchi identity (2.4) should also
be promoted to their non-commutative versions
b = 0,
DE
b = 0.
DB
(3.12)
A number of comments are in order here. The new Bianchi identity follows immediately
b. However, it is not clear to us whether it constitutes a locally
from the definition of F
sufficient integrability condition (as it does in an ordinary gauge theory) which ensures
b can be written as the covariant derivative of a gauge potential. The Gauss law
that F
above is nothing else than one of the equations of motion. However, a rigorous proof that
it is a constraint in the non-commutative theory would require a careful analysis, which
is difficult again due to its non-locality. Nevertheless, there are two observations which
are worth noticing. First of all, at least in the case of purely magnetic , that is, when
0a = 0, the non-commutative theory becomes local in time and the Hamiltonian analysis
is straightforward [14]. In this case one can prove that the Gauss law (which contains
only first order time derivatives in its Lagrangian form) becomes a Hamiltonian constraint.
6 The case with cos = 1 in (3.11), which corresponds to a monopole, was studied in [11] in the context of a
U (2) gauge group. The non-commutative monopole equation was solved to first order in . The solution exhibits
a certain non-locality corresponding to the tilt of the D-string ending on the brane.
149
Second, both the Bianchi identity and the Gauss law are required in order to rewrite the
electric and magnetic charges as surface integrals:
Z
Z
b E,
b
b B.
b
Zmag = dS Y
(3.13)
Zel = dS Y
As in the ordinary case, this is what ensures that they have a topological nature, which
in turn guarantees that the saturation of the bound automatically implies the equations of
motion.
Now we are ready to finally write down the non-commutative BIon solution. It is simple
to check that the configuration
()
()
b= q ,
A = 0,
(3.14)
A 0 = Y
4||
solves both the non-commutative BPS equations (3.11) (with sin = 1 as before) and
the non-commutative Gauss law (3.12). In (3.14) we have = ( a ), and the superscript
() denotes that the above are the components of the gauge potential one-form in the ()
coordinate system, namely that A = A 0 d 0 . Its components in the -coordinate system
are, by virtue of (3.2),
p
( )
( )
()
A = 0.
(3.15)
A 0 = 1 b2 A 0 ,
()
Hereafter we will denote A 0 simply by A 0 until the moment in which we have to compare
with (2.12).
The solution (3.14) is the announced non-commutative BIon. Its charge is again easily
computed as we did in the previous section. The result is again (2.9) with
zel = q 2
(3.16)
150
and similarly for the scalar fields. As proved in [13], 7 these conditions are in general not
satisfied. This means that the evolution of the fields in -space determined by integrating
Eqs. (1.8) between some initial and final values 0 and 1 depends on the path followed
from 0 to 1 . The choice of path should be made according to some physical input. 8
In our case, we are interested in the -evolution of the BIon (3.14) starting from an
initial which is purely electric, and ending at = 0. This suggests that we should restrict
ourselves to the hyperplane ab = 0 in -space, and consider paths contained within this
hyperplane. This restriction can be physically further motivated as follows. Suppose that
we start with a D3-brane in the absence of any B-field. In this situation only the ordinary
description is available. Now we smoothly turn on the electric components of B. As soon
as B is different from zero, both the ordinary and the non-commutative descriptions are
available. It follows from its definition (1.5) that, in the latter description, will also be
purely electric. We can now imagine following the evolution of the fields as a function
of (or, equivalently, of B) as it increases. We see that in this physical situation we are
following a path which lies in the ab = 0 hyperplane.
The restriction ab = 0 simplifies the -evolution equations dramatically, because it
implies that the -product of any two time-independent functions f and g reduces to their
ordinary product, namely that f g = fg. With this simplification Eqs. (1.8) become
A 0
= A 0 b A 0 ,
(4.2)
b
1
1
A a
= A 0 b A a + A 0 a A b A b a A 0 ,
(4.3)
b
2
2
b
Y
b,
= A 0 b Y
(4.4)
b
where a 0a . Note that in the above equations b / b .
The unique solution of (4.3) with the initial condition that A a vanishes at some initial
value of is that it vanishes for all values of , regardless of the chosen path. Furthermore,
taking the derivative of (4.2) and using (4.2) itself, one can easily check that the crossed
derivatives of A 0 coincide. Finally, (4.2) and (4.4), together with the initial condition
b, show that A 0 and Y
b remain equal for all values of . We thus conclude that,
A 0 = Y
in the hyperplane ab = 0, the -evolution of the BIon (3.14) is path-independent, and that
to determine it we need only solve (4.2) with the initial condition
q
,
(4.5)
A 0 (, ) = =
0
4||
where = ( a ) and 0 = (b, 0, 0).
(4.2) constitutes a system of three quasi-linear partial differential equations which can be
solved by the method of characteristics: 9 instead of solving (4.2) directly, one introduces
a new three-vector t = (t a ) and solves
7 I thank Joan Simn for drawing my attention to this reference.
8 As explained in [13] there is no path dependence for the case of a U (1) gauge group if terms of O( F
b) are
neglected.
9 I am grateful to Emili Elizalde for help at this point.
b
A 0
b
b
b
=
,
=
,
=0
A
0
a
a
t a
t a
t a
with initial conditions that depend on a further three-vector s = (s a ), namely
q
, at t = 0.
A 0 =
= s,
= 0,
4|s|
151
(4.6)
(4.7)
A 0 A 0 b A 0 b A 0
0 A0 .
=
+
=
+
A
t a
b t a
b t a
a
a
(4.9)
It is also obvious from (4.7) that A 0 satisfies the required initial condition (4.5).
We see from (4.8) how to interpret t: it is simply the difference between the initial and
the final values of the non-commutativity parameter. Therefore, in our case we will be
interested in examining the values of the fields at
t = (b, 0, 0).
(4.10)
So we see that (4.8) provides the solution (namely the gauge potential and the embedding
of the brane in target-space) in parametric form.
In order to check that the solution (4.8) at = 0 is, as we claim, precisely the same as
(2.12), we have to express (4.8) in the same coordinate system as (2.12).
First, we have to undo the rescaling (3.2), because the closed string metric g takes the
Minkowski form in the -coordinates, but not in the -coordinates. We thus have, using
(3.2), (3.15), (4.8) and (4.10), that, at = 0:
1
b
q
+
s1,
1 =
2
1 b 4|s|
1 b2
p
q
,
A = 0.
A0 = 1 b 2
4|s|
Y=
q
,
4|s|
(4.12)
(4.13)
Note that we have dropped the hats on the fields, because the above are already their
values in the ordinary description. Note also that, although we did not write the superscript
152
( ) explicitly, A0 and A now denote the components of the gauge potential in the coordinate system.
We have drawn the solution (4.12) (which is still expressed in parametric form) in Fig. 3.
For |s| we have Y 0. Therefore the brane is asymptotically flat and extends along
the directions labeled by 1 , 2 and 3 , as shown in the figure. On the contrary, for |s| 0,
we see that
1 b2 1
.
(4.14)
Y
b
This means that the spike comes out of the region | | 0 in a direction at an angle with
the Y -axis (see Fig. 3 again), where
tan =
b
1 b2
(4.15)
We have labelled this direction by X, and the orthogonal direction in the 1 Y plane
by 1 . Recall that the ordinary BIon solution (2.12) is written in the static gauge and
in a coordinate system in which the spike points along the transverse scalar field. In our
case, this happens precisely in the X coordinate system. Therefore, the last step we have
to take is to rotate the solution (4.12) by an angle . Defining
1
1
cos
sin
=
(4.16)
X
sin cos
Y
we find that 1 = s 1 and that
b
1
q
s1.
X=
1 b2 4|s|
1 b2
Finally, we see from (3.1) and (4.16) that
p
BX = b2 .
B = b 1 b2,
(4.17)
(4.18)
Therefore, using (3.17), we arrive at the final form of our solution, expressed in the static
gauge:
A0 =
e
,
4| |
X=
1
B
e
1 .
1 + BX 4| | 1 + BX
(4.19)
5. Discussion
In this last section we wish to address a number of issues which were not fully discussed
in the text.
The first one concerns the interpretation of the scalar fields in the non-commutative
theory. In the ordinary description of a single D-brane, the scalars unambigously determine
the embedding of the brane in spacetime. This is the reason why many phenomena in field
theories acquire a clearer geometrical interpretation when such theories are realized as
worldvolume theories of branes. In the non-commutative description, this interpretation
153
is much less clear. The reason is that the scalar fields, even in the case of one single Dbrane, are no longer gauge-invariant but gauge-covariant quantities; namely they transform
b = i[ , X].
b This problem is related to the fact that all gauge-invariant quantities in
as X
non-commutative gauge theories seem to be non-local, obtained after integrating some
gauge-covariant quantity. Presumably, one can determine global properties of the brane
embedding from the non-commutative scalar fields, such as winding numbers, etc., by
integrating appropriate expressions, but not the local details of the embedding. The reason
why we did not have to resolve this problem in our discussion of the non-commutative
BIon is that we were only interested in identifying a solution of the non-commutative
theory which was exactly mapped to the ordinary BIon in the presence of a B-field. Since
the SeibergWitten map maps gauge orbits into gauge orbits and the ordinary scalar fields
are gauge-invariant, any scalar field configuration in the non-commutative theory which is
gauge-equivalent to (3.14) would have also been mapped to the ordinary BIon. We simply
b in its gauge-equivalence class.
chose the simplest representative of Y
The second point we wish to discuss here is whether it makes sense to consider a
non-commutativity matrix with non-vanishing electric components. 10 In the ordinary
description, it certainly makes sense to consider B0a 6= 0, which leads through (1.5) to
0a 6= 0. Moreover, even if in one coordinate system B has no electric components, we
can always choose to describe the physics in a frame in which B does have electric
components. The only question is whether the equivalence between the ordinary and the
non-commutative descriptions still holds in this frame. It seems that our result can be
regarded as evidence in favour of the affirmative answer to this question.
We would like to close this paper with a little digression on the role of S-duality in
the non-commutative theory, and more generally of the full SL(2, Z) duality group of
type IIB string theory. Consider the BIon solution in the ordinary description of D3branes and in the absence of any B-field. As we have explained, this is the worldvolume
realization of the spacetime configuration in which a fundamental string ends on the D3brane. The S-duality of string theory maps this configuration into one in which a D-string
ends on the D3-brane. Thus, from the worldvolume point of view of the latter, S-duality
is an inherited symmetry which corresponds to electromagnetic duality. It maps the BIon
into the monopole. These two objects are therefore equivalent, in the sense that they are
related by a symmetry of the theory. One might think that this is also the case in the noncommutative worldvolume description, perhaps with a further exchange of the electric and
the magnetic components of . However, this is not true. The reason is that a BIon in
the non-commutative theory corresponds, as we have seen, to a fundamental string ending
on the D3-brane in the presence of a B-field. S-duality maps this configuration into a Dstring ending on the D3-brane in the presence now of a RamondRamond C-field. Clearly,
this does not correspond to a monopole in the non-commutative theory, which should
instead correspond to a D-string ending on the D3 in the presence of a B-field. These
considerations raise the following interesting question. There exists an SL(2, Z)-covariant
worldvolume action for D3-branes coupled to a supergravity background [15], which can
10 I would like to thank Michael B. Green and Paul K. Townsend for a conversation on this point.
154
consist, in particular, of a flat background with constant B- and C-fields. The case of
vanishing C-field, for which we know that a non-commutative description is possible, is
mapped to the generic one by SL(2, Z). Since SL(2, Z) is a symmetry of IIB string theory,
should there not exist an SL(2, Z)-covariant non-commutative description of D3-branes in
the presence of both constant B- and C-fields?
Note added
While this paper was being written, I learned about [16], which has some overlap with
Section 3.
Acknowledgments
I am grateful to Joaquim Gomis for the suggestion which motivated this work, and
to both him and Joan Simn for illuminating discussions and useful comments on the
manuscript. I am also grateful to Selena Ng for her helpful comments on a previous
version of this paper. Finally, I would like to thank Emili Elizalde, Josep I. Latorre, Josep
M. Pons, Toni Mateos and Jordi Molins for discussions. This work has been supported
by a fellowship from the Comissionat per a Universitats i Recerca de la Generalitat de
Catalunya.
References
[1] A. Connes, M. Douglas, A. Schwarz, Non-commutative geometry and Matrix theory: compactification on tori, JHEP 9802 (1998) 003, hep-th/9711162.
[2] M. Douglas, C. Hull, D-branes and the non-commutative torus, JHEP 9802 (1998) 008, hepth/9711165.
[3] N. Seiberg, E. Witten, String theory and non-commutative geometry, JHEP 9909 (1999) 032,
hep-th/9908142.
[4] C.G. Callan, J.M. Maldacena, Brane dynamics from the BornInfeld action, Nucl. Phys. B 513
(1998) 198212, hep-th/9708147.
[5] G.W. Gibbons, BornInfeld particles and Dirichlet p-branes, Nucl. Phys. B 514 (1998) 603
639, hep-th/9709027.
[6] J.P. Gauntlett, J. Gomis, P.K. Townsend, BPS bounds for worldvolume branes, JHEP 9801
(1998) 033, hep-th/9711205.
[7] L. Thorlacius, BornInfeld string as a boundary conformal field theory, Phys. Rev. Lett. 80
(1998) 15881590, hep-th/9710181.
[8] J. Polchinski, Dirichlet-branes and RamondRamond charges, Phys. Rev. Lett. 75 (1995) 4724,
hep-th/9510017.
[9] K. Hashimoto, BornInfeld dynamics in uniform electric field, JHEP 9907 (1999) 016, hepth/9905162.
[10] A. Hashimoto, K. Hashimoto, Monopoles and dyons in non-commutative geometry, JHEP 9911
(1999) 005, hep-th/9909202.
[11] K. Hashimoto, H. Hata, S. Moriyama, Brane configuration from monopole solution in noncommutative super YangMills theory, JHEP 9912 (1999) 021, hep-th/9910196.
155
[12] S. Lee, A. Peet, L. Thorlacius, Brane-waves and strings, Nucl. Phys. B 514 (1998) 161176,
hep-th/9710097.
[13] T. Asakawa, I. Kishimoto, Comments on gauge equivalence in non-commutative geometry,
JHEP 9911 (1999) 024, hep-th/9909139.
[14] O. Dayi, BRST-BFV analysis of equivalence between non-commutative and ordinary gauge
theories, hep-th/0001218.
[15] M. Cederwall, A. Westerberg, World-volume fields SL(2, Z) and duality: the type IIB 3-brane,
JHEP 9802 (1998) 004, hep-th/9710007.
[16] H. Hata, S. Moriyama, String junction from non-commutative super YangMills theory, hepth/0001135.
Abstract
Freed, Harvey, Minasian and Moore (FHMM) have proposed a mechanism to cancel the
gravitational anomaly of the M-theory fivebrane coming from diffeomorphisms acting on the normal
bundle. This procedure is based on a modification of the conventional M-theory ChernSimons
term. We apply the FHMM mechanism in the ten-dimensional type IIA theory. We then analyze
the relation to the anomaly cancellation mechanism for the type IIA fivebrane proposed by Witten.
2000 Elsevier Science B.V. All rights reserved.
PACS: 11.25.Sq; 04.65.+e
1. Introduction
In our journey towards M-theory there appears an object that for a long time has been
considered to be rather mysterious, the M-theory fivebrane or M5-brane. The M5-brane
was discovered for the first time as a solution to the classical equations of motion of
eleven-dimensional supergravity [1]. The fivebrane mystery originated in part because
people did not know how to write down an action for this six-dimensional theory of (2, 0)
antisymmetric tensor-multiplets. This problem has been solved in [2,3] and [4] for the
case of one single fivebrane. However, the action for N coincident M5-branes is still
not known because apparently it is not possible to write down the action for a theory
of non-abelian tensor-multiplets. For a recent discussion on the subject see [5]. Progress
towards understanding this system was made by Harvey, Minasian and Moore [6] who
computed the gravitational anomalies of the non-abelian tensor-multiplet theory through
anomaly cancellation of a system of N coincident M5-branes. The calculation of [6] is
based on the anomaly cancellation mechanism for a single M5-brane proposed by Freed,
1 beckerk@theory.caltech.edu
2 mbecker@theory.caltech.edu
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 5 3 - X
157
Harvey, Minasian and Moore (FHMM) [7]. Some earlier attempts to understand anomaly
cancellation for the M5-brane were made in [811].
The perturbative anomaly cancellation for a single fivebrane in type IIA theory was
found by Witten [12] who suggested an anomaly cancellation mechanism based on
a fivebrane worldvolume counterterm. Since the type IIA theory can be obtained by
dimensional reduction of M-theory we expect that both anomaly cancellation mechanisms
can be related by dimensional reduction. It is the purpose of this note to understand the
relation between both mechanisms. In Section 2 we review some basic ideas which one
has to keep in mind for the next sections. In Section 3 we will start with the technically
simpler case of a chiral string in five dimensions. Here the normal bundle is described by an
SO(3) gauge theory. This string originates in compactifications of M-theory on a Calabi
Yau manifold when the M5-brane wraps a non-trivial four-cycle of the CalabiYau. It can
potentially have gauge and gravitational anomalies. We will show that after dimensional
reduction to four dimensions the FHMM ChernSimons term gives a string worldvolume
counterterm that cancels the four-dimensional anomaly. This exactly coincides with the
lower-dimensional version of the anomaly cancellation mechanism proposed in [12]. In
section 4 we consider the M5-brane, where the normal bundle is described by an SO(5)
gauge theory. The basic idea is the same as in the simpler string example. After dimensional
reduction to ten dimensions we will obtain the fivebrane worldvolume counterterm of [12].
Our conclusions are presented in Section 5 and some useful formulas are collected in
Appendix A. In this paper we will study perturbative anomalies. The calculation of global
anomalies in M-theory still remains an open question.
158
a two-form potential with anti-self-dual field strength which is a singlet under SO(5). The
anomaly can be calculated with the descent formalism. It is expressed in terms of an eight(0)
(0)
(1)
form, I8 = dI7 and the gauge transformation is given by I7 = dI6 . The anomaly is
then given by a six-form on the fivebrane worldvolume whose explicit form appears in [12]
Z
(2.1)
2 I6zm (1).
W6
The second source of anomalies comes from ChernSimons couplings of the bulk theory
Z
G4 I7b .
(2.2)
M11
Here G4 is the four-form field strength and I7b is a gravitational ChernSimons seven-form
that can be expressed in terms of the eleven-dimensional Riemann tensor. This interaction
was discovered in [15] by a one-loop calculation in type IIA string theory and in [16]
from anomaly cancellation of the tangent bundle. In the presence of fivebranes the gauge
invariance of this coupling is broken. Under infinitesimal diffeomorphisms x I x I + v I
where is an infinitesimal parameter and v is a vector field, I7b transforms as I7b
I7b + dI6b(1) , where I6b(1) is a six-form. Taking the gauge variation of (2.2) and integrating
by parts we obtain
Z
b(1)
dG4 I6 .
(2.3)
S =
M11
In the presence of fivebranes the four-form is no longer closed but roughly speaking obeys
the Bianchi identity
dG4 = 25 .
(2.4)
Here 5 is a five-form which integrates to one in the directions transverse to the fivebrane
and has a delta function support on the fivebrane. The total gravitational anomaly was
computed in [12] with some standard formulas appearing in [17]
Z
Z
p2 (N) (1)
zm
b (1)
= 2
.
(2.5)
I6 + I6
2
24
W6
W6
Here p2 (N) is the second Pontrjagin class of the normal bundle. This is in agreement with
the result found in [16] who checked that the anomaly in diffeomorphisms of the tangent
bundle cancel.
A new mechanism was needed to cancel the remaining anomaly. Until this point there
existed two mechanisms in the literature that did not seem to have an obvious relationship.
First, Witten [12] introduced a counterterm that lives on the fivebrane worldvolume and
that cancels the anomaly in diffeomorphisms of the normal bundle for the ten-dimensional
type IIA theory. This mechanism did not seem to have an obvious generalization to
M5-branes [12]. On the other hand there exists the mechanism proposed in [7] that is
159
(3.1)
Here G2 is the two-form field strength which couples to the string 3 . However, the
expression (3.1) is not well defined when one is dealing with interactions that are nonlinear in the field strength. So for example, order (0)2 -terms may appear when one is
checking the supersymmetry of the Lagrangian. The appearance of (0)2 -contributions in
the Lagrangian was discussed some time ago in [18] for compactifications of M-theory
on a manifold with boundary. In fact, there are some striking similarities between both
theories. As in the formula above, the Bianchi identity of [18] contains a delta function
whose support is at the boundary of space-time. According to [18] the appearance of (0)2 terms is a symptom of attempting to treat in classical supergravity what really should be
3 Here and in the following we absorb a factor 1/2 in field strengths and potentials.
160
treated in quantum M-theory. In other words, some degrees of freedom are missing in the
classical picture.
In the case at hand the authors of [7] made a clever proposal in order to effectively
include the missing degrees of freedom. FHMM proposed to smooth out the fivedimensional Bianchi identity by making the following ansatz
e2
(3.2)
dG2 = d .
2
The right-hand side of this identity is the so-called Thom class of the normal bundle [19].
= (r) is a smooth function of the radial direction away from the string. It takes values
(r) = 1 for sufficiently small r and (r) = 0 for sufficiently large r. The delta function
of (3.1) can be obtained as a limiting function of the bump form d. The global angular
form e2 has integral two over the fiber and is closed
de2 = 0,
(3.3)
because the Euler class of an odd bundle is zero. The explicit form of e2 can be found in
the Appendix of [7] 4 :
1
abc D y a D y b y c F ab y c .
(3.4)
e2 =
4
Here y a = y a /r for a = 1, 2, 3 are the coordinates on the fiber. Using the connection
ab = ba one can define a covariant derivative
D y a = dy a ab y b ,
(3.5)
and a two-form
F ab = d ab ac cb .
(3.6)
and y a = ab y b .
(3.7)
An equivalent formula for e2 which will turn out to be useful for explicit calculations is
1
abc dy a dy b y c d ab y c ,
(3.8)
e2 =
4
which is identical to the expression for e2 appearing in [20] that we have collected in
Appendix A.
Since the angular form is closed and gauge invariant one can apply the descent formalism
(0)
e2 = de1
(0)
(1)
and e1 = de0 .
(3.9)
One can now use the above forms to find the solution to the Bianchi identity and the
five-dimensional ChernSimons term. The solution to the Bianchi identity (3.2) which is
non-singular on the string is given by
1
(0)
(3.10)
G2 = dC1 d e1 .
2
4 This corrects a factor 2 in [7]. This factor has to be corrected in the explicit expressions of all angular forms
appearing in [7] and [6].
161
Performing a gauge transformation and demanding that G2 should be invariant one obtains
that C1 transforms under gauge transformations of the normal bundle
1
(1)
(3.11)
C1 = d e0 .
2
In order to have compatibility between the equations of motion and the Bianchi identity
(3.2) one has to correct the space-time action. FHMM proposed that one way of doing this
is by introducing a modified ChernSimons term
Z
(C1 1 ) d(C1 1 ) d(C1 1 ).
(3.12)
SCS = 12D lim
0
M5 D (W2 )
Here the integration is over the five-dimensional space-time without a tubular region of
(0)
radius around the string, 1 = e1 /2 and D is a constant related to the central charge
of the system whose precise definition can be found in [21] and [7].
The ChernSimons interaction is not gauge invariant because of the gauge transformation
1
(3.13)
(C1 1 ) = d e0(1) .
2
The variation of the ChernSimons interaction is then given by a boundary term
2
Z
1
e0(1) G2 e2 ,
(3.14)
SCS = 6D lim
0
2
S (W2 )
after applying Stokes theorem. Here one has to use the fact that the boundary of the fivedimensional space without the tubular region D (W2 ) is an S 2 -sphere bundle over the
worldvolume of the string W2 . This is denoted as S (W2 ). Since G2 and are smooth
functions near the brane one obtains in the limit of small
Z
Z
3
e2 e2 e0(1) = 3D p1(1) (N).
(3.15)
SCS = D
2
S (W2 )
W2
The relevant formula used to carry out the remaining integration is collected in the
Appendix. The above anomaly precisely cancels the contributions from chiral zero-modes
plus anomaly inflow coming from bulk interactions.
3.2. Dimensional reduction to four dimensions
This is the lower-dimensional counterpart of the fivebrane in ten dimensions that was
considered by Witten [12]. The string in four dimensions is described by a one-form field
strength and a zero-form potential. The FHMM Bianchi identity for this case is
1
dH1 = d(e1 ).
2
e1 is the SO(2) invariant global angular form which satisfies
de1 = 2(F ),
(3.16)
(3.17)
162
where (F ) = ab F ab /(4) is the Euler class of the SO(2) bundle. From (3.16) we see
that the Bianchi identity on the brane is given by
(3.18)
dH1 W = (F ),
2
which is equivalent to the Bianchi identity used in [12]. 5 The explicit form of e1 is
1
1
1 ab
a b
a b
y = ab dy y
e1 = ab (D y)
.
(3.19)
2
In order to make contact with the formalism of [12] we can locally write
e1 /2 = d0 + 1 ,
(3.20)
0 = .
(3.23)
Using the forms 0 and 1 it is easy to find the non-singular solution of the Bianchi
identity. It is given by
H1 = dB0 + 1 d0 .
(3.24)
Note that a term of the form e1 is not allowed since e1 is singular on the brane. To see this
note that the integral of e1 over the fiber is non-vanishing even if the volume of the fiber
tends to zero. Since H1 should be gauge invariant under the SO(2) transformations of the
normal bundle we see that B0 has a transformation
B0 = .
(3.25)
Let us now work out the relation between the 5d and 4d ChernSimons terms. For that
purpose we would like to consider an SO(3) bundle N of the form N = N 0 O, where N 0
is an SO(2) bundle and O is a trivial bundle. We will then assume that the 3-component of
all connections is equal to zero, 3a = 0 for a = 1, 2. The expression for e2 then becomes
e2 =
1
1
(F )y 3 +
ab ab dy 3 ,
2
4
(3.26)
5 In order to relate the present formulation with the one of [12] one has to change the sign of . This explains
the minus sign in front of (F ) in formula (3.18).
163
where is the volume element of the SO(3) fiber. Next, notice that the five-dimensional
anomaly followed from the Bott and Cattaneo formula
e23 = 2p1 (N),
(3.27)
where denotes the integration over the fiber. Inserting (3.26) into this expression one
obtains after integration over the fiber p1 (N 0 ) = (F )2 , which is the correct relation
between the first Pontrjagin class and the Euler class of an even bundle. The formula that
FHMM actually used to compute the five-dimensional anomaly is
(3.28)
e2 e2 e0(1) = 2p1(1) (N).
In four dimensions the first Pontrjagin class is p1 (N 0 ) = (F )2 = d(1 (F )). After a
gauge transformation the right-hand side of the previous equation becomes (1 (F )) =
d( ). This determines p1(1) (N 0 ) = (F ). This is an important piece of information
because this together with (3.26) and (3.30) determines the value of e0(1)
3
(1)
e0 = y3 .
(3.29)
2
This choice guarantees that the integration over the fiber correctly reproduces the value of
(1)
p1 (N 0 )
(3.30)
e2 e2 e0(1) = 2p1(1) (N 0 ) = 2 (F ).
(0)
A is a boundary term
Using Stokes theorem we see that SCS
Z
1
1
B0 y3 G2 e2 G2 e2 .
9D lim
0
2
2
(3.34)
S (W2 )
Since the fields B0 and G2 are smooth near the string we obtain after carrying out the limit
and the integration over the fiber
Z
Z
Z
9
D lim
B0 e2 e2 y3 = 3D B0 (F ) = 3D H1 1 .
(3.35)
0
4
S (W2 )
W2
W2
164
To carry out the integration over the fiber we have used (3.26) and formula (6.6) of
the Appendix. The result (3.35) is precisely the lower-dimensional analogue of the
worldvolume counterterm proposed by Witten for the case of the type IIA fivebrane [12].
In addition there are four-dimensional space-time interactions which are invariant under
gauge transformations of the normal bundle. We will not determine invariant terms by a
direct calculation because even in 5d the complete set of invariant terms is not known.
This would require a microscopic derivation of the interaction presented by FHMM. This
derivation is still missing. However, in order to have a consistent 4d theory we expect an
additional term in space-time of the form
Z
1
C1 H1 e1 G2 .
(3.36)
2
M4
This can be easily seen because once the Bianchi identities are modified H1 is no longer
closed. Replacing H1 by H1 e1 /2 guarantees that the Bianchi identities and the
equations of motion of this theory are compatible.
1
a1 a5 (D y)
a2 (D y)
a3 (D y)
a4 y a5
a1 (D y)
2
32
2F a1a2 (D y)
a3 (D y)
a4 y a5 + F a1 a2 F a3 a4 y a5 ,
(4.2)
where ai , i = 1, . . . , 5, labels the fiber coordinates. One can again apply the descent
formalism and introduce the notations
(0)
e4 = de3
(0)
(1)
and e3 = de2 .
(4.3)
(4.4)
165
where 3 = e3(0) /2. This term is not invariant under diffeomorphisms. Its variation is
obtained by using the anomalous gauge transformation of C3 which determines
(1)
(4.5)
(C3 3 ) = d e2 /2 .
The result for the gauge transformation of the action is then
Z
Z
(1)
(1)
e4 e4 e2 =
p2 (N).
SCS =
24
12
S (W6 )
(4.6)
W6
The last identity can be obtained by using the corresponding version of the formulas by
Bott and Catteneo [20]. We have collected the relevant formulas in Appendix A.
4.2. Dimensional reduction to ten dimensions
The anomaly cancellation for the fivebrane in type IIA theory has been verified in [12].
In ten dimensions the fivebrane is described by a three-form field strength H3 with a twoform potential B2 . The modified Bianchi identity is
1
dH3 = d(e3 ).
2
The angular form e3 is given by
1
1 ab
1
a
b
c d
c d
(D y)
(D y)
y F (D y)
y ,
e3 = 2 abcd (D y)
3
2
2
(4.7)
(4.8)
where a = 1, . . . , 4 labels the SO(4) fiber coordinates. Notice that the sign between both
terms is different than in the expression given in [6]. This can be written in the form
e3
= d2 + 3 ,
(4.9)
2
where 2 = 2 (yi , ) is a function of the fiber and brane coordinates. It contains besides
other terms the volume form of S 4 . Its explicit form is not needed in the following. 3 is
the ChernSimons three-form
1
2
abcd ab dcd ab cx xd ,
(4.10)
3 =
3
32 2
which is related to the Euler class by
1
abcd F ab F cd .
32 2
Therefore the global angular form e3 is related to the Euler class
(F ) = d3 =
(4.11)
de3 = 2(F ),
(4.12)
(4.13)
166
(4.15)
In the same way as for the chiral string in 4d there is only one solution of the Bianchi
identity which is non-singular on the fivebrane
H3 = dB2 + 3 d 2 .
(4.16)
(4.17)
In order to relate the 11d and 10d theories we will assume that the SO(5) bundle N is of
the form N = N 0 + O, where N 0 is an SO(4) bundle and O is trivial. The connections
involving the five-component are then vanishing. Recall that the second Pontrjagin class
and the Euler class of the SO(4) bundle are related as p2 (N 0 ) = (F )2 , so that we obtain
p2 (1) (N 0 ) = . In order to satisfy the Bott and Cattaneo formula
(1)
(4.18)
e4 e4 e2 = 2p2 (1)(N 0 ) = 2 (F ),
(1)
total derivatives for e3 have to be chosen in such a way that e2 (1) becomes
(1)
e2 = 45 y5 .
(4.19)
To carry out the integration over the fiber we have used the formula (6.11) of the Appendix.
The SO(4) gauge transformation of the potential appearing in the eleven-dimensional
ChernSimons term is then
1
45
(4.20)
(C3 3 ) = d(e2(1)) = d( y5 ).
2
2
The anomalous term of the eleven-dimensional ChernSimons interaction can be expressed
in terms of the ten-dimensional potential B2 as
45
(4.21)
C3 3 = d(B2 y5 ) + I,
2
where I is an invariant under SO(4) gauge transformations. Because of the appearance
of the total derivative in the previous expression we are able to write the anomalous
contribution to the ChernSimons interaction as a boundary term, exactly as we had done
in the lower-dimensional case
Z
15
1
1
A
lim
=
y5 B2 G4 e4 G4 e4 .
(4.22)
SCS
2 0
2
2
S (W6 )
6 There is again a change of sign in .
167
Since B2 and G4 are smooth functions near the fivebrane the only non-vanishing
contribution to the above integral is
Z
15
lim
B2 e4 e4 y5 .
(4.23)
8 0
S (W6 )
Using formula (6.11) of the Appendix we can carry out the integration over the fiber
Z
Z
A
=
B2 (F ) =
H 3 3 .
(4.24)
SCS
12
12
W6
W6
This is precisely the counterterm found by Witten [12] that cancels the anomaly from SO(4)
transformations of the normal bundle of a type IIA fivebrane.
Of course, as in the lower-dimensional example, in order to have compatibility between
the equations of motion and the Bianchi identity (4.7) one should have a modified spacetime ChernSimons term of the form
Z
1
C3 H3 e3 G4 .
(4.25)
2
M10
However, this does not contribute to the anomaly because it is gauge invariant. It is very
satisfying to see that in ten dimensions this anomalous contribution to the space-time
interaction can indeed be expressed as a worldvolume counterterm as proposed in [12].
5. Conclusion
In this paper we have considered a string and a fivebrane embedded in five and
eleven dimensions respectively. These theories are not invariant under diffeomorphisms
of the normal bundle. This results in an anomaly that can be expressed in terms of the
corresponding Pontrjagin class of the normal bundle. Until now there have been two
anomaly cancellation mechanisms in the literature whose relation had not been worked
out until now. The mechanism proposed by Freed, Harvey, Minasian and Moore [7] is
formulated for theories with an odd fiber dimension while the mechanism proposed by
Witten [12] is useful for theories with an even fiber dimension. In this paper we saw that
both mechanism are actually not different. We have shown that after dimensional reduction
the FHMM anomaly cancellation mechanism [7] becomes equivalent to the one proposed
in [12]. This is very satisfactory and provides further understanding of both anomaly
cancellation mechanisms.
Even thought a microscopic derivation of the FHMM ChernSimons term is still missing
we believe that this interaction provides a way of effectively dealing with the presence
of N coincident M5-branes. So for example, the proposed interaction explains the N 3
contributions to the black hole entropy found in [21] from a detailed reduction of the
fivebrane tensor-multiplet. The N 3 growth of the entropy of a system of multiple M5branes was first discovered in [22].
168
Acknowledgements
We are grateful to Jeff Harvey, Petr Horava, Greg Moore, John Schwarz and Edward
Witten for useful discussions. This work was supported by the US Department of Energy
under grant DE-FG03-92-ER40701.
Appendix A
In this appendix we would like to collect some formulas that are useful when computing
the integrals involved in the calculation of the anomalies presented in this paper. We would
like to begin with integrals involving e2 . First e2 can be rewritten in the form
+ d( c yc )
2
where is the volume element
e2 =
(6.1)
(6.2)
(6.3)
(6.4)
The explicit evaluation of the Bott and Cattaneo formula goes along the same lines [20]
2
3
(2)3 e23 = 3 d a ya + d a ya
(6.5)
= 3 d a d b ya yb 3 d a b c ya dyb dyc .
169
e42 = 0,
e43 = 2p2 .
(6.8)
(6.9)
8 2
ab .
(6.10)
15
This equation implies that after imposing the condition that the 5-component of the
connection vanishes we have
2
(6.11)
e4 e4 y 5 = (F ),
45
as a simple evaluation of the involved integrals implies.
(ya yb ) =
References
[1] R. Gven, Black p-brane solutions of D = 11 supergravity theory, Phys. Lett. B 276 (1992) 49.
[2] P. Pasti, D. Sorokin, M. Tonin, Covariant action for a D = 11 five-brane with the chiral field,
Phys. Lett. B 398 (1997) 41; hep-th/9701037.
[3] I. Bandos, K. Lechner, A. Nurmagambetov, P. Pasti, D. Sorokin, M. Tonin, Covariant action for
the superfive-brane of M-theory, Phys. Rev. Lett. 78 (1997) 4332; hep-th/9701149.
[4] M. Aganagic, J. Park, C. Popescu, J. Schwarz, Worldvolume action for the M-theory fivebrane,
Nucl. Phys. B 496 (1997) 191; hep-th/9701166.
[5] X. Bekaert, M. Henneaux, A. Sevrin, Deformations odd chiral two-forms in six dimensions,
hep-th/9909094.
[6] J.A. Harvey, R. Minasian, G. Moore, Nonabelian tensor multiplet anomalies, JHEP 9809 (1998)
004; hep-th/9808060.
[7] D. Freed, J.A. Harvey, R. Minasian, G. Moore, Gravitational anomaly cancellation for M-theory
fivebranes, Adv. Theor. Math. Phys. 2 (1998) 601; hep-th/9803205.
[8] S.P. de Alwis, Coupling of branes and normalization of effective actions in string/M theory,
Phys. Rev. D 56 (1997) 7963; hep-th/9705139.
[9] L. Bonora, C.S. Chu, M. Rinaldi, Perturbative anomalies of the M5-brane, hep-th/9710063.
[10] L. Bonora, C.S. Chu, M. Rinaldi, Anomalies and locality in field theories and M-theory, hepth/9712205.
170
[11] M. Henningson, Global anomalies in M-theory, Nucl. Phys. B 515 (1998) 233; hep-th/9710126.
[12] E. Witten, Five-brane effective action in M-theory, J. Geom. Phys. 22 (1997) 103; hepth/9610234.
[13] E. Witten, Five-branes and M-theory on an orbifold, hep-th/9512219.
[14] E. Witten, Flux quantization in M-theory, J. Geom. Phys. 22 (1997) 1; hep-th/9609122.
[15] C. Vafa, E. Witten, A one-loop test of string duality, Nucl. Phys. B 447 (1995) 261; hepth/9505053.
[16] M.J. Duff, J.T. Liu, R. Minasian, Eleven-dimensional origin of string/string duality: A one loop
test, Nucl. Phys. B 452 (1995) 261; hep-th/9509084.
[17] L. Alvarez-Gaum, E. Witten, Gravitational anomalies, Nucl. Phys. B 234 (1983) 269.
[18] P. Horava, E. Witten, Eleven-dimensional supergravity on a manifold with boundary, Nucl.
Phys. B 475 (1996) 94.
[19] R. Bott, L.W. Tu, Differential Form in Algebraic Topology, Springer-Verlag, New York, 1982.
[20] R. Bott, A.S. Cattaneo, Integral invariants of 3-manifolds, dg-ga/9710001.
[21] J. Maldacena, A. Strominger, E. Witten, Black hole entropy in M-theory, hep-th/9711053.
[22] I.R. Klebanov, A. Tseytlin, Entropy of near extremal P-branes, Nucl. Phys. B 475 (1996) 164;
hep-th/9604089.
[23] T. Banks, W. Fischler, S.H. Shenker, L. Susskind, M-theory as a matrix model: A conjecture,
Phys. Rev. D 55 (1997) 5112.
[24] N. Hitchin, Lectures on special Lagrangian submanifolds, math.dg/9907034.
[25] D.S. Freed, E. Witten, Anomalies in string theory with D-branes, hep-th/9907189.
Abstract
The non-commutative geometry is revisited from the perspective of a generalized D-p-brane. In
particular, we analyze the open bosonic string world-sheet description and show that an effective noncommutative description on a D-p-brane corresponds to a re-normalized world-volume. The worldvolume correlators are analyzed to make a note on the non-commutative geometry. It is argued that
the U (1) gauge symmetry can be viewed in a homomorphic image space for the non-commutative
space of coordinates with an SO(p) symmetry. In the large B-field limit, the non-commutativity
reduces to that among the zero modes and the world-volume description is precisely in agreement
with the holographic principle. 2000 Elsevier Science B.V. All rights reserved.
PACS: 11.25.-w; 11.15.-q
Keywords: Open string theory; D-branes; Non-commutative geometry
1. Introduction
Dirichlet (D) p-branes are non-perturbative [1] topological defects seen in string theory.
They have been studied extensively to explore the non-perturbative domain [24] using
mathematical tools familiar in string theory. Among various aspects of a generalized D-pbrane, the non-commutative geometry [5,6] on its world-volume has significantly played a
role to enhance our understanding of the gauge theory. In a recent analysis [7], a precise
relation between the non-commutative and commutative gauge symmetries on a D-p-brane
world-volume has been established.
The idea of non-commutativity is quite natural in the theory of open strings. For
instance, its world-sheet boundary is an one-dimensional ordered space which is evident
from its vertex operators insertion. In fact, under a rotation of the world-sheet, the vertex
operators retain the cyclic symmetry and implies a non-commutative geometry there. In the
supriya@fy.chalmers.se
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 2 2 - X
172
context of open string field theory, the non-commutative geometry has also been addressed
[8]. Since the dynamical aspect of a D-p-brane has its origin in open string physics,
interestingly, the longitudinal collective coordinates for a generalized D-p-brane also
become non-commutative. This implies that the world-volume geometry can be analyzed
perturbatively as a deformation to the commutative one. The (convergent) series expansion
has been shown to be identical with the Moyal product following a boundary conformal
field theory technique [9]. In the recent past, non-commutative geometry in the realm
of a D-p-brane have been explored in great details by various authors [1021]. Further
investigation in this direction is believed to provide a conceptual understanding of the Dbrane geometry and may enlighten a formulation for the quantum gravity.
In this article, we focus on the non-commutative description on a world-volume of
a generalized D-p-brane with an emphasis on the zero modes. We consider the renormalized world-volume to bring a note on the effective non-commutativity as seen in
a physical process, such as scattering phenomena. We study the world-volume correlators
for the renormalized D-p-brane to comment on the non-commutative and commutative
symmetries. By diagonalizing the world-volume into two-dimensional block diagonal
forms, we discuss the possible geometry on a rotated D-p-brane. Finally, the large B limit
is analyzed and the holographic correspondence is argued to relate the non-commutative
and commutative geometries. In this limit, the world-volume becomes non-local due to
the zero modes and the ground state wave function for the D-particle can be seen to be
degenerate for the Landau level.
We plan to present the article as follows. In Section 2.1, we briefly discuss the D-p-brane
dynamics in the open string channel to obtain the boundary conditions. In Section 2.2, we
present the non-commutative aspect of a D-p-brane. The re-normalization of the worldvolume is performed in Section 2.3. Subsequently in Section 3.1, we obtain the worldvolume correlators for a generalized D-p-brane. Section 3.2, deals with the world-volume
rotational symmetry SO(p) for a D-p-brane and the role of zero modes are discussed. The
large B-field limit is discussed in Section 3.3 and we conclude the article with discussions
in Section 4.
where the second integral denotes the U (1) gauge (field: A ) interaction at the
boundary, .
173
Now consider an arbitrary D-p-brane 1 with p-spatial coordinates describing its worldvolume in the open string theory (1). The background fields can be seen to induce metric h
and the antisymmetric B-field on the world-volume of the D-p-brane. In a gauge, h0i = 1
and B0i = 0 for i = 1, 2, . . . , p, they can be given by
b i X j X
hij = G
b i X j X .
and Bij = B
(2)
The gauge choice allows an independent treatment of the time component (X0 ) from the
remaining spatial ones. On the boundary, it satisfies the Neumann condition n X0 = 0,
where n denotes the normal coordinate. The B-field is a (p p)-matrix of rank r = p/2
and is magnetic due to the gauge choice. In fact, the B-field cannot be gauged away in
presence of a D-p-brane and the gauge invariance can be maintained in combination with
the U (1) gauge field in the theory. Since a D-p-brane is defined with Dirichlet boundary
conditions, X = 0, along the transverse directions ( = (p + 1), . . . , 25), the effective
dynamics (1) turns out to be the one on its world-volume. Then the induce world-volume
bij and Bij = B
bij ) of
fields (2) can be identified with the longitudinal components (hij = G
the background fields. Also, the gauge field on the world-volume can be identified with the
longitudinal components of the U (1) field.
The general nature of the world-volume fields is very difficult to treat and one needs to
adopt an approximation. We consider geodesic expansions for the fields (hij , Bij and Ai )
around their zero modes and they can be given by
Ai (X) = 12 Fij Xj + O(F ),
bij Xk + O( 2 G)
b
bij (X) = ij + k G
G
and
bij Xk + O( 2 B).
b
bij (X) = Eij + k B
B
(3)
For slowly varying world-volume fields, the derivative corrections (3) can be ignored and
the effective action (1) for the world-volume dynamics (with a diagonal metric) becomes
Z
Z
1
i a j
0
i
j
h
X
+
(B
+
2
F
)
X
X
.
(4)
S'
ij
a
ij
4 0
+ 2 0 F )
(5)
SWZ = Qp dt d p eB C.
Now the Neumann boundary conditions for a D-p-brane are derived from the effective
action (4). They are modified due to the B-field and can be expressed as
hij n Xj + B ij t Xj = 0,
(6)
174
(7)
In presence of B-field, the momenta (7) receive correction and is known to be responsible
for the non-commutative description on the D-p-brane world-volume. The commutation
relation between the conjugate coordinates at the boundary can be derived (6) from that
in the bulk among the canonical variables with an invertible B-field. In general, the
commutators on the boundary ( = 0 and ) are subtle to define. In a time ( ) ordered
space, the commutators can be expressed as
i
j
ij
ij
XL ( ), XL ( 0 ) = i 2S A E( 0 ) and
i
j
ij
ij
(8)
XR ( ), XR ( 0 ) = i 2S + A E( 0 ),
i = X i (, 0) and Xi = X i (, ) are the open string operators defined, respecwhere XL
R
tively, in the left and right ends of the open string Xi (, ) on the D-p-brane world-volume.
Under the reversal of the parametrization, , the corresponding non-zero modes
in the Fourier expansion for the operators (XL XR ) can be seen to remain invariant. The
right hand sides (8) for the commutators are c-numbers and contain a symmetric (i j )
ij
ij
S and an antisymmetric A matrix parameters. Explicitly, they can be expressed as
ij
S
h
h
h + B h B
ij
and
ij
A
1 1
B
h + B h B
ij
.
(9)
The left and the right ends of the open string are separated by a geodesic of string length
and are out side each other light cone. This in turn implies that no physical effect can be
propagated between the left and right space-like modules (AL and AR ). The commutators
between them become
i
j
(10)
XL ( ), XR ( ) = 0.
The expression for the commutators (8) involve a discrete value function E(y) which
takes +1 or 1, respectively for positive and negative y. In a general description, E(y)
ensures the ordering in the phase space. A close observation on the non-commutativity (8)
ij
indicates that the diagonal elements, S , are ordered in time, , where as the off-diagonal
ij
elements A can be ordered with respect to the space-like coordinates Xi ( ). To avoid the
naive ambiguity (if any), while defining the commutators (8) at equal time ( = 0 ), one
can alternately define the operators, Xi ( ), with a time ordering, > 0 , such that | 0 |
is infinitesimally small . As a result, the non-commutativity in the space-like directions,
175
i 6= j , can be interpreted intuitively as a time ordered space. Then the commutators (8) in
the left and right modules 2 can be expressed,
i
(11)
X ( ), Xj ( ) = T Xi ( )Xj (0 ) Xi ( )Xj (+0 ) .
Since the world-volume dynamics involves open string modules, it describes a phase
space for a D-p-brane. The algebra A of functions (in left and right modules) are generated
by the operators U i = exp(iXi ) and satisfy the C ? algebra 3 with multiplication of
functions on the non-commutative space. It can be given by
ij
ij
U i U j U i ? U j = ei(2S A ) U i U j .
(12)
S ,A
n>n0
l
Y
n=1
Vn
(13)
A =0
P
Using momentum conservation, n kn = 0, it can be checked that the symmetric part
ij
S in the phase factor (13) does not contribute substantially. Thus the effective nonij
commutative parameter is due to the antisymmetric matrix elements A which in turn
implies the presence of B-field. Then the effective phase responsible for the world-volume
non-commutativity reduces to
X ij
0
0
A kin kj n E(n n ) .
(14)
exp 2i
n>n0
This expression implies that the non-commutativity on the world-volume is solely due to
the broken phase of cyclic symmetry in the time ordered space. This in turn implies that
ij
the antisymmetric parameter, A , plays a vital role for the non-commutative description
on the world-volume.
The illustration (14) shows that the world-volume fields should be re-defined to absorb
ij
the symmetric matrix elements, S . In fact, the diagonal matrix elements are associated
with a short distance divergence [17,25,26], which can be seen while defining the
2 Henceforth, we drop the index L and R, since the left module is a commutant of the right module and viceversa. Physics is identical in both the modules except for a signature (8).
3 E.g., U i (X) ? U j (X) = exp(i ij ) U i (X + y)U j (X + z)|
y=0=z . The ? product is associative and
A y i zj
leads to Moyal bracket description.
176
correlators in Section 3.1. The divergence can be regulated by introducing a cut-off. This
in turn implies that the world-volume should be properly re-normalized to absorb the
symmetric matrix elements completely. If X i denote the renormalized coordinates
Xi ( ) = X i ( ) + X i ( ),
(15)
then X i contain the divergence and can be regulated by introducing a cut-off. The
coefficient of the divergence piece (15) is symmetric in (i j ) and can be intuitively
obtained from Ref. [17]
X (S + S 0 )K [divergence],
(16)
where the extrinsic curvature K can be obtained by considering the Lie derivative on the
The additional symmetric elements, ij0 , can be expressed as
induced fields (h, B).
S
ij
B
B
ij
.
(17)
S 0 =
h + B h B
As a result (15), the world-volume fields (2) become renormalized, i.e., h h and
4 Then the commutators (8) in terms of the renormalized coordinates in the time
B B.
ordered space become
i
ij
(18)
X ( ), X j ( 0 ) = iA E( 0 ).
Since the short distance divergence is renormalized, one can also, alternately, rewrite the
commutators (18) at equal time ( = 0 ) where ordering is implicit among the space-like
coordinates X i ( ). For the renormalized coordinates, the commutators takes the form:
i
ij
(19)
X ( ), X j ( ) = iA .
On the other hand, the boundary interaction (4) naively seems to be invariant under an
infinitesimal U (1) gauge transformation Ai = i . However, the gauge field Ai (X) itself
possesses a dependence on the non-commuting coordinates. Thus a proper treatment of the
gauge field is necessary to account for the string coordinates. In fact, such an approximation
(geodesic expansion around its zero modes) has already been taken into account (4) for
slowly varying B-field. As a result the U (1) gauge sector is constrained by the collective
coordinates A i ' 12 B ij X j . This in turn implies that the gauge transformation can be
viewed as a reparameterization invariance
A i = 12 B ij X j + O(B).
(20)
For the renormalized D-p-brane world-volume, the equal-time commutator in the Abelian
( being its generator) gauge sector becomes
i2 kk 0
A Bik Bk 0 j + O(A2 ).
A i , A j =
4
(21)
4 The detail of the computation for the re-normalization can be obtained from Ref. [17] by a suitable conformal
transformation.
177
(22)
The explicit expression for the propagator Gij (, 0 ) on the world-volume can also be
obtained by a conformal transformation from our earlier work [17]. The boundary value
propagator defines the world-volume correlators and can be given by
i ij
E( 0 ).
(23)
2 A
As discussed in Section 2.2, the limit 0 can be considered with a time ordering > 0
and when | 0 | is kept infinitesimally small. Then the propagator in the limit, 0 ,
becomes
i ij
ij
.
(24)
Gij (, ) = 2S ln
2 A
Gij (, 0 ) = 2S ln | 0 |
ij
It shows that in presence of B-field, 5 a finite contribution is associated with the offij
diagonal matrix elements, A , which is antisymmetric. For = 0 in Eq. (23), the
ij
the symmetric elements, S , are associated with a logarithmic divergence and the
antisymmetric elements are accompanied with a discrete value function E() which is
not well defined there. Then, a priori, the PauliVillars regularization does not seem to
be appropriate in presence of B-field. However in the limit, 0 , the propagator (23)
is well behaved which essentially defines the point-splitting regularization. As a result,
different operators (i 6= j ) are not allowed at the same time ( 6= 0 ). It is necessarily due
the antisymmetric property of B-field which defines the point-splitting scheme.
Once we obtain the necessary renormalization for the D-p-brane world-volume (16), the
symmetric elements can be gauged away in a physical process (14). Then the renormalized
propagator can be seen to be constant (antisymmetric) matrix and rewritten as
ij (, 0 ) = i ij E( 0 ).
(25)
G
2 A
In principle, the renormalized D-p-brane dynamics [17] receive 0 -corrections (due to the
ij
curvatures) associated with the antisymmetric matrix elements A . The higher order terms
0
associated with can be argued for the corrections to the commutative geometry. This
in turn implies that the 0 corrections (accompanied with the antisymmetric property)
5 If B = 0, the propagator (23) is well defined and can be seen to contain a logarithmic divergence ln .
As a standard practice, the loop divergences contained in hXi ( )Xi ( )i can be regulated by PauliVillars
regularization scheme. In this approach, one renormalizes the world-volume fields (15) at the expense of the
vertex operators. Then every element in the propagator precisely corresponds to that of a D-particle.
178
correspond to the deformation of the YangMills theory on the world-volume and lead
to an associative algebra in the left and right modules. The simple analysis explains two
different geometrical aspects for a generalized D-p-brane which is in agreement with the
recent analysis [7].
3.2. SO(p) symmetry: homomorphic image space
In general, a rotation among the spatial coordinates changes the topology leading to
equivalent geometries. For instance, under a rotation, a (diagonal) metric and an antisymmetric two-form field can be transformed to anisotropic geometry in presence of a new
metric and vanishing two-form field. As a result, the non-commutative geometry can be
viewed, alternately, with a non-local description on the (rotated) D-p-brane world-volume.
In other words, a renormalization, in the context, leads to a world-volume re-definition and
can be seen as a rotation among the (spatial) collective coordinates defining the D-p-brane.
In fact the constant matrix coefficients in the regulated propagator (25) correspond to
a rotation in the Cartan sub-algebra. Then the world-volume can be described in terms of
p/2 independent blocks. The metric and the B-field can be expressed as
hij =
p/2
M
hr
r=1
0
hr
p/2
M
0
and Bij =
Br
r=1
Br
0
.
(26)
Now consider the zero modes on the world-volume, x i = Xi i , which can be seen
to play an important role in presence of B-field. These (constant) modes are characterized
by a group of translations on the world-volume and is responsible for the inhomogeneous
space. On the other hand, the non-zero modes are described by a group of rotation, SO(p),
for a D-p-brane. Under an inhomogeneous transformation
Xi R ij j + x i ,
(27)
where R ij denotes the rotation matrix [17] and can be written in the (2 2)-block diagonal
form:
p/2 r
M
2iAr
S + Sr 0
ij
.
(28)
R =
2iAr
Sr + Sr 0
r=1
It is little tricky to define the corresponding rotation angles, r , as the metric is not flat.
Finally, the angle for each (2 2)-block can be written as
r = sin1 2iAr
(29)
= cos1 Sr + Sr 0 .
Now the world-volume geometry can be analyzed from that of a real plane. 6 The new
coordinates, X ( ) = x + , describe a two-dimensional phase space and can be given
by
6 Henceforth, we omit the index r unless it is required.
179
+ = 1 sin + 2 cos ,
= 1 cos 2 sin .
(30)
Since the rotation angle, , is not fixed, it gives rise to a non-local description on the D-pbrane world-volume.
In the new frame, the boundary condition (6) turns out to be
D ( ) = 0,
where
D = (hn iBt ).
(31)
The resultant directions () intertwine the Neumann and Dirichlet conditions [17]. The
nature of the differential, D , is determined by the B-field.
For a renormalized D-p-brane world-volume, the commutator (19) in the new frame can
be given by
r
r
( ) = iAr .
(32)
X ( ), X +
Since the presence of zero modes make the world-volume an inhomogeneous space, the
translation group does not commute with the rotation SO(p). Then the commutator (32)
can be expressed in terms of the zero and non-zero modes of the un-twisted world-volume:
(33)
X , X + = x 1 , x 2 + x 1 , 2 + 1 , x 2 + 1 , 2 .
All four of the commutators (33) contribute (independently) towards the non-commutativity on the world-volume. Under a homomorphic transformation, the zero modes can
be modded out to obtain a homogeneous description on a D-p-brane. As a result, the
homomorphic image space is commutative among the zero and non-zero modes though
the world-volume retains the ordering (independently) among the zero and non-zero mode
sectors. Then the commutator (32) can be given by
(34)
x , x+ + , + = iA .
The analysis shows that the non-commutative symmetry can be mapped onto a commutative one under a homomorphic transformation. Thus the U (1) sector of the world-volume
describes the homomorphic image space for the non-commutative space which is described
by the SO(p) rotation group.
In this frame, the constant renormalized matrix propagator (25) turns out to be diagonal
and satisfies
D (, 0 ) = 0.
D G
Explicit expression for the propagator can be given by
1
0
D (, 0 ) = 2 A
E( 0 ).
G
0
12 A
(35)
(36)
The diagonal form of the renormalized propagator further confirms (following the
discussions in Section 2.3) the commutative aspect of a rotated D-p-brane world-volume.
In the limit 0 , the propagator (36) contributes a finite constant (with antisymmetric
property). This in turn, can be seen to introduce a constant shift on the world-volume fields
leading to quantum description on a D-p-brane world-volume.
180
(38)
It shows that the string modes, Xi , are identified with the zero modes, x i , in the large
B limit. The expression (38) corresponds to the Dirichlet condition for D-particles on a
D-p-brane world-volume. This in turn implies that the D-p-brane world-volume becomes
spatially non-localized and represents a high density of D-particles. In addition, the worldvolume non-commutativity (8) can be seen only among the zero modes, x i , independently
in the left and right modules.
ij
Here, the non-commutative parameter A = (1/B)ij and the equal-time commutator
(19) reduces to that among the zero modes of the collective coordinates (Xi , A i ). It takes
the form:
i j
x , x = i(1/B)ij ,
i j
i
(39)
x , A = hij .
2
It shows that the collective coordinates in the gauge sector becomes the canonical momenta
to the conjugate coordinates or vice-versa in presence of a B-field. In general, noncommutative description (8) holds good for any value of B-field though in the limit, a clear
7 In fact, the closed string channel is T-dual to the open string channel. In closed string sector, the left and right
modules overlaps due to the periodicity in its boundary value and the non-commutative description (18) cancels
against each other.
181
picture of D-particles described by the zero modes emerge out leaving the feebly coupled
D-p-brane.
In the new frame discussed Section 3.2, the picture can be easily identified with the
motion of a large number (37) of D-particles in an uniform and strong magnetic field
(B-field). Since the D-particles are described only by the zero modes, the wave function
+ , is
representing its ground state, = exp(ik x ) for the momentum k = 1 Bx
associated with an additional phase due to the non-commutative description, [x , x + ] =
iB 1 . It can be given by
i
|(strong B) = e B x |(B=0) .
(40)
Since the world-volume is a compact space, the single-value condition on the wavefunction can be imposed by introducing a periodicity in one of the spatial coordinate (x ).
Then its conjugate momentum becomes (Dirac) quantized: k = 2n for an integer n and
the phase factor (40) reduces to: exp(2inx ). Since the Landau levels are independent
of the momentum, k , the D-particle wave function degenerates in presence of a strong
magnetic field leading to a non-local description. Though there are some subtleties due to
the boundary value of the coordinate field, the illustration shows that the non-commutative
description leads to non-localization of D-particles on the D-p-brane world-volume.
Then the non-commutative YangMills symmetry on the world-volume may be better
understood from the study of a simple physical model for the D-particles.
4. Discussions
To summarize, we have shown that a re-normalization of the generalized D-p-brane
is essential to account for a precise non-commutative geometry on its world-volume.
We learned that the B-field is essential, due to its antisymmetric property, for the
non-commutative aspect of a D-p-brane. Due to the subtleties in defining the noncommutativity, we have considered a time ordering among the operators as an alternative
to the non-commutative description. The world-volume geometry is analyzed with respect
to the SO(p) rotational symmetry group. The zero modes were seen to introduce the noncommutative feature between the group of translations and the rotation. A homomorphic
map from the non-commutative space was argued for the Abelian gauge symmetry.
Large B limit was analyzed and the zero modes non-commutativity on the D-p-brane
world-volume were discussed. In the large B limit, the correspondence between the
non-commutative and commutative symmetries can be obtained by implementing the
holographic idea. The existence of such a holographic connection between different
geometries may enhance our understanding of quantum gravity.
In the actual computations, though the world-volume fields are slowly varying, the nonij
commutative parameter matrix A is assumed to be a constant for simplicity. In principle,
ij
A is not fixed. It would imply that the deformation parameter should be expanded
(a convergent series) around its zero mode. However such a generalization is out of reach
in the present context.
182
Acknowledgments
I would like to thank M. Cederwall, G. Ferretti and B.E.W. Nilsson for various useful
discussions. The work is supported by the Swedish Natural Science Research Council.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
Abstract
The GreenSiegel central extension of superalgebras for BPS branes is studied. In these cases
commutators of usual bosonic brane charges only with the broken supersymmetry charges allow this
central extension. We present an interpretation of these fermionic central charges as fermionic brane
charges, and show that they take nonzero values for a nontrivial fermionic boundary condition. Such
fermionic coordinates solutions are determined by equations of motion in a suitable gauge condition
which manifests NambuGoldstone fermionic modes as well as bosonic modes in the static gauge.
We also show that some modes of dilatino fields couple with the fermionic brane currents. 2000
Elsevier Science B.V. All rights reserved.
PACS: 11.17.+y; 11.30.Pb
Keywords: Superalgebra; SUSY central extension; BPS states; D-brane
1. Introduction
For supersymmetric theories superalgebras are powerful tool to explore non-perturbative
aspects about BPS states [13]. About a decade ago Green showed the possibility of the
central extension of the Super-Poincare algebra [4], where the momentum charge does not
commute with the supercharge and gives rise to the fermionic central charge: 3
{Q , Q } = 2(C/
P ) ,
[Pm , Q ] = (Zm ) .
(1.1)
Siegel proposed a good use of this algebra that the WessZumino action of the Green
Schwarz superstring can be obtained as a local bilinear form [5], namely an element of the
ChevalleyEilenberg cohomology [6]. Recently this idea has been applied to superbrane
theories and super-D-brane theories [712]. As well as a superstring these objects require
1 mhatsuda@post.kek.jp
2 sakaguch@yukawa.kyoto-u.ac.jp
3 Q, P , Z are a supercharge, a momentum and a fermionic central charge. and C = i 0 are gamma matrix
and a charge conjugation matrix, a slash represents contraction with m , e.g. P/ = Pm m .
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 3 5 - 8
184
the WessZumino actions to include the kappa symmetry. The WessZumino actions for pbranes and D-p-branes are higher rank tensors and/or containing U(1) fields, so integrating
and obtaining their local expressions are complicated [13,14]. These complications can be
simplified, if Siegels method is used. As long as the algebra (1.1) tells the truth in this
sense, its physical interpretation will be the next question.
It is interesting to compare the global superalgebra with the local superalgebra. In
order to realize the global superalgebra one needs the local supersymmetry constraints
F = 0 to remove extra fermionic degrees of freedom. They anticommute with the
global supercharges, {F, Q} = 0 [5,15]. (Q , Pm ) are global charges, while (F , Dm ) are
covariant derivatives satisfying following algebras:
D) ,
{F , F } = 2(C/
[Dm , F ] = (W m ) .
(1.2)
W is named the super-YangMills field strength [1618]. It is also expected that W and
Z are a global charge and a covariant derivative respectively in a larger superspace. In
order to see the physical role of Z , it is useful to take a concrete model. For a superstring
case charges and covariant derivatives are given in terms of spacetime coordinates X
and , for example
W 0 ,
0
(1.3)
. Eq. (1.3)
where stands for the derivative with respect to worldsheet spacial coordinate
R 0
suggests that the new fermionic central charges are realized by Z . This is a surface
term, so nontrivial Z will require nontrivial boundary values of the fermionic coordinates.
Bergshoeff and Sezgin applied the Siegels method to a M2 brane where commutators of
supercharges and not only the momentum charge but also brane charges allow fermionic
central extension [7,8]. Further studies have been done for NS1 (IIA, IIB) and D1 [9],
for D3, D5 and NS5 [12] and for D2 and M5 [11]. In the work on SL(2, R) SO(2, 1)
covariant central extension [10], the momentum and the NS/NS and R/R brane charges
are treated equally as SO(2, 1) triplet elements. Then the SO(2, 1) triplet fermionic central
charges appear. It was shown that there exists the RaritaSchwinger type constraint on the
new fermionic central charges and one of the triplet elements is redundant. In other words,
the momentum or the brane charge can be chosen to be super-invariant. In this paper we
will consider the super-invariant momentum and the super-noninvariant brane charges. In
[712] the super-noninvariant momenta can be realized by introducing auxiliary fields.
Instead, in this paper we will examine the super-noninvariant brane charges by introducing
a nontrivial fermionic boundary condition without any auxiliary fields.
Organization of the present paper is as follows. In Section 2 we discuss the fermionic
central extension of the superalgebra from the BPS state point of view. The Jacobi identity
of three supercharges restricts that the commutator of the brane charge and the broken
supersymmetry charge allows the fermionic central extension. In Section 3 we will justify
this by examining equations of motion in a suitable gauge condition. We show that there
exists nontrivial fermionic solution which can give non-zero values of new fermionic
brane charges. In Section 4 we will also consider the supergravity coupling with this
state. The quantum states and the vertex operators for D-branes are not known because
185
of the difficulty of the soliton quantization. On the other hand classical soliton property
may suggest how to couple with the supergravity fields. Since it is supposed that Dbranes are static objects, the nonrelativistic approximation will be a good approximation.
So we begin by the classical Gauss law equation, then examine its consistency under the
broken supersymmetry. The unbroken supersymmetry transformation of a purely bosonic
soliton solution has been studied in [20]. Usually, including this reference, the fermionic
supergravity fields are set to be zero. However we rather consider nontrivial fermionic
supergravity fields. It turns out that there can be nonzero dilatino fields whose some mode
couple to the new fermionic brane currents.
=q
0
d Xm
= q I (XF XI )m ,
(2.3)
where F,I are D-string end points. The massiveness of a D-string is reflected to its infinity
length of L|L .
One may notice that there is no way to centrally extend of [P , Q],
Z
Z
/ 0 (2 )q
d1 pm (1 ), d2 X
[Pm , Q] =
Z
Z
(2.4)
d1 (1 2 ) (2 )m q = 0
= d2 2
with q = q I I = q NS 3 + q R 1 (I = NS, R or 3, 1). On the other hand, the brane charges
I allow to have the central extension in a commutator with the supercharges:
Z
Z
I
I
d1 1
m , Q = q
d2 (2 )m (1 2 )
= Z I m .
4 Notation follows [21] where (X, ) are spacetime coordinates and their conjugates are (p, ).
(2.5)
186
(2.6)
M = Tq L
(2.7)
{S, S} 6= 0
(2.9)
then
Q|0i = 0,
S|0i 6= 0,
(2.10)
for a ground state |0i. The projection operators P defined as (2.7) become simple form 5
for a ground state in the static gauge
(2.11)
P = 12 1 1AB (01 ) (q )AB , q = q /|q |.
It is shown that the central extension (2.5) is possible only for broken SUSY
[m , S ] = (Zm ) ,
(2.12)
[m , Q ] = 0
(2.13)
since
(2.14)
with A( B) = A B + A B .
The algebra discussed above is summarized as follows:
{Q, Q} = {Q, S} = 0,
I
m , Q = 0,
[Pm , Q] = 0,
{S, S} = 2CG ,
I
m , S = Z I m ,
[Pm , S] = 0.
X
antisymmetrized m
(2.15)
m1 mN .
187
(3.1)
by using the kappa invariance. The local superalgebra for a D-string is given by [21]
{FA (1 ), FB (2 )} = 2(C )AB (1 )(1 2 ),
m
1AB p m (q )AB (1 )m ,
( )AB =
(3.2)
where
F = + (p
/ 1 q ) 12 ( 0 q + q 0 ),
p m = pm + m q 0 ,
0
m 0 .
(1 )m = Xm
(3.3)
is a rank half and nilpotent operator so that a half of F are first class constraints
FA (F )A = 0 generating the kappa symmetry and another half are second class.
The counting of the physical degrees of freedom is following: the degrees of freedom of
A and A are 16 2 2. The number of the second class constraints is 16. The number
of the first class constraints is 16, and the one for the gauge fixing conditions is 16. So
totally surviving degrees of freedom are 16 2 2 (16 + 16 2) = 16, so 8 s and
8 s are physically dynamical variables.
For a ground state in the static gauge, the right hand side of the (3.2) becomes the
same form of the unbroken SUSY projection operator C = 2iMP of (2.11). This
coincidence occurs to guarantee the universal property, {Q, F } = 0. In order to solve BPS
fermionic solutions explicitly, we restrict ourself to discuss only on the ground state in the
rest of this section. It should be noticed that the kappa projection is not C but . The
kappa symmetry is generated by
Z
Z
Z
Z
0
0
0 0
(3.4)
F = F = F P+ (2M ) = F P+ + ,
therefore the kappa parameter is projected along the unbroken SUSY direction. The gauge
condition (3.1) can be possible by using the kappa transformation with + = + = + .
Under the global translation, Xm is transformed as a Xm = am . Analogously under
the global SUSY, is transformed = , while + = 0 is preserved by the kappa
transformation ( + )+ = + + + with + = + . In this gauge Xm is transformed
under the global SUSY and the kappa transformation
( + )Xm = m + + m + m = m + 2+ m ,
(3.5)
where is nothing but the broken SUSY parameter. Of course the static gauge is always
recovered by the reparametrization invariance. The transverse coordinates Xi and are
NambuGoldstone modes associated with the broken translation invariance and the broken
supertranslation by a brane [22].
188
(3.6)
3
m = Xm m , F = F []
,
I
m I ) m + 1 m (I = 1, 3).
= (
G = m m ,
(3.7)
1
,
LW Z |p=1 = T
(3.8)
and we will discuss general D-p-brane cases in the last section. Under an arbitrary variation
of it becomes
TG
I
G G + q I
,
L =
2 det(G + F )
G = ( ( ) ( ) )
+ 2( )( ) 2( )( ),
(3.9)
I
= ( q q )
q I
+ 12 ( q )( ) ( q )( )
( )( q ) + ( )( q ) .
For a ground state where G and F are constant and terms in O( 3 ) vanishes, the
equation of motion L/ = 0 becomes quite simple
( q ) = 0,
where q I are given by
(
q NS = q 3 = E 1 = T
F01
det(G+F )
(3.10)
qR = q1 = T ,
(3.11)
(3.12)
with a constant spinor , then the equation of motion (3.10) reduces into
(1 q 0 )1 = 1 (1 + q 01 ) = 0.
(3.13)
189
Then there exists nontrivial ground state solution of as the form of (3.12) with
P+ = 0.
(3.14)
The boundary term of the equation motion also allows the solution (3.12)
( q ) |boundary = 0.
(3.15)
=q
0
d
= q I L |L .
(3.16)
It is natural to set the volume of Z to be infinity, since it is obtained from infinity volume
brane charge by the supertransformation.
The choice = 0 or = const. brings back to the usual Super-Poincare algebra which
is usually considered as a ground state solution.
4. Supergravity coupling
A D-string is described by the NS/NS and R/R worldvolume currents:
Z
(J I )nl (x) = q I d d X[n Xl] (10) x X(, ) .
(4.1)
= 0.
(4.2)
(4.3)
(4.4)
I are fluctuations.
where Hlmn
Now let us consider the broken supersymmetry transformation of (4.4). Because the
brane charge ( I )m transforms into the fermionic brane charge (Z I ) under the broken
190
supersymmetry and because of (4.3), a bosonic brane current (J I )mn also transforms into
a fermionic brane current (J I )m :
Z
(J I )mn (x) = q I d d X[m Xn] + X[m Xn] (10)(x X)
Z
I
d d X[m n] (10) (x X)
= q
(J I )[m|| (x)( n] ) .
(4.5)
It is noted that under the broken supersymmetry brane coordinates transform into Xm =
m = m . The fermionic currents J m
Z
I n
I
(J ) (x) = q
(4.6)
d d Xn (10) x X(, )
are also conserved currents
Z
n (J I )n (x) = q I d d Xn n (10) x X(, )
n
Z
I
X
d d
(10) x X(, )
= q
n
X
Z
= q I d d (10) x X(, ) = 0,
(4.7)
(4.8)
If the right hand side of Eq. (4.4) transforms under the broken supersymmetry of the
brane, the left hand side should also transform. In another word the total supersymmetry
charge is sum of the supergravity part and the brane part. Transformation rules of
supergravity fields are given in [24], and
mn I ,
(4.9)
(B I )mn = 2 [m n]
where is a gravitino and is a dilatino. In the first order of this test brane perturbation the
background metric is not affected, since the energy momentum tensor which is a source
of the metric contains square of H as the fluctuation. So gravitinos must be set to zero
a
m a = 0. In order to have nontrivial transformation rule of
m = 0 because em
I
Bmn with the broken SUSY parameter , we will consider nonzero + so that B01 =
+ 01 6= 0 as their fluctuations.
Under the broken global SUSY a dilatino does not transform (, H ) = 0 in
a flat background ( = 0, H = 0). Then the left hand side of (4.4) transforms into
l (H I )lmn = (B I )mn
= 1 + [m I n]
2
(4.10)
191
in the Lorentz gauge m Bmn = 0. As a result of (4.5) and (4.10), under the broken SUSY
(4.4) transforms into
1
I n (x) = (J I )n (x).
(4.11)
2 +
Therefore this new fermionic brane charge becomes a source of dilatino fields.
It is also confirmed that another current does not follow. Even if one more broken
supersymmetry is performed on the fermionic brane current, it vanishes by (4.2)
Z
(4.12)
(J I )n = q I (10) (x X) ( n ) = 0.
This concludes that a D-string action allows the fermionic brane currents coupled with
some modes of dilatino fields as well as the usual brane current coupling Bmn J mn , when a
D-string has a nontrivial fermionic boundary condition.
5. Discussions
Generalization to arbitrary D-p-branes is straightforward. The global supersymmetry
and the local supersymmetry algebras for D-p-branes are given in [13,14], as the same
form as (2.1) and (3.2). The projection (2.11) has the form of
P = 12 (1 b),
(5.1)
(5.3)
For D-p-brane cases, brane charges are p-rank tensors. Each commutator with the broken
supersymmetry charges replaces a vector index of the brane charges by a spinor index.
When has nontrivial solution of (5.3), totally p + 1 kinds of brane charges with vector
indices or spinor indices have nonzero values. For example a D2 brane carries totally 3
kinds of brane charges, among which 2 kinds of bosonic brane charges mn and Z are
obtained in the static gauge as
12 = T L2 ,
Z = T L2 1( 2) ,
(5.4)
192
Z2 = T L2 1 .
(5.5)
The supergravity coupling will follow the argument of the Section 4 replacing Bmn by a
(p + 1)-rank gauge field Cm1 mp+1 for a D-p-brane respectively.
It is curious that considering that Eq. (4.11) is second order although usual equation
of motion for a spinor field is first order. Even in a flat background this fermionic brane
current will couple to some mode of dilatino. So the coupling will be with the dilaton
potential mode (/
D/) or nonlocal. Further studies are necessary to clarify these points
and to apply many other systems.
Acknowledgements
M.H. wishes to thank Joaquim Gomis, Ken-ji Hamada, Nobuyuki Ishibashi and Shunya
Mizoguchi for fruitful discussions, and M.S. would like to express his gratitude to Hiroshi
Kunitomo and wishes to thank the theory group of KEK for the kind hospitality. We also
thank Kiyoshi Kamimura for helpful discussions, and Mitsuko Abe and Nathan Berkovits
for a question what is Z ? which is our motivation of this work.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
193
[23] M. Hatsuda, S. Yahikozawa, P. Ao, D.J. Thouless, Phys. Rev. B 49 (1994) 15870, condmat/9309018.
[24] J.H. Schwarz, Nucl. Phys. B 226 (1983) 269.
Abstract
In the framework of nonrelativistic QCD, we compute the leading double-logarithmic corrections
of order s3 ln2 (1/s ) to the heavy-quarkantiquark bound-state wave function at the origin,
which determines the production and annihilation rates of heavy quarkonia. The phenomenological
implications for the topantitop and systems are discussed. 2000 Elsevier Science B.V. All
rights reserved.
PACS: 12.38.Bx; 12.39.Jh; 14.40.Gx
1. Introduction
Nonrelativistic quantum chromodynamics (NRQCD) [13] is a powerful tool for the
investigation of heavy-quark threshold dynamics. Recent developments of the NRQCD
effective-theory approach [414] led to the complete next-to-next-to-leading-order 1
(NNLO) description of the production of heavy quarkantiquark pairs at threshold.
Important applications include sum rules and toponium phenomenology [1527]. A
review of the recent progress in the perturbative study of heavy quarkantiquark systems
may be found, for example, in Ref. [28]. In view of the surprising significance of the NNLO
corrections, it appears indispensable to also gain control over the next-to-next-to-next-toleading order (N3 LO) in order to improve the reliability of the theoretical predictions and
our understanding of the structure and the peculiarities of the threshold expansion.
Some specific classes of N3 LO corrections were analyzed in literature. The one-loop
renormalization of the O(1/m2q ) operators was obtained in Refs. [5,29]. The retardation
Permanent address: Institute for Nuclear Research, Russian Academy of Sciences, 60th October Anniversary
Prospect 7a, Moscow 117312, Russia.
1 In the NRQCD effective theory, there are two expansion parameters, the strong coupling constant and the
s
heavy-quark velocity , and the perturbative order of some correction is determined by their total power. For
example, terms of O(s2 ), O(s ), and O( 2 ) contribute at NNLO.
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 3 9 - 5
198
effects arising from the emission and absorption of virtual ultrasoft gluons by the heavy
quarks were studied in Ref. [30]. The leading logarithmic O(s3 ln(1/s )) corrections to
the heavy-quark bound-state energies En , where n is the principal quantum number, were
obtained in Ref. [31].
In this paper, we take the next step in this direction and investigate a particular class
of N3 LO corrections, namely the leading logarithmic O(s3 ln2 (1/s )) corrections to the
wave functions at the origin n (0) of the heavy quarkantiquark bound states which are not
generated by the renormalization group (RG). As is well known, n (0) are key parameters
in the analysis of the creation and annihilation of heavy quarkonia. The origin of these
logarithmic corrections is the presence of several scales in the threshold problem. In fact,
ln(1/s ) appears as a logarithm of a ratio of scales. These corrections are related to the
anomalous dimensions of the operators in the effective Hamiltonian. They can be found by
analyzing the divergences of the effective theory or by direct inspection of the regions of
the logarithmic integration. We shall verify that both methods lead to the same result. As a
by-product of our analysis, we shall also reproduce the O(s3 ln(1/s )) corrections to En
recently obtained in Ref. [31].
In N3 LO, there are in general also logarithmic corrections of the form s3 lnm (/s mq )
(m = 1, 2, 3), where s mq represents the soft or potential scales (see Section 2). These
corrections may be directly extracted from the NNLO result via the RG equation, and they
may be resummed by an appropriate choice of the normalization point, s mq . We note
in passing that, in the MS scheme, the optimal choice in NNLO is few units s mq
[21,26].
This paper is organized as follows. In Section 2, we recall the potential-NRQCD
(pNRQCD) formalism, from which En and n (0) can be extracted. In Section 3, we
evaluate the non-RG O(s3 ln2 (1/s )) and O(s3 ln(1/s )) corrections to n (0) and En ,
respectively, using the effective-theory approach. In Section 4, we repeat this calculation
in the conventional approach, by inspecting the logarithmically divergent phase-space
integrals. In Section 5, we present a numerical analysis and discuss phenomenological
implications of our results. Section 6 contains our conclusions.
(e+ e q q)
,
+
+
(e e )
(2)
199
by
R(s) = 12Q2q Im (s + i),
(3)
where E = s 2mq is the q q energy counted from the threshold, Ch (s ) is the square
of the hard matching coefficient of the nonrelativistic vector current, and the ellipsis stands
for the higher-order terms in . G(x, y, E) is the nonrelativistic Green function, which
sums up the (s /)n terms singular near the threshold. It is determined by the Schrdinger
equation which describes the propagation of the nonrelativistic quarkantiquark pair in
pNRQCD,
(H E)G(x, y, E) = (3) (x y),
(5)
x2
+ V (x) + ,
mq
V (x) = VC (x) + .
(6)
X
n (x)n (y)
d k k (x)k (y)
+
,
(7)
G(x, y, E) =
En E
(2)3 k 2 /mq E
n=1
200
where m and k are the wave functions of the q q bound and continuum states,
respectively.
Below the threshold, the (perturbative) vacuum-polarization function of a stable heavy
quark is essentially determined by the bound-state parameters. For the leading-order
Coulomb (C) Green function, the energy levels and wave functions at the origin read
EnC =
2s
,
mq n2
3
C 2
(0) = s ,
n
n3
(8)
where s = s CF mq /2. We are interested in the corrections to |nC (0)|2 . Note that, for the
study of bound-state parameters, we have s , so that we are only dealing with one
expansion parameter.
,
(9)
10 H =
3
2mq x 2
2m2q x x
m2q
4m3q
where CA = 3 is the eigenvalue of the quadratic Casimir operator of the adjoint
representation of the colour group, S represents the spin of the quarkantiquark system,
and {, } denotes the anticommutator. The first term of Eq. (9) is the non-Abelian potential
[3234], and the rest is the standard Breit potential and the kinetic-energy correction.
Note that the representation (9) of the nonrelativistic Hamiltonian is not unique and can
be related to other representations found in literature by the use of the equations of motion.
However, the corresponding corrections to the Green function [18,19,22] are independent
of the specific representation. In the vicinity of the nth bound-state pole, the resulting
correction to the Green function reads [22]
|nC (0)|2 2s
4
4 S(S + 1) CF + 2CA
1G(0, 0, E) EE C = C
n
En E m2q
3
GC (0, 0, E) + ,
(10)
where GC (x, y, E) is the Coulomb Green function. The spin-independent term proportional to CF in Eq. (10) comes about as (4 1 + 1)CF , where the contributions arise from
the anticommutator, -function, and 1/m3q terms of Eq. (9). Writing
n (0)2 = C (0)2 1 + 1 2 (0) ,
n
we have
1n2 (0) =
2s
m2q
4
4 S(S + 1) CF + 2CA GC (0, 0, En ) + .
3
(11)
(12)
201
These corrections are singular, since the Coulomb Green function at the origin is ultraviolet
(UV) divergent. In dimensional regularization, with d = 4 2 space-time dimensions, it
takes the form [26]
mq s 1
1 2 mq s
+ =
+ ln
+ ,
(13)
GC (0, 0, k) =
2 k
2
2
2
k
where k 2 = mq E and the ellipsis stands for the nonlogarithmic contribution. The pole
term in Eq. (13) is canceled by the O(s2 ) infrared (IR) pole of the hard matching
coefficient Ch (s ) [16,17]. Thus, the scale in the logarithm is to be identified with mq ,
and the sought correction reads [19,21]
2
1
(14)
10 n2 (0) = CF s2 2 S(S + 1) CF + CA ln .
3
s
The residual operators in the NNLO effective nonrelativistic Hamiltonian which are not
contained in Eq. (9) correspond to the purely perturbative corrections to the static Coulomb
potential. The corresponding corrections to the wave functions at the origin [19,21,26]
contain RG logarithms of the form s2 lnm (/(s mq )) (m = 1, 2), which vanish for =
s mq . This also holds in N3 LO for the RG logarithms of the form s3 lnm (/(s mq )) (m =
1, 2, 3) because the ultrasoft effects enter the stage only in N3 LO, so that the corresponding
running of the strong coupling constant at the ultrasoft scale only becomes relevant in
N4 LO. An important point here is that, starting from NNLO, the hard matching coefficient
Ch (s ) receives a nonvanishing anomalous dimension. Therefore, starting from N3 LO,
not only the running of s should be taken into account, but also the effective-theory RG
should be used for the evolution of the hard matching coefficient from = mq down to
= s mq [23].
At N3 LO, the non-RG leading logarithmic corrections to the wave functions at the origin
are produced by the one-loop renormalization of the operators in Eq. (9). In dimensional
regularization, the pole part of the correction is
1 CF s
2
4
CA s2 2 CA s 2 1
00
CF + CA
+
x ,
1 H=
2
3
3
mq x 2 3 m2q
x
8
16
s
CF CA
(x)
3
3
m2q
17 7 2
s
2
CF +
S CA
(x) ,
(15)
+
3
3
3
m2q
where the first three terms contained within the parentheses represent the IR divergence,
while the fourth one embodies the UV divergence of the potential. The IR poles are
canceled by the ultrasoft contribution, with the characteristic scale s2 mq , and may be
read off from Refs. [30,31], while the UV poles are canceled by the IR poles of the hard
coefficients and may be extracted from Refs. [5,29]. Evaluating the corrections to the wave
functions at the origin due to Eq. (15) in the same way as Eq. (12) was obtained, we find
the N3 LO non-RG leading logarithmic corrections to the wave functions at the origin to be
202
100 n2 (0) =
7
CF s3 3 2
41
2
1
CF +
2
12 12
3
s
(16)
In the derivation of this result, the factor (/k)2 in Eq. (13) needed to be expanded up
to O( 2 ). The contributions to Eq. (16) related to the IR and UV poles of Eq. (15) are of
opposite signs. This may be understood by observing that the IR poles introduce a factor
ln(E0C /s ) ln s , while the UV poles contribute a factor ln(mq /s ) ln(1/s ). The
Abelian CF3 term in Eq. (16) agrees with the QED result of Ref. [35]. In contrast to the
QED case, the leading logarithmic QCD corrections are spin dependent.
In the remainder of this section, we revisit the leading logarithmic corrections to the
bound-state energy levels, which may be obtained in a way similar to the case of the wave
functions at the origin. These corrections start from N3 LO, since there are no relevant
singularities in NNLO. In addition to Eq. (15), we now have to take into account the IRsingular contribution to the Coulomb potential [36,37],
1 CF CA3 s4
,
2 12x
(17)
which is canceled by the ultrasoft contribution [30]. Obviously, this term gives no
contributions to the wave functions. Writing
En = EnC + 1En ,
(18)
we find
7
2
41
4
1 3
s3 3 3
2
2
C +
S(S + 1) 2 CF CA +
CF CA + CA
1En = En
n F
6n 6n
3n
6
3n
1
(19)
ln ,
s
in agreement with the result for l = 0 of Ref. [31].
n2 (0) =
2s
m2q
Z 3
|kC (0)|2
d k
4
.
4 S(S + 1) CF + 2CA
3
(2)3 k 2 /mq En
(20)
The integral over k logarithmically diverges at large momentum and should be cut
at scale mq , where the nonrelativistic approximation becomes inapplicable. At low
momentum, the logarithmic integration over k is effectively cut at the Coulomb scale
203
s mq . Inserting in Eq. (20) the well-known expression for the Coulomb wave function
at the origin,
C 2 2s
1
s
(0) =
=1+
+ ,
(21)
k
k 1 exp (2s /k)
k
where only the second term is relevant for our purposes, we find, with logarithmic accuracy,
1
n2 (0) = CF s2
Zmq
dk
2
+ .
2 S(S + 1) CF + CA
3
k
(22)
s mq
00
n2 (0) = CF
s3
8 2
4
C + 4CA CF + CA2
3 F
3
Zmq
dk
k
s mq
1 2
17 7
CF +
S(S + 1) CF CA
+
3
6
6
Zmq
Zk
E
dk
k
s mq
dk 0
k0
Zmq
dk 0
.
k0
(23)
Here k is the momentum of the potential heavy quark, which can be as small as s mq , while
k 0 is the momentum of the virtual gluon, which can even be ultrasoft, of order s2 mq . The
integrals over k have the same origin as those in Eq. (22). The integrals over k 0 represent the
logarithmic corrections to the potential due to the virtual-gluon exchanges [5,29,30]. In the
first integral on the right-hand side of Eq. (23), the quark momentum k plays the role of an
UV cutoff, and the quark energy E = k 2 /mq acts as an IR cutoff, so that this contribution
corresponds to the IR poles of Eq. (15). In the second integral, the momentum k acts as
an IR cutoff, and this contribution corresponds to the UV poles of Eq. (15). Integrating
Eq. (23) over k 0 and k, we arrive at Eq. (16). A similar analysis for QED bound states may
be found in Ref. [35].
The N3 LO leading (single) logarithmic corrections to the energy levels may be obtained
by the method introduced in Refs. [3840], where the regions of the virtual-photon
momentum which lead to logarithmic contributions have been studied in the QED case.
In this way, we obtain
3
1En = En s
Zs mq 0
2
31
4
1 3
8 3
dk
2
2
C +
CF CA + CA
CF CA +
3n F
6n 3n2
3n
6
k0
s2 mq
7
2
1 3
17
2
C +
S(S + 1) 2 CF CA
+
3n F
6n 6n
3n
Zmq
dk 0
,
k0
(24)
s mq
where again the first (second) integral corresponds to the IR (UV) poles of Eq. (15). After
integration, we recover Eq. (19).
204
5. Phenomenological applications
Our results affect two processes of primary interest, namely the threshold production
of top and bottom quarkantiquark pairs. The relatively large width t of the top quark
serves as an efficient IR cutoff for long-distance effects. Because the relevant scale
mt t is much larger than the asymptotic scale parameter QCD , but comparable to the
Coulomb scale, the cross section in the threshold region may be described by the NRQCD
perturbation expansion if singular Coulomb effects are properly taken into account [41].
In order to analyze the significance of the N3 LO leading logarithmic corrections to the
cross section, we start from the NNLO calculation of Ref. [26] and add the contributions
from Eqs. (16) and (19). In Fig. 1, the normalized cross section R thus obtained is
compared with the pure NNLO result. The input parameters are taken to be s (MZ ) =
0.118, mt = 175 GeV, and t = 1.43 GeV. The soft renormalization scale is determined
from the condition s = 2s (s )mt , and hard renormalization scale is chosen to be
h = mt . The effect of the N3 LO leading logarithms is twofold. The normalization
of the cross section is reduced by about 7% around the 1S peak and below, and the
energy gap between the 1S peak and the nominal threshold is decreased by roughly
10%.
In the case of bottom quarkantiquark production, the nonperturbative effects are much
more significant, and one is led to use the sum rule approach [4244] to get them under
control. Specifically, appealing quark-hadron duality, one matches the theoretical results
for the moments of the spectral density,
Fig. 1. Normalized cross section R(E) of e+ e t t in NNLO (dotted line) and with the leading
logarithmic N3 LO corrections included (solid line), as a function of the centre-of-mass energy E
counted from the nominal threshold at 2mt .
n dn
12 2
(s)
4m2q
n
s=0
n!
ds
Z
n
R(s)
= 4m2q
ds n+1 ,
s
205
Mn =
(25)
with their experimental counterparts evaluated from the expressions in the second line
of Eq. (25). For large n, the moments are saturated by the near-threshold region. Then,
the main contribution to the experimental moments comes from the resonances, which
are measured with high precision. On the other hand, for n of O(1/s2 ), the Coulomb
effects should be properly taken into account on the theoretical side. In order to analyze
the N3 LO leading logarithmic corrections to the sum rules, we upgrade the NNLO
result of Ref. [21] by including Eqs. (16) and (19). We fix the strong coupling constant
by s (MZ ) = 0.118 and focus on the determination of the bottom-quark mass. At present,
this appears to be the most interesting application of Eq. (25). We find that the inclusion
of the N3 LO leading logarithms in the sum rules leads to a reduction of the extracted
mass value by approximately 30 MeV for moderate values of n, 5 < n < 15, and if
the soft renormalization point s is chosen to be s = 4s (s )mb . This result does not
depend on whether the energy denominators of the Green function are expanded around
the Coulomb values or not (see Ref. [21] for details) and on which mass parameter, pole or
MS mass, is considered. On the other hand, the result essentially depends on s because
the s dependence of s is compensated only by higher-order terms. For example, for
s = 2s (s )mb , the correction to the mass parameter reaches 70 MeV. However, the
perturbative result can hardly be trusted at such a low renormalization point [21,26].
For completeness, we also present the numerical corrections to the parameters of the
1S resonance. For mb = 4.8 GeV and s = 4s (s )mb , the wave-function correction is
19%, and the correction to the binding energy is 25 MeV.
206
which is usually assigned to this kind of bottom-quark mass determination on the basis
of the renormalization scale dependence of the strong coupling constant in the NNLO
result. In fact, the non-RG leading logarithmic contributions are about two times smaller
than the contribution from the N3 LO RG logarithms, which may be estimated from the
renormalization scale dependence of the NNLO result [21,26]. Thus, the resummation
of the RG logarithms [25] seems to be very justified. At the same time, the scale of the
corrections is close to the estimate given in Ref. [30] on the basis of the analysis of the
ultrasoft contributions.
Although, in contrast to QED, ln(1/s ) is not a big number, especially for the case of
bottom, the leading logarithmic terms can be considered as typical representatives of the
N3 LO corrections. For comparison, at NNLO, the leading logarithmic term accounts for
approximately one half (third) of the total correction to the n = 0 wave function at the
origin in the case of top (bottom). Obviously, the N3 LO corrections are comparable to the
NNLO ones and reach 10% in magnitude, even in the case of top, where s 1/10. This
tells us that the NRQCD threshold expansion is not a fast convergent series for the physical
value of the strong coupling constant.
A final comment refers to the resummation of the non-RG leading logarithmic
corrections to the wave functions at the origin. A part of the non-RG leading logarithmic
corrections to the heavy-quark threshold cross section was resummed in Ref. [23] by
using the RG equation of the effective theory for the evolution of the hard matching
coefficient Ch (s ) from scale mq down to scale mq . This evolution equation is obtained
by studying the dependence of Ch (s ) on the scale , which cancels the dependence
of Eqs. (12) and (13). This effectively sums up the higher-order NRQCD corrections due
to the tree-level operators of Eq. (9). The terms resummed in this way are of the form
s (s ln(1/))2n1 , with n = 1, 2, . . . , i.e., they include only even powers of s . For the
bound-state parameters, we have s , and Eq. (16) gives the first term of this series.
In order to resum all corrections of the form s (s ln(1/s ))n , it is necessary to add
the terms with odd powers of s , of the form s (s ln(1/s ))2n , which are generated by
Eq. (15). For this end, one has to take into account not only the evolution of Ch (s ),
but also the evolution of the potential (9) from the hard scale down to the ultrasoft
scale. Numerically, however, the effect of the resummation may not be essential for
phenomenological applications.
Acknowledgements
This work was supported by the Bundesministerium fr Bildung und Forschung under
Contract No. 05 HT9GUA 3, and by the European Commission through the Research
Training Network Quantum Chromodynamics and the Deep Structure of Elementary
Particles under Contract No. ERBFMRXCT980194. The work of A.A.P. was supported
in part by the Volkswagen Foundation under Contract No. I/73611, and by the Russian
Academy of Sciences through Grant No. 37.
207
References
[1] W.E. Caswell, G.P. Lepage, Phys. Lett. B 167 (1986) 437.
[2] G.P. Lepage, L. Magnea, C. Nakhleh, U. Magnea, K. Hornbostel, Phys. Rev. D 46 (1992) 4052.
[3] G.T. Bodwin, E. Braaten, G.P. Lepage, Phys. Rev. D 51 (1995) 1125; Phys. Rev. D 55 (1997)
5855 (Erratum).
[4] M. Luke, A.V. Manohar, Phys. Rev. D 55 (1997) 4129.
[5] A.V. Manohar, Phys. Rev. D 56 (1997) 230.
[6] A.H. Hoang, Phys. Rev. D 56 (1997) 5851.
[7] B. Grinstein, I.Z. Rothstein, Phys. Rev. D 57 (1998) 78.
[8] M. Luke, M.J. Savage, Phys. Rev. D 57 (1998) 413.
[9] P. Labelle, Phys. Rev. D 58 (1998) 093013.
[10] A. Pineda, J. Soto, Nucl. Phys. Proc. Suppl. 64 (1998) 428.
[11] A. Pineda, J. Soto, Phys. Lett. B 420 (1998) 391; Phys. Rev. D 59 (1999) 016005.
[12] M. Beneke, V.A. Smirnov, Nucl. Phys. B 522 (1998) 321.
[13] H.W. Griesshammer, Preprint No. NTUW-98-22, hep-ph/9810235, October 1998.
[14] A. Czarnecki, K. Melnikov, A. Yelkhovsky, Phys. Rev. A 59 (1999) 4316.
[15] A. Pineda, F.J. Yndurain, Phys. Rev. D 58 (1998) 094022.
[16] A. Czarnecki, K. Melnikov, Phys. Rev. Lett. 80 (1998) 2531.
[17] M. Beneke, A. Signer, V.A. Smirnov, Phys. Rev. Lett. 80 (1998) 2535.
[18] A.H. Hoang, T. Teubner, Phys. Rev. D 58 (1998) 114023; Phys. Rev. D 60 (1999) 114027.
[19] K. Melnikov, A. Yelkhovsky, Nucl. Phys. B 528 (1998) 59; Phys. Rev. D 59 (1999) 114009.
[20] J.H. Khn, A.A. Penin, A.A. Pivovarov, Nucl. Phys. B 534 (1998) 356.
[21] A.A. Penin, A.A. Pivovarov, Phys. Lett. B 435 (1998) 413; Nucl. Phys. B 549 (1999) 217; Nucl.
Phys. B 550 (1999) 375.
[22] A.H. Hoang, Phys. Rev. D 59 (1999) 014039; Phys. Rev. D 61 (2000) 034005.
[23] M. Beneke, A. Signer, V.A. Smirnov, Phys. Lett. B 454 (1999) 137.
[24] O. Yakovlev, Phys. Lett. B 457 (1999) 170.
[25] T. Nagano, A. Ota, Y. Sumino, Phys. Rev. D 60 (1999) 114014.
[26] A.A. Penin, A.A. Pivovarov, Preprint No. MZ-TH/98-61, hep-ph/9904278, December 1998, to
appear in Sov. J. Nucl. Phys. [Yad. Fiz.].
[27] M. Beneke, A. Signer, Phys. Lett. B 471 (1999) 233.
[28] M. Beneke, Preprint No. CERN-TH/99-355, hep-ph/9911490, November 1999.
[29] A. Pineda, J. Soto, Phys. Rev. D 58 (1998) 114011.
[30] B.A. Kniehl, A.A. Penin, Nucl. Phys. B 563 (1999) 200.
[31] N. Brambilla, A. Pineda, J. Soto, A. Vairo, Phys. Lett. B 470 (1999) 215.
[32] S.N. Gupta, S.F. Radford, Phys. Rev. D 24 (1981) 2309; Phys. Rev. D 25 (1982) 3430.
[33] S.N. Gupta, S.F. Radford, W.W. Repko, Phys. Rev. D 26 (1982) 3305.
[34] S. Titard, F.J. Yndurain, Phys. Rev. D 49 (1994) 6007.
[35] S.G. Karshenboim, Sov. Phys. JETP 76 (1993) 541 [Zh. Eksp. Teor. Fiz. 103 (1993) 1105].
[36] T. Appelquist, M. Dine, I.J. Muzinich, Phys. Rev. D 17 (1978) 2074.
[37] N. Brambilla, A. Pineda, J. Soto, A. Vairo, Phys. Rev. D 60 (1999) 091502; Nucl. Phys. B 566
(2000) 275.
[38] W.E. Caswell, G.P. Lepage, Phys. Rev. A 20 (1979) 36.
[39] A.S. Elkhovsky, A.I. Milstein, I.B. Khriplovich, Sov. Phys. JETP 75 (1992) 954 [Zh. Eksp.
Teor. Fiz. 102 (1992) 1768].
[40] K. Melnikov, A. Yelkhovsky, Phys. Lett. B 458 (1999) 143.
[41] V.S. Fadin, V.A. Khoze, JETP Lett. 46 (1987) 525 [Pisma Zh. Eksp. Teor. Fiz. 46 (1987) 417];
Sov. J. Nucl. Phys. 48 (1988) 309 [Yad. Fiz. 48 (1988) 487].
[42] V.A. Novikov, L.B. Okun, M.A. Shifman, A.I. Vainshtein, M.B. Voloshin, V.I. Zakharov, Phys.
Rev. Lett. 38 (1977) 626; Phys. Rev. Lett. 38 (1977) 791 (Erratum); Phys. Rep. C 41 (1978) 1.
208
[43] M.B. Voloshin, Sov. J. Nucl. Phys. 36 (1982) 143 [Yad. Fiz. 36 (1982) 247].
[44] M.B. Voloshin, Yu.M. Zaitsev, Sov. Phys. Usp. 30 (1987) 553 [Usp. Fiz. Nauk 152 (1987) 361].
Abstract
For moments of leptoproduction structure functions we show that all dependence on the
renormalization and factorization scales disappears provided that all the ultraviolet logarithms
involving the physical energy scale Q are completely resummed. The approach is closely related
to Grunbergs method of effective charges. A direct and simple method for extracting MS from
experimental data is advocated. 2000 Elsevier Science B.V. All rights reserved.
1. Introduction
The problem of renormalization scheme dependence in QCD perturbation theory
remains on obstacle to making precise tests of the theory. In a recent paper [1] one of
us pointed out that the renormalization scale dependence of dimensionless physical QCD
observables, depending on a single energy scale Q, can be avoided provided that all
ultraviolet logarithms which build the physical energy dependence on Q are resummed.
This was termed complete Renormalization Group (RG)-improvement in Ref. [1]. It
was stressed that standard RG-improvement, as customarily applied with a Q-dependent
scale = xQ, omits an infinite subset of these logarithms. One should rather keep
independent of Q, and then carefully resum to all-orders the RG-predictable ultraviolet
logarithms. In this way all -dependence cancels between the renormalized coupling and
the logarithms of contained in the coefficients, and the correct physical Q-dependence
is built. At next-to-leading order (NLO) the result is identical to the Effective Charge
approach of Grunberg [2,3]. We wish to extend this argument to processes involving
factorization of operator matrix elements and coefficient functions, where a factorization
1 c.j.maxwell@durham.ac.uk
2 abolfazl.mirjalili@durham.ac.uk
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 8 4 - X
210
scale M arises in addition to the renormalization scale . We shall use the prototypical
factorization problem of moments of leptoproduction structure functions as a specific
example. We shall identify the logarithms of , M, and Q which occur, and will show
explicitly that on resumming all the ultraviolet logarithms the and M dependence
disappears. We shall organize the paper so that we review the treatment of Ref. [1]
whilst showing how it generalizes for the moment problem. We begin in Section 2
by giving some basic definitions for the moments of structure functions. Section 3
considers the dependence of the perturbative coefficients on the parameters which label
the renormalization procedure in both cases. Section 4 deals with the complete RGimprovement of the structure function moments and identifies and resums the physical
ultraviolet logarithms. Finally, in Section 5 we discuss a more straightforward way of
motivating this approach, and consider how to directly extract MS from data. We also
give our conclusions in Section 5.
x n2 F (x) dx,
(1)
Mn (Q) = On (M) Cn Q, a(), , M .
(2)
Here M is an arbitrary factorization scale and a() is the RG-improved coupling s ()/
defined at a renormalization scale . The operator matrix element hOn (M)i has an Mdependence given by its anomalous dimension,
M hOi
= O (a) = da d1 a 2 d2 a 3 d3 a 4 + .
(3)
hOi M
For simplicity we shall from now on suppress the n-dependence of terms in equations,
as we have done in Eq. (3). For a given moment d is independent of the factorization
convention, whereas the higher di (i > 1) depend on it. In Eq. (3) the coupling a is
governed by the -function equation
a
= (a) = ba 2 1 + ca + c2 a 2 + c3 a 3 + .
(4)
M
M
Here b = (33 2Nf )/6, and c = (153 19Nf )/12b, are the first two coefficients of
the beta-function for SU(3) QCD with Nf active flavours of quark. They are universal,
whereas the subsequent coefficients c2 , c3 , . . . are scheme-dependent. Eq. (3) can be
integrated to [4,5]
#
"Za
Z (1)
(x)
(x)
dx
dx ,
(5)
O(M) = A exp
(x)
(2) (x)
0
211
where (1) and (2) denote these functions truncated at one and two terms, respectively.
The factor A is scheme-independent [5] and can be fitted to experimental data. The second
integral in Eq. (5) is an infinite constant of integration. In Eq. (2) C(Q, a(), , M) is the
coefficient function and has the perturbation series
C(Q, a,
, M) = 1 + r1 a + r2 a 2 + r3 a 3 + .
(6)
We shall use a to stand for a() and a for a(M). After combining the integrals in Eq. (5)
one obtains
d/b
ca
exp I(a) 1 + r1 a + r2 a 2 + r3 a 3 + ,
(7)
M=A
1 + ca
where I(a) is the finite integral
Za
I(a) =
dx
0
(8)
which can be readily evaluated numerically. The coupling a( ) itself, where b ln(/),
is obtained as the solution of the transcendental equation [6]
ca
1
+ c ln
=
a
1 + ca
where
B(x) x 2 (1 + cx
Za
0
1
1
+ 2
,
dx
B(x) x (1 + cx)
(9)
+ c2 x 2 + c3 x 3 + ).
(10)
(11)
r2
= 1,
c2
r2
= 0,
c3
...,
(12)
212
(13)
(14)
= r1 b,
= 2r2 b + r1 bc,
r2
r1
= d,
M
= d1 + dr1 dL,
M
M
M
r3
= d2 + d1 r1 + dr2 dr1 L 2d1 L dL2 ,
M
M
r2
L r1
1
c
r1
,
= ,
=
d1
b
d1 2b b
b
c2
(c r1 )
r2
c2
L2
cr1
r3
+
L +
,
=
d1
2b
3b
b
b
3b
b
r3
L
r1
r2
1
c
r1
,
= 0,
= ,
=
d2
d2
2b
d2 3b
b
2b
r1
r2
r3
1
= 0,
= 0,
= ,
d3
d3
d3
3b
r3
dL
dr1
r2
3d
4d1
cd
r1
,
+3
+3
r1 5 ,
= 0,
=
=
c2
c2
2b
c2
3b
b
2b
3b
r1
r2
r3
5d
.
(15)
= 0,
= 0,
=
c3
c3
c3 6b
Here we have defined for convenience L b ln(M/). Consistently integrating the partial
derivatives of r1 yields
d
d1
X1 (Q),
(16)
r1 = M
b
b
213
r12 + r1 r1 +
+ X2 ,
r2 =
2 2d
d
2b
2b
2b
2
2
b
b
3b 1 r13
b 2
bc 2bd1
+
+
r
+
+
r
+
r1 r1
r3 =
1
d
d 2 2d 2 3
d2 d 1
d2
d12
d2 d2
bc bd1 d1 2
dc2 cd1
r +
+
+ X2 +
+
c2 r1
+ 2 +
2d
d 1
2b
2b
2db d
2b
d
2
d
d2 cd1 2bX2
b2
d1 c2 2d1X2
+
+
r1 + 2 r1 r12 +
+
+ 12
d
d
d
3b
d
d
d
3
2
cd
d
d3
dc3 2d1 c2 d2 c
dcc2
+ 1 +
+
+ X3
+ 12 +
3bd
3b
2db 3d
6b
3b
3b
..
.
(18)
analogous to Eqs. (13) in the single scale case. Notice that we could equally use r1 and L
as parameters instead of r1 and r1 , since L = (b/d)(r1 r1 ). As in the single scale case
there are constants of integration Xn representing the RG-unpredictable part of rn . They
are Q-independent and FRS-invariant.
In the single scale case parametrizing the RS-dependence using r1 , c2 , c3 , . . . means that
given a complete Nn LO calculation X2 , X3 , . . . , Xn will be known. Using Eqs. (13) to
sum to all-orders the RG-predictable terms, i.e., those not involving Xn+1 , Xn+2 , . . . , with
coupling a(r1 , c2 , c3 , . . .) is equivalent to Nn LO perturbation theory in the scheme with
r1 = c2 = c3 = = 0, and yields the sum
R(n) = a0 + X2 a0 2 + X3 a0 3 + + Xn a0 n ,
(19)
where a0 a(0, 0, 0, . . .) is the coupling in this scheme. From Eqs. (9) and (11) it satisfies
ca0
Q
1
+ c ln
= b ln
.
(20)
a0
1 + ca0
R
In fact the solution of this transcendental equation can be written in closed form in terms
of the Lambert W -function [9,10], defined implicitly by W (z) exp(W (z)) = z,
1 Q b/c
1
,
z(Q)
.
(21)
a0 =
c[1 + W (z(Q))]
e R
A similar expansion to Eq. (19), but motivated differently, has been suggested in Ref. [11].
214
In the moment problem by an exactly similar argument, with the chosen parametrization
of FRS, given a complete Nn LO calculation (i.e., a calculation of r1 , r2 , . . . , rn and the
d1 , d2 , . . . , dn and c2 , c3 , . . . , cn in some FRS) the invariants X2 , X3 , . . . , Xn will be
known. Using Eqs. (18) to sum to all-orders the RG-predictable terms not involving
Xn+1 , Xn+2 , . . . , will be equivalent to working with an FRS in which all the FRS
parameters are set to zero. r1 = 0 means that = M. Setting r1 = 0, d1 = 0 in Eq. (17)
yields M = b ln(Q/M ), so that a = a = a0 , given by Eq. (21) with R replaced
by M . Further, with ci = di = 0 the integral I(a) in Eq. (8) vanishes, so that finally
the sum of all RG-predictable terms for the moment problem at Nn LO will be
d/b
ca0
1 + X2 a02 + X3 a03 + + Xn a0n ,
(22)
M=A
1 + ca0
with an extremely similar structure to the single scale case in Eq. (19). Substituting for a0
in terms of the Lambert W -function using Eq. (21) we then obtain
b/d
1 + X2 a02 + ,
M = A W z(Q)
1 Q b/c
.
z(Q)
e M
(23)
b
cR
M=A
b
1 + cR
215
d/b
.
(24)
(25)
This is similar to Grunbergs proposal [3] to associate an effective charge with M so that
b d/b . The ri are built from the ci , di , M and , and are RS-dependent, but
M = A(cR)
b can be RG-improved as in the single scale case. We have,
FS-independent. Effectively R
for instance,
b
d1
(26)
= X1 (Q),
r1 = b ln / b ln M/ r1 +
d
d
b as a single
where we have used Eq. (17). Comparing with Eq. (11) we see that treating R
scale problem we have 0 (Q) = X1 (Q). This further implies that R
b = M and so the
b will be of the
corresponding CORGI couplings are identical. The CORGI expansion for R
form
b = a0 + X
b2 a02 + X
b3 a 3 + .
R
0
(27)
Inserting this result in Eq. (24) and reexpanding in a0 will reproduce the CORGI expansion
in Eq. (22).
4. Complete RG-improvement
In the single scale case using Eq. (11) one can write
Q
.
r1 = b ln ln
R
(28)
The first -dependent logarithm depends on the RS, whereas the second Q-dependent UV
logarithm will generate the physical Q-dependence and is RS-invariant. If one makes the
simplification that c = 0 and sets c2 = c3 = = 0, then the coupling is given by
a() =
b ln(/)
(29)
The sum to all-orders of the RG-predictable terms from Eqs. (13), given a NLO calculation
of r1 , simplifies to a geometric progression,
R = a + r1 a + r1 2 a 3 + + r1 n a n+1 + .
(30)
216
= xQ, but then the resulting Q-dependence involves the arbitrary parameter x. In
contrast using Eqs. (28), (29) and summing the geometric progression in Eq. (30) gives,
1
Q
1
,
=
a()
R(Q) a() 1 b ln b ln
R
b ln(Q/R )
(31)
1
+ O 1/[b ln(Q/R )]3 .
b ln(Q/R )
(32)
M
Given a NLO calculation of r1 we wish to see how the physical Q-dependence of M(Q)
arises on resumming to all-orders the UV logarithms contained in the RG-predictable terms
from Eqs. (18). If we make similar approximations, so that c = 0 and the di and ci are set
to zero, then
M = A ca(M)
d/b
1 + r1 a() + r2 a()2 + .
(34)
We retain the overall factor of cd/b . The task is then to show that on resumming the
and ln(/)
d/b
1
b ln(Q/M )
d/b h
1 + O 1/ ln(Q/M )
2 i
(35)
Again, the complete RG-improvement summing over all UV logarithms is forced on one
if and M are held independent of Q.
The algebraic structure of the resummation of RG-predictable terms for the moment
problem is considerably more complicated than the geometric progression of Eq. (30)
encountered in the single scale case. With the simplifications c = 0, ci = 0, di = 0 the
first two RG-predictable coefficients from Eqs. (18) are
b
1
b
r 2 + r1 r1 ,
2 2d 1 d
2
2
b
b
3b 1 r13
b 2
b2
+
+
+
r
+
r1 r12 .
r3 =
1
1
d 2 2d 2 3
d2 d
d2
r2 =
(36)
(37)
Suitably generalizing the partial derivatives in Eqs. (15) one can arrive at a general form
for the RG-predictable terms. It is useful to arrange them in columns,
r1
r2
r
3
r4
r
5
0
b
d r1 r1 a
1 2
b
d r1 r1 a
2 3
b
d r1 r1 a
3 4
b
d r1 r1 a
4 5
b
d r1 r1 a
..
.
0
2
b r1
1 d 2 a 2
r2
2 db r1 1 db 21 a 3
r2
3 db r1 )2 1 db 21 a 4
3
r2
4 db r1 1 db 21 a 5
..
.
0
0
b r13 3
a
1 db 12
d 3 3
r
3 db r1 1 db 12 db 31 a 4
2
r3
6 db r1 1 db 12 db 31 a 5
..
.
217
...
...
...
(38)
...
...
..
.
The idea will be to resum each column separately. Denoting the sum of the mth column by
Sm , one finds
2
3
4
b
b
b
b
2
3
4
r1 r1 a +
r1 r1 a +
r1 r1 a +
r1 r1 a 5 +
S1 = r1 a +
d
d
d
d
2
3
4
b
b
b
b
r1 a +
r1 a +
r1 a +
r1 a +
= r1 a 1 +
d
d
d
d
1
b
.
(39)
= r1 a 1 r1 a
d
Careful examination of the pattern of terms in Eq. (38) leads to the general result for Sm
for m > 1,
m
S1
b 1
b 1
b
1
b
1
+ +
.
(40)
Sm = (1)2m1
d
d 2
d 3
d m1 m
Finally the resummed RG-predictable terms in the coefficient function will follow
from C = 1 + S1 + S2 + S3 + + Sn + . Introducing for convenience x S1 =
1
db r1 a)
, we find
r1 a(1
2
x
b
b 1 x3
b
1
+
1
C =1+x
d
2
d
d 2 3
b 1
b 1 x4
b
1
d
d 2
d 3 4
2
3
bx
d
bx
1 d d
1 d d
d bx
1
1
2
+
+
+
=1+
b d
2! b b
d
3! b b
b
d
b d/b
.
(41)
= 1+ x
d
Substituting for x yields
1 d/b
1 db r1 a + db r1 a d/b
b
b
=
.
r1 a 1 r1 a
C = 1+
d
d
1 db r1 a
We can write the numerator in Eq. (42) as
(42)
218
r1 r1
b
b
= 1 + aL,
1 r1 a + r1 a = 1 + ab
d
d
d
(43)
a
b
a
=
.
(44)
C = 1 r1 a
d
a
a
Since a = a() = 1/ we can rearrange Eq. (16) to obtain
Q
d 1
d ln
,
r1 =
b a
M
and substituting this result into Eq. (44) we find
d/b
1
a d/b .
C=
b ln(Q/M )
(45)
(46)
Combining this with the anomalous dimension part (ca)d/b we reproduce the physical Qdependence of M(Q) in Eq. (35).
5. Discussion and conclusions
An alternative and more straightforward way of understanding the CORGI proposal is
as follows. Given a dimensionless observable R(Q), dependent on the single dimensionful
scale Q, we clearly must have, on grounds of generalized dimensional analysis [12]
R(Q) = (/Q),
(47)
= 1 R(Q) ,
(48)
Q
where 1 is the inverse function. This is indeed the basic motivation for Grunbergs
Effective Charge approach [2,3]. We are assuming massless quarks here. The extension if
one includes masses has been discussed in [3,13]. The structure of 1 is [14,15]
(49)
F R(Q) G R(Q) = R /Q,
where
c/b
F R(Q) e1/(bR) 1 + 1/(bR)
(50)
(51)
which follows from Eq. (11), with r r1MS ( = Q) the NLO MS coefficient. Note that
r is Q-independent. The tilde over reflects the convention assumed in integrating the
219
beta-function equation to obtain Eq. (9) [6], and MS = (2c/b)c/b MS in terms of the
standard convention. The function G(R(Q)) has the expansion
X2
R(Q) + O R2 + .
(52)
G R(Q) = 1
b
Here X2 is the NNLO RS-invariant constant of integration which arises in Eqs. (13).
Assembling all this we finally obtain the desired inverse relation between R and , the
universal dimensional transmutation parameter of the theory
(53)
QF R(Q) G R(Q) er/b (2c/b)c/b = MS .
Notice that all dependence on the subtraction scheme chosen resides in the single
factor er/b , the remainder of the expression being independent of this choice. This
corresponds to the observation of Celmaster and Gonsalves [16], that s with different
subtraction conventions can be exactly related given a one-loop (NLO) calculation. If only
a NLO calculation has been performed G = 1 since X2 will be unknown, so that the best
one can do in reconstructing MS is
(54)
QF R(Q) er/b (2c/b)c/b = MS .
This is precisely the result obtained on inverting the NLO CORGI result R = a0 given by
Eq. (21).
The essential point is that the dimensional transmutation scale is the fundamental
object. In contrast the convention-dependent dimensionful scales and M are ultimately
irrelevant quantities which cancel out of physical predictions if one takes care to resum all
of the ultraviolet logarithms that build the physical Q-dependence in association with .
Our purpose has been to indicate that the unphysical and M dependence of conventional
fixed-order perturbation theory reflects its failure to resum all of these RG-predictable
terms. We have analyzed how Eq. (54) is built by explicitly resumming the conventiondependent logarithms together with the ultraviolet logarithms. Having done this, however,
one can simply use Eq. (53) to test perturbative QCD. Given at least a NLO calculation for
an observable R(Q) one simply substitutes the data values into Eq. (53), where G(R(Q))
can include NNLO and higher corrections if known, and obtains MS . To the extent that
remaining higher-order perturbative and possible power corrections are small, one should
find consistent values of MS for different observables. There is no need to mention
or M in this analysis, let alone to vary them over an ad hoc range of values. For the moment
problem the result corresponding to Eq. (53) is
M r /b
(2c/b)c/b = MS ,
(55)
QF M
A G A e
where r r1MS ( = Q) is defined in Eq. (26). The modified functions F and G are most
b in Eq. (24) is directly related to M/A and also satisfies
easily obtained by noting that R
Eq. (53). One finds
c/b
F (x) = exp bc 1 x b/d 1 + bc x b/d 1
,
b/d
x
X2
+ .
(56)
G(x) = 1
d c(1 x b/d )
220
Where X2 is the NNLO FRS-invariant which arises in Eqs. (18). The scheme-independent
parameter A reflects a physical property of the operator On in Eq. (2). An and MS should
be fitted simultaneously to the data for Mn (Q) using Eq. (55).
We hope to report direct fits of data to MS as outlined above, for both e+ e jet
observables [17] and structure functions and their moments [18], in future publications.
Acknowledgements
A.M. acknowledges the financial support of the Iranian government and also thanks
S.J. Burby for useful discussions.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
Abstract
We derive an evolution equation describing the high energy behavior of the cross section for the
single diffractive dissociation in deep inelastic scattering on a hadron or a nucleus. The evolution
equation resums multiple BFKL pomeron exchanges contributing to the cross section of the events
with large rapidity gaps. Analyzing the properties of an approximate solution of the proposed
equation we point out that at very high energies there is a possibility that for a fixed center of mass
energy the cross section will reach a local maximum at a certain intermediate size of the rapidity
gap, or, equivalently, at some non-zero value of the invariant mass of the produced particles. 2000
Elsevier Science B.V. All rights reserved.
PACS: 12.38.-t; 12.38.Cy; 24.85.+p
Keywords: Diffractive dissociation; BFKL pomeron; Rapidity gaps
1. Introduction
Some time ago an evolution equation was derived in the framework of Muellers dipole
model [14] which resums multiple Balitsky, Fadin, Kuraev and Lipatov (BFKL) [5,6]
pomeron exchanges for deep inelastic scattering (DIS) in the leading ln 1/x approximation
[7]. In terms of the usual Feynman diagram formulation the equation in [7] sums up the socalled pomeron fan diagrams contributing to the total cross section of a quarkantiquark
pair on a hadron or nucleus (see Fig. 1), and turns out to be a generalization of the Gribov,
Levin and Ryskin (GLR) equation [8]. Similar nonlinear equation for multiple pomeron
exchanges was also obtained by Balitsky [9] in the effective Lagrangian approach.
The physical picture of DIS presented in [7] is the following: virtual photon splits into a
quarkantiquark pair, which, by the time it reaches the target develops a cascade of dipoles,
Permanent address.
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 2 5 - 5
222
Fig. 1. An example of the pomeron fan diagram, which were considered in deriving (2).
each of which independently interacts with the target. All the QCD evolution was included
in the developed cascade of the dipoles in the large Nc limit, similar to [1,3].
The equation in [7] was written for the elastic amplitude of the scattering of the qq pair
of transverse size x located at impact parameter b on a hadron or nucleus, N0 (x, b, Y ),
where Y ln 1/x is the rapidity variable of the slowest particle in the qq pair. The targets
structure function F2 , as well as the total DIS cross section, could be easily expressed in
terms of N0 (x, b, Y ):
Z 2
Q2
d x01 dz
(x01 , z) d 2 b 2N0 (x01 , b, Y ),
(1)
F2 (x, Q2 ) =
4
4 2 EM
where the quark and antiquark are located at transverse coordinates x0 and x1 correspondingly and x01 = x1 x0 . Q2 is the virtuality of the photon, z is the fraction of the pairs
momentum carried by a quark. (x01 , z) is the probability of virtual photon fluctuating
into a q q pair, which we will refer to as virtual photons wave function. The exact expression for (x01 , z) is well known and is given in [7] as well as in a number of other
references.
In [7] the following equation was written for N0 (x, b, Y ):
x01
4CF
ln
Y
N0 (x01 , b, Y ) = (x01 , b) exp
ZY
x01
4CF
CF
ln
dy exp
(Y y)
+ 2
2
x01
2N0 x02 , b + 12 x12 , y
2
2
x02 x12
N0 x02 , b + 12 x12 , y N0 x12 , b + 12 x02, y ,
d 2 x2
(2)
where the initial condition (x01 , b) was given by the GlauberMueller interaction formula
of the quarkantiquark pair with the nucleus
2 2
(x01 , b0 ) = 1 exp
x01 AxG x, 1/x201 .
2Nc S
223
(3)
Here, for a cylindrical nucleus, S = R 2 is the transverse area of the hadron or nucleus,
A is the atomic number of the nucleus, and xG is the gluon distribution in a nucleon in
the nucleus, which was taken at the two gluon (lowest in ) level [10]. In Eq. (2) is an
ultraviolet regulator, which never appears in N0 (x01 , b, Y ).
There have been several attempts to solve Eq. (2) analytically [11,14]. Unfortunately it
turned out to be very hard to construct an analytical solution describing simultaneously the
behavior of N0 (x01 , b, Y ) both outside and inside of the saturation region. Instead Eq. (2)
was solved separately outside [k > Qs (Y )] and inside [k < Qs (Y )] the saturation region
[11,14], where approximately [11]
exp[(P 1)Y ]
,
Qs (Y ) 2 A1/3
14Nc (3)Y
(4)
F
with P 1 = 4C
ln 2. In [11,14] it was concluded that for not very large energies,
corresponding to rapidities of the order of Y 1/ (or, equivalently, for k > Qs (Y )),
the solution of Eq. (2) behaves with energy like a single BFKL pomeron exchange, with
multiple pomeron corrections setting in as energy increases and slowing down the growth
of N0 (x01, b, Y ). For very high energies (Y > P11 ln 12 or k < Qs (Y )) the solution
of Eq. (2) saturates to a constant, namely N0 (x01, b, Y ) = 1, which corresponds to the
blackness of the total cross section of the quarkantiquark pair on the nucleus [11,14].
Thus, as one can see at least the qualitative features of N0 (x , b, Y ) as a function of Y and
x are known. For a better quantitative understanding of the behavior of N0 (x , b, Y ) one
should, probably, solve Eq. (2) numerically, which would be a very interesting and useful
project to perform.
In [7] the object of interest was the total inclusive DIS cross section, where no
restrictions are imposed on the final state of the process. In this paper we are going to
study the cross section of the single diffractive dissociation. The physical picture of the
process we are going to consider is the following: in DIS the virtual photon interacts with
the hadron or nucleus breaking up into hadrons and jets in the final state. At the same time
the target hadron (nucleus) remains intact. The particles produced as a result of hadrons
breakup do not fill the whole rapidity interval, leaving a rapidity gap between the target and
the slowest produced particle. The process is depicted in Fig. 2. In diffractive dissociation
one imposes a restriction on the final state the existence of a rapidity gap.
Below we are going to employ the techniques of Muellers dipole model [14], similarly
to the way they were applied in [7], to construct a cross section of the single diffractive
dissociation in DIS which would include the effects of multiple pomeron exchanges.
Analogous to [7,11] we will neglect the diagrams with pomeron loops, i.e., the diagrams
where a pomeron splits into two pomerons and then the two pomerons merge into one
pomeron. As was argued in [7,1113] these diagrams are suppressed compared to the
pomeron fan diagrams of Fig. 1 in DIS on a large nucleus by factors of A1/3 . Thus one
1
1
can safely neglect them up to rapidities of the order of Y >
P 1 ln 2 A1/3 . For the case
of DIS on a proton, strictly speaking there is no similar argument allowing one to neglect
224
Fig. 2. Single diffractive dissociation process considered in the paper. The interaction between the
target and virtual photon is represented schematically by an exchange of a color singlet object.
Fig. 3. Traditional description of single diffractive dissociation. Dash-dotted line represents the final
state.
pomeron loop diagrams. Therefore, whereas the fan diagram evolution dominates in DIS
on a nucleus, it could only be considered as a model for DIS on a hadron.
In the traditional description of the single diffractive dissociation one usually considers
triple pomeron vertex [2,1518], where the pomeron above the vertex is cut, and the two
pomerons below the vertex are on different sides of the cut, thus producing a rapidity gap,
as shown in Fig. 3. That way the particles are produced by the cut pomeron in the rapidity
interval adjacent to the virtual photons fragmentation region and no particles are produced
with rapidities close to the final state of the target due to uncut pomerons. In this paper we
want to enhance this picture by including multiple pomeron exchange diagrams. We would
like to understand the behavior of the diffractive dissociation at very high energies, which
may include some effects of saturation of hadronic or nuclear structure functions.
An example of a graph in the usual Feynman diagram language which we will consider
225
Fig. 4. A diagram contributing to the diffractive dissociation with rapidity gap Y0 as considered in
the text.
below is given in Fig. 4. Dash-dotted line corresponds to the final state, i.e., to the cut.
As one can see in Fig. 4 some of the pomerons are cut, some remain uncut. In the region
of rapidity where pomerons are cut we have particles being produced. In the notation of
Fig. 4 this corresponds to the interval in rapidity from Y Y0 to Y . In the region where the
pomerons are uncut (rapidity interval from 0 to Y0 ) nothing is produced, which corresponds
to a rapidity gap. That way Fig. 4 demonstrates a generalization of the traditional picture
of Fig. 3, in which the cut pomeron splits into two pomerons, which later branch into
two uncut pomerons each. Our goal in this paper is to resum all the diagrams where the
cut pomeron can split into any number of cut pomerons via fan diagrams, and the cut
pomerons in turn split into uncut pomerons, which can also branch into any number of
uncut pomerons interacting with the target below.
In Section 2 we employ the dipole model formalism [14] to write down an equation
for the cross section of the single diffractive dissociation of a quarkantiquark pair on
a hadron or nucleus with the rapidity gap bigger or equal to Y0 . The task is not very
straightforward, since the dipole model provides us with the dipole wave function of
the q q pair by the time it hits the target. In the rest frame of the target the typical time
scale associated with the interaction is negligibly small compared to the typical lifetime
of the QCD evolved wavefunction of the pair. Thus one could say that the interaction is
instantaneous and happens at the time t = 0 [19]. However, after the interaction, the q q
state with many gluons in it can undergo several transformation until all the partons in it
reach the mass shell at t = . Dipole model does not provide us with any information
about these final state interactions. Nevertheless one can gain control over the final state
through the unitarity condition and cancelation of certain classes of final state interactions,
as will be shown below. As a result we obtain a non-linear evolution equation for the
diffractive cross section, shown in formula (9).
We show how Eq. (9) can be rederived by applying the AGK cutting rules to the pomeron
fan diagrams in Section 3. Since the cut or uncut pomerons precisely specify the final state
226
in the traditional language, we, therefore, proved the consistency of our treatment of the
final states in the dipole model.
Even though the exact analytical solution of that equation seems to be very complicated
in Section 4 we analyze the properties of the solution. As a toy model we consider a
simplified version of the equation with the kernel independent of transverse coordinates.
We show that the simplified equations solution exhibits a remarkable property at very
high energies. We plot the cross section of diffractive dissociation events with rapidity gap
Y0 at fixed large center of mass energy as a function of the size of the rapidity gap Y0 .
This is equivalent to plotting the cross section as a function of the invariant mass of the
produced particles MX2 , since, as could be easily seen from Fig. 2, Y0 = Y ln MX2 /Q2 . At
not very large Y0 (large MX2 ) the cross section increases with the increase of rapidity gap,
which agrees with the result of the traditional triple pomeron description of the diffractive
dissociation of [2,15]. However, as Y0 gets very high comparable to the values of rapidity
at saturation, which corresponds to a very small produced mass MX2 , the cross section
reaches a maximum and starts decreasing. That way if one would be able to measure the
single diffractive cross section for a range of different invariant masses of the produced
particles, one should be able to observe the maximum and turnover of the cross section for
small masses if the energy is high enough for the saturation effects to become important.
If the exact, probably numerical, solution of the equation confirms our conclusion, which
was reached by approximate methods, then the single diffractive cross section is a new and
independent observable which can be used to test whether the saturation region has been
reached in DIS experiments.
The summary of our results is presented in Section 5.
227
Fig. 5. An example of elastic scattering of the quarkantiquark pair on the nucleus as pictured in the
text.
(5a)
with S(b) the S-matrix of the scattering process at a fixed impact parameter b, then the
elastic cross section is [20,22,23]
Z
(5b)
el = d 2 b [1 S(b)]2 .
Using Eqs. (5a), (5b) it was shown in [11,20] that the elastic cross section for the scattering
of the quarkantiquark pair generated by a virtual photon in DIS on a nucleus or hadron
could be expressed in terms of the elastic amplitude N0 (x01 , b, Y ), or, equivalently, in
terms of the imaginary part of the forward scattering amplitude N0 (x01 , b, Y ) d 2 b. (The
forward amplitude is purely imaginary.) The result was [11,20,21]
N el (x01, b, Y ) = N0 (x01, b, Y )2
(6)
and is illustrated in Fig. 5. The cut in Fig. 5 corresponds to the final state, when the particles
reached the mass shell and t = . Thus in that picture we explicitly impose the constraint
that not only the q q system developed a dipole cascade and the cascade in turn interacted
with the target at t = 0, but it also recombined back into a q q pair by the time the system
reached the final state at t = . Eq. (6) allows us to calculate the elastic cross section of
the scattering of any (not necessarily original) dipole in the dipole cascade on the nucleus
including multiple pomeron evolution.
The second observation which we have to make before starting to derive our equation
is based on the work of Chen and Mueller [4] and concerns the so-called final state
interactions. By that we mean the interactions such as branching or recombination of
partons in the dipole cascade after t = 0, i.e., after the interaction with the target. Chen
and Mueller [4] succeeded in proving that certain classes of final state interactions cancel
(see Fig. 6).
We denote a gluon by the double line in Fig. 6 corresponding to the large Nc limit. In
each diagram of Fig. 6 we assume that the summation over all possible connections of the
gluon to the quark and antiquark lines is performed. In Fig. 6A we consider the situation
228
when the gluon is emitted in a dipole after the interaction with the target at t = 0, which is
denoted by dotted line. The three different positions of the t = final state cut, denoted
by dash-dotted line in Fig. 6A correspond to the cases when the emitted gluon is present in
the final state and when the gluon recombines back in the amplitude or complex conjugate
amplitude leaving the dipole intact by t = . As was proved in [4] the sum of the three
cuts of Fig. 6A is zero.
In Fig. 6B we consider the situation when the gluon was developed in the dipole wave
function before the interaction with the target and is already there by the time t = 0. Then it
can either remain in the final state t = or it can recombine back, leaving only the original
dipole in which it was produced in the final state. In Fig. 6B we explore the case when the
gluon in the final state becomes a gluon emitted after t = 0 in the complex conjugate
amplitude (first diagram in Fig. 6B). Chen and Mueller [4] showed that this diagram is
canceled by the second diagram in Fig. 6B.
Diagrams of Fig. 6 demonstrate that the contribution of gluons produced or absorbed
after t = 0 cancels out in calculation of inclusive quantities such as the total cross section.
We can now conclude that the total inclusive DIS cross section can be calculated using
just t = 0 formalism, as was done in [7]. Really, as one can see from Fig. 6 all the final
state interactions cancel, effectively leaving the t = 0 state unchanged by t = . This
conclusion is intuitively easy to understand: one just needs to have some interaction of the
projectile with the target to include the diagram in the total cross section, independent of
what happened after that interaction.
Now we are ready to write down our equation for diffractive cross section. Let us start by
defining the object for which the equation will be written. We denote by N D (x , b, Y, Y0 )
the cross section of single diffractive dissociation of a dipole of transverse size x , rapidity
Y and impact parameter b on a target hadron or nucleus. The process has a rapidity gap
covering a rapidity interval adjacent to the target (see Figs. 2, 4) which could be greater or
equal than Y0 . The corresponding diffractive structure function, similarly to Eq. (1), is
Z 2
Q2
d x01 dz
(x01 , z) d 2 bN D (x01, b, Y, Y0 ).
(7)
F2SD (x, Q2 , Y0 ) =
2
4 EM
4
229
If one wishes to obtain a cross section of diffractive dissociation of the dipole with a fixed
rapidity gap Y0 one has to differentiate N D with respect to Y0 , as will be discussed later.
When Y = Y0 the rapidity gap fills out the whole rapidity interval, turning the process of
dipoles dissociation into an elastic scattering. Taking the answer for elastic process from
Eq. (6) we obtain an initial condition for the evolution of N D
N D (x , b, Y = Y0 , Y0 ) = N02 (x , b, Y0 ).
(8)
The initial condition of Eq. (8) is illustrated in Fig. 5. At this point we have to
make a clarifying comment. One might think that multiple interactions are included
in N0 (x , b, Y0 ) only through the multiple pomerons which are produced via pomeron
splitting vertices as shown in Fig. 5. However, this is not correct. We should not forget
that each of the dipoles produced by our evolution interacts multiply with the nucleons
in the target nucleus. This interaction is given by the GlauberMueller formula of Eq. (3)
and is included in Eq. (8). One can see that this interaction is included in N D of Eq. (8)
by recalling the initial condition for N0 (x , b, Y0 ) at Y0 = 0 from Eq. (2). It follows that
N0 (x , b, Y0 = 0) = (x , b). Thus, in the absence of the small-x evolution (Y0 = 0) the
structure function F2D would be given by the multiple rescatterings of the original quark
antiquark pair involving only two gluon exchanges. The final state of the process would
consist of the intact target nucleus or proton and of the q q pair. The process like this is
called quasi-elastic scattering and has been considered before [20,21,2430]. Our analysis
here incorporates the contribution of this process.
Note that Eq. (8) insures that after the interaction with the target at t = 0 the system
of developed dipoles only recombines back into the original dipole by t = . All
other possible final state fluctuations have been excluded because we explicitly put
N02 (x , b, Y0 ) as the amplitude of the elastic process.
To construct N D one has to require that nothing is produced in the t = final state
in the rapidity interval from 0 to Y0 , where the target hadron or nucleus is situated at
rapidity 0. This does not restrain the rapidity gaps from being greater than Y0 . That way
N D would include all events with rapidity gap greater or equal to Y0 . The condition of the
rapidity gap up to Y0 is satisfied by Eq. (8).
As was mentioned above our physical picture of the diffractive dissociation event is
the following: before hitting the target the virtual photon develops a dipole cascade,
which at the time t = 0 interacts with the target. After that some partons in the cascade
may recombine, producing rapidity gaps, some may not. By imposing the recombination
condition in Eq. (8) we made sure that the dipoles with rapidities y < Y0 will recombine
by t = , producing a rapidity gap. The dipoles with y > Y0 are free to either recombine
into the dipoles off which they were produced or to remain unchanged till t = . As
long as y > Y0 there are no restrictions on the final state like rapidity gaps. Therefore
the dipoles are free to recombine after t = 0. However, for that situation the cancelation
rules proven by Chen and Mueller [4] apply. The graphs when the dipoles recombine to
either increase the size of the rapidity gap Y0 or to produce extra rapidity gaps at y > Y0
are included, but as one can see from Fig. 6B they cancel. For the asymmetric case when
the dipole in consideration interacts with the target in the amplitude and then recombines
230
back into the dipole off which it was produced which has no interaction with the target
in the complex conjugate amplitude we employ a different cancelation mechanism: the
contributions when the dipole recombines in the complex conjugate amplitude at the times
t > 0 and t < 0 cancel each other, leaving us with the diagram of similar to the second
graph of Fig. 6B, which could be incorporated in the elastic interaction of the original
dipole with the target N0 (x , b, y). That way the asymmetric graphs may include the
dipoles of any rapidities y > Y0 interacting elastically with the target via Eq. (8). For
the symmetric case when the dipole at y > Y0 interacts elastically with the target in
the amplitude and then recombines into the original dipole which interacts with the target
elastically in the complex conjugate amplitude we applied the cancelation of Fig. 6B. Thus
the symmetric case reduces to the situation when the same dipole interacts elastically in
the amplitude and complex conjugate amplitude at y = Y0 and is put only in the initial
condition of our equation given by N02 (x , b, Y0 ).
Dipole splitting is a little more complicated. The softest dipoles in the cascade interact
the way it is described in Eq. (8), which fixes the final state for them. Now let us consider
a dipole which is not one of these soft dipoles and, therefore, does not interact with the
target. The dipole can split into two dipoles either before or after t = 0. This event is
allowed only when the produced real gluon has the rapidity greater or equal than Y0 , so
that it would not destroy the rapidity gap. In that case the real gluon emitted before t = 0
is canceled by a virtual correction in the same dipole before t = 0. The real gluon emitted
after t = 0 is canceled by the virtual term after t = 0, similarly to Fig. 6A. Finally, this noninteracting dipole can develop a virtual corrections with the gluon being softer than Y0 , i.e.,
having smaller rapidity than the rapidity of the gap. These gluons could not be canceled
by the real contributions, since those are excluded by the condition of the gaps existence.
However, since the virtual corrections could happen either before or after t = 0, these
two combinations come with different signs. The absolute values of both t > 0 and t < 0
contributions in that case are the same and, therefore, they cancel each other. Thus we have
shown that the final state dipole splitting is canceled together with the virtual corrections
in the non-interacting dipoles.
The above discussion is summarized in Fig. 7. The change of the cross section of
diffractive dissociation N D in one step of the rapidity evolution could be generated by
231
several terms on the right hand side of the equation in Fig. 7. The first terms corresponds
to the usual single BFKL pomeron evolution, while the rest of the terms correspond to
the triple pomeron vertex, similarly to the equation written in [1,7]. Effectively Fig. 7
states that in one step of evolution the color dipole splits into two dipoles. The subsequent
evolution can happen in one of the dipoles, which corresponds to the first term on the
right hand side of Fig. 7. It could also happen that we continue the evolution leading to
the cross section of diffractive dissociation in both dipoles (the second terms in Fig. 7).
The third and forth terms correspond to the asymmetric cases. As was discussed above in
the asymmetric case the dipole can interact elastically with the target in the amplitude but
not in the complex conjugate amplitude and vice versa at any rapidity Y > Y0 . Thus we
include the cases when only one of the produced dipoles interacts asymmetrically with the
target (the third term in Fig. 7) and when both of them do so (the fourth term in Fig. 7).
The coefficients in front of different terms come from combinatorics. Note that they match
the coefficients of the Abramovskii, Gribov, Kancheli (AGK) [16] cutting rules for two
pomeron contributions.
Recalling the initial condition of Eq. (8) we can turn the picture of Fig. 7 into a formula.
For that we also have to include the virtual corrections similarly to the way they were
included in [1]. The resulting equation is
N D (x01 , b, Y, Y0 )
= N02 (x01 , b, Y0 ) e
Z
x2
d 2 x2 2 012
x02 x12
4CF
ln
x01
(Y Y0 )
CF
+ 2
ZY
dy e
4CF
ln
x01
(Y y)
Y0
2N D x02 , b + 12 x12 , y, Y0
+ N D x02, b + 12 x12 , y, Y0 N D x12 , b + 12 x02, y, Y0
4N D x02 , b + 12 x12, y, Y0 N0 x12 , b + 12 x02 , y
i
+ 2N0 x02 , b + 12 x12 , y N0 x12 , b + 12 x02, y .
(9)
The shifts in the impact parameter dependence of the functions on the right hand side of
Eq. (9) have occurred because the centers of mass of the produced dipoles are located at
different impact parameters than the center of mass of the initial dipole.
Eq. (9) describes the small-x evolution of the cross section of the single diffractive
dissociation for DIS N D and is the central result of this paper.
232
Fig. 8. Different pomeron cuts contributing to the cross section of diffractive dissociation which lead
to different terms on the right-hand side in Eq. (9).
pomeron vertices in the theory. This assumption is true only in the large Nc limit, as was
shown in the dipole model.
AGK cutting rules [16] apply to the BFKL pomerons in QCD (see [31] for a detailed
discussion). For the BFKL pomeron one can write
BFKL
= GBFKL
,
2 Im ael
in
(10)
BFKL is the one BFKL pomeron contribution to the forward scattering amplitude
where ael
BFKL
is the total inelastic cross section. In writing Eq. (10) we have neglected the
and Gin
elastic contribution to the cross section, which is a justified assumption for the BFKL
pomeron exchange [31]. Eq. (10) tells us that the inelastic cross section in one pomeron
exchange approximation is twice the forward amplitude of the process, since the latter is
purely imaginary.
Different cuts of pomeron fan diagrams corresponding to the terms on the right-hand side
of Eq. (9) are displayed in Fig. 8. The first (linear) term on the right-hand side of Eq. (9)
corresponds to just usual linear BFKL evolution with the factor of 2 arising from Eq. (10)
and is not shown in Fig. 8. The non-linear evolution corresponds to a pomeron splitting
into two via the triple pomeron vertices. Let us now concentrate on the first (topmost)
triple pomeron vertex in the diagrams of Fig. 8. This is where one step of non-linear
evolution takes place in our picture. The graph of Fig. 8A corresponds to the case when a
cut pomeron splits into two cut pomerons, which later on produce rapidity gaps. That is the
cross section of diffractive dissociation ND splits into two similar cross sections. Therefore
2 term in Eq. (9). The factor in front of this term results from
this term gives rise to ND
multiplying 2 from the cut pomeron on top of the vertex by the symmetry factor of 1/2. The
diagram of Fig. 8B corresponds to a cut pomeron splitting into another cut pomeron, which
then develops a rapidity gap (ND ), and an uncut pomeron, corresponding to the forward
scattering amplitude N0 . Since there are two of such diagrams, multiplying the factor of 2
of the cut pomeron by 2 and putting a minus sign since one of the resulting pomerons is
completely on one side of the cut we will get 4ND N0 , which is the contribution of that
diagram in Eq. (9). Finally the graph of Fig. 8C corresponds to the case when both of the
pomerons below the vertex are uncut. Each uncut pomeron gives a factor of N0 . The factor
of 2 arises from the cut pomeron above [Eq. (10)].
233
Therefore we were able to reproduce all the terms on the right hand side of Eq. (9) from
the usual pomeron fan diagrams using the AGK cutting rules. We note that a similar result
was obtained in the framework of reggeon theory in [32]. The cut (or uncut) pomeron
specifies t = final state very precisely: if the pomeron is cut it insures that there will
be particles produced in the final state. Uncut pomeron insures that nothing is produced
in the t = final state. Thus in this section we have demonstrated the consistency of our
analysis of the evolution of the dipole state after the interaction with the target which was
employed in the previous section.
2
(x
x
)
ln
=
2
01
02
2
2
2
x02 x12
CF
2
d 2 x2
2
h
x01
2 x2
x02
12
N D x02 , b + 12 x12 , Y, Y0 N D x12 , b + 12 x02, Y, Y0
4N D x02 , b + 12 x12, Y, Y0 N0 x12 , b + 12 x02 , Y
i
+ 2N0 x02 , b + 12 x12 , Y N0 x12 , b + 12 x02 , Y ,
(11)
with Eq. (8) being the initial condition for the differential equation (11).
Let us define the following object
F (x01 , b, Y, Y0 ) = 2N0 (x01, b, Y ) N D (x01 , b, Y, Y0 ) (Y Y0 ),
(12)
which has the meaning of the cross section of the events with rapidity gaps less than Y0 .
The theta-function that multiplies N D in Eq. (12) insures the trivial fact that the rapidity
gap can not be larger than the total rapidity interval. One can see that Eq. (11) can be
rewritten for Y0 6 Y in terms of F as
F (x01, b, Y, Y0 )
Y
2
Z
x01
x01
2CF
2
2
d
x
2
(x
x
)
ln
F x02 , b + 12 x12 , Y, Y0
=
2
01
02
2
2
2
x02 x12
CF
2
Z
d 2 x2
2
x01
2
2
x02 x12
F x02, b + 12 x12 , Y, Y0 F x12 , b + 12 x02 , Y, Y0 ,
(13)
234
(14)
If one differentiates Eq. (2) with respect to Y one would obtain the same differential
equation as Eq. (13) for N0
N0 (x01 , b, Y )
Y
2
Z
x01
x01
2CF
2
2
N0 x02, b + 12 x12 , Y
d
x
2
(x
x
)
ln
=
2
01
02
2
2
2
x02x12
CF
2
Z
d 2 x2
2
x01
2
2
x02 x12
N0 x02 , b + 12 x12 , Y N0 x12 , b + 12 x02 , Y .
(15)
The initial condition for Eq. (15) is different from that for Eq. (13)
N0 (x01 , b, Y = 0) = (x01 , b)
(16)
with (x01 , b) given by Eq. (3). From Eq. (12) one can see now that in order to find
N D (x01, b, Y, Y0 ) one has to solve the same equation [(13) or (15)] for two different initial
conditions given by formulae (14) and (16), which would yield us with the results for N0
and F . Then N D could be found using Eq. (12).
As was mentioned above the exact analytical solution of Eq. (13), or, equivalently, (15)
has not been found [11,14]. Nevertheless the qualitative behavior of the solution of this
equation is understood very well, with some quantitative estimates performed [11,14]. It
turned out that the solution of this equation can be qualitatively well approximated by the
solution of the equation with suppressed transverse coordinate dependence
N0 (Y )
= (P 1) N0 (Y ) (P 1) N0 (Y )2 .
(17)
Y
Eq. (17) is a toy model of the full Eq. (15), with all the transverse coordinate integrations
suppressed and the dipole BFKL kernel substituted by its eigenvalue at the BFKL saddle
point. The fact that the coefficients in front of the linear and quadratic terms in Eq. (17)
are identical could, for instance, be understood in the double logarithmic limit (x02, x12
x01 ), in which the kernel in front of the quadratic term in Eq. (15) produces an extra factor
of two and becomes equal to the kernel in front of the linear term [11,14]. However it is
a more general property of the solution of Eq. (15), which is valid not only in the double
logarithmic limit.
In this paper we will try to understand the qualitative behavior of the solution of Eq. (9)
using the toy model of Eq. (17). An exact, probably numerical, solution of Eq. (15)
would give more precise results for N D . However, we believe that the qualitative features
provided by the toy model of Eq. (17) will be preserved in the exact solution.
Let us suppose that N0 (Y ) is a solution of Eq. (15), or, in our toy model, of Eq. (17).
Then F (Y, Y0 ) will be a solution of the same equation with different initial condition set
at a different value of rapidity (see Eq. (14)). Thus, if we neglect the transverse coordinate
dependence, the solution for F (Y, Y0 ) could be obtained from the solution for N0 (Y ) just
by a shift in rapidity. Define the shift variable 1Y by
235
(18)
so that
F (Y, Y0 ) = N0 (Y + 1Y ).
(19)
Now one can see that since N0 (Y ) satisfies the differential equation (15) (or Eq. (17) in
our toy model), then F (Y, Y0 ) given by Eq. (19) satisfies the same equation (17) with the
initial condition of Eq. (14). Of course this is true only when one neglects the transverse
coordinate dependence of N0 and F , but could also be a reasonable approximation for the
case of weak transverse coordinate dependence.
Solving Eq. (17) with the initial condition N0 (Y = 0) = we obtain
e(P 1)Y
,
1 + (e(P 1)Y 1)
which employing Eq. (18) yields us with the following expression for 1Y
1
ln 2 +
e(P 1)Y0 .
1Y =
P 1
1
N0 (Y ) =
e(P 1)Y 2 + 1
e(P 1)Y0
.
F (Y, Y0 ) =
1 + e(P 1)Y 2 + 1
e(P 1)Y0
(20)
(21)
(22)
Now we can explore the qualitative behavior of our result. First we note that for not very
large rapidities, Y 1/, we can put denominators in Eqs. (20) and (22) to be equal to 1,
which through Eq. (12) provides us with
N D (Y, Y0 )
(P 1) 2 e(P 1)(Y +Y0 ) + (Y Y0 )N02 (Y0 ).
(23)
Y0
The first term on the right-hand side of Eq. (23) is the usual result of the lowest order
approach employing only one triple pomeron vertex [2,15] (see Fig. 3). The second (deltafunction) term in Eq. (23) corresponds to the contribution of the elastic cross section when
the rapidity gap fills the whole rapidity interval.
Remember that N D (Y, Y0 ) is the cross section of having a rapidity gap greater or equal
to Y0 . If one wants to obtain the cross section for a fixed size of the rapidity gap one has to
differentiate N D with respect to Y0 and take a negative of the result, since N D is obviously
a decreasing function of Y0 . This is what was done in arriving at Eq. (23). That way we have
verified that the toy model solution of our Eq. (9) maps onto the old and well established
triple pomeron vertex result [15].
At the lowest order 2 A1/3 . Thus the first term on the right-hand side of Eq. (23),
which is usually calculated in the perturbative approaches [15], is of the order of 5 1.
We note that using the non-linear evolution equation (2) we were able to obtain control over
the kinematic regions where the coupling is still small but the cross sections can become
very large, with N0 1. Thus the traditional perturbative calculation of Eq. (23) appears
like a small perturbation compared to our large cross section result of Eq. (20), which was
still obtained in the small coupling constant limit.
236
At very high energies, when Y , one can see from Eqs. (20), (22) and (12) that
N D (Y, Y0 ) 1,
Y .
(24)
This result is easy to understand. As we know from quantum mechanics at very high
energies, when the total cross section is black, only the elastic and inelastic contributions
to the cross section survive to give half of the total cross section each. The total cross
section becomes tot = 2R 2 , whereas totally inelastic and elastic cross sections will be
inel = el = R 2 . All the intermediate contributions with finite size rapidity gaps covering
fraction of the total rapidity interval go to zero. Since N D (Y, Y0 ) is the cross section of
having rapidity gap greater or equal than Y0 for any finite non-zero Y0 it includes the
contribution of the elastic cross section, which corresponds to the case when the rapidity
gap covers the whole rapidity interval. At the same time N D (Y, Y0 ) does not include the
totally inelastic contribution, when there is no rapidity gap at all. Thus as at Y only
the elastic piece in N D survives to give half of the total cross section, which corresponds
to N D 1 in our notation, where everything has to be multiplied by R 2 .
Let us define the cross section of single diffractive dissociation with the fixed rapidity
gap Y0 , which, for Y0 6 Y , in the light of the above discussion is
N D (x01 , b, Y, Y0 ) F (x01 , b, Y, Y0 )
=
Y0
Y0
2
+ (Y Y0 )N0 (x01, b, Y0 ).
R(x01 , b, Y, Y0 ) =
(25)
+ (Y Y0 )N02 (Y0 ).
(26)
Now one can see that the triple pomeron vertex result of Eq. (23) could be recovered from
the first term of Eq. (26) by putting the denominator to be 1 and neglecting with respect
to 1 in the numerator. Also the quantum mechanical result discussed above is illustrated
explicitly: for any finite Y0 the cross section of a fixed size gap R(Y, Y0 ) goes to zero as
Y , as could be derived from Eq. (26).
Finally, let us plot the cross section of diffractive dissociation R(Y, Y0 ) of Eq. (26) as a
function of the size of rapidity gap Y0 for a fixed center of mass rapidity Y excluding the
elastic cross section (delta-function) contribution. This is equivalent to plotting the cross
section as a function of the invariant mass of the particles produced in the virtual photons
decay, since Y0 = Y ln MX2 /Q2 (see Fig. 2). The plot is shown in Fig. 9. There we put
P 1 = 0.5, = 0.03, Y = 10 and we divided Y0 by Y on the horizontal axis.
Fig. 9 demonstrates that the cross section increases with the size of the rapidity gap
over most of the rapidity interval. This is what one would expect from the lowest order
formula (23). Also that implies that it is more advantageous to produce particles with
smaller invariant mass MX2 . However at some very large size of the rapidity gap the cross
section R(Y, Y0 ) reaches a maximum and starts decreasing. The maximum of R(Y, Y0 ) in
Eq. (26) is reached at the size of the rapidity gap
237
Y0max
2(1 ) (1 )2 (P 1)Y
1
ln
+
=
e
.
P 1
(27)
From Eq. (27) we can immediately conclude that if the target nucleus is very large so
that each of the dipoles undergoes many rescatterings making 1 the maximum will
be pushed all the way into the region of negative Y0 , making the cross section of Fig. 9
just a decreasing function of Y0 for all rapidities of interest. Thus, if the effects of multiple
rescatterings of individual dipoles become important then the qualitative behavior of the
cross section of diffractive dissociation will change completely. The effect of multiple
rescattering is to push the maximum of Eq. (27) towards the smaller values of rapidity,
which are easier to reach experimentally.
If the multiple dipole rescatterings are not very important we can just use the lowest
order (one rescattering) 2 A1/3 1 and consider two limits. At not very large values
of rapidity, Y 1/, when the single BFKL pomeron exchange is most important the
maximum can never be reached since
Y0max
1
1
1
ln 2 1/3
.
P 1 A
P 1
(28)
1
1
ln 2 1/3
P 1 A
(29)
238
reader that our results in this section are based on the toy model and a more elaborate
numerical analysis of Eq. (2) is needed to achieve the required level of rigor in these
conclusions. Also Eq. (2) resums fan diagrams, leaving out the pomeron loop contributions.
Even though pomeron loops are always suppressed by powers of A compared to fan
diagrams, they will become important at the rapidities of the order of Y P11 ln 12 .
This is still above the saturation energy of Eq. (29) and does not affect much the kinematic
region considered above. Nevertheless for real life nuclei the suppression of pomeron loop
diagrams is not that large and they may play an important role already at the rapidities of
interest.
5. Conclusions
In this paper we have derived the small-x evolution equation for the cross section of
single diffractive dissociation (9). It resums multiple BFKL pomeron exchange diagrams
contributing to the diffractive cross section. The equation is not derived in the framework
of any model of multiple exchanges: it was directly derived from QCD. We made use of
the fact that in DIS with large Q2 the strong coupling constant is small. We also employed
the leading logarithmic approximation resumming logarithms of Bjorken x [5,6]. Finally
we employed the large Nc limit of QCD at high energies Muellers dipole model [14].
In our approach we have resummed all possible contributions to the single diffractive
dissociation in DIS corresponding to the different final states. In the absence of the
QCD evolution the final state would consist of the original quarkantiquark pair and an
intact nucleus, which would correspond to the quasi-elastic scattering case [20,21,2428].
The evolution of Eq. (9) would add one or more gluons to the final state. The gluons
will be treated in the color dipole approximation of the small-x evolution. However,
this approximation gives a good description of the experimental data (see [29,30] and
references therein).
The obtained Eq. (9) is nonlinear and most likely can not be solved analytically. We
solved a simplified model of the equation (Eqs. (20) and (22)), in which the transverse
coordinate dependence was suppressed. We believe that this approximation preserves
the qualitative features of the solution. In this toy model we observed an interesting
phenomenon: the diffractive cross section has a maximum as a function of the rapidity
gap Y0 (Fig. 9). This effect can not be obtained from the usual single triple pomeron
vertex approach of [2,15] and if experimentally observed would signify the importance
of non-linear effects in QCD evolution of structure functions. The effect should be seen
in the physically measurable quantity DIS diffractive cross section and it would be an
independent evidence of the onset of saturation.
Acknowledgments
The authors would like to thank Larry McLerran and Al Mueller for several helpful and
encouraging discussions. We have also benefited from interesting discussions with Errol
239
Abstract
Over a large fraction of phase space a combination of an operator product and heavy quark
expansions effectively turn the decay B D ()0 e+ e into a short distance process, i.e., one
in which the weak and electromagnetic interactions occur through single local operators. These
processes have an underlying W-exchange quark diagram topology and are therefore Cabibbo
allowed but suppressed by combinatoric factors and short distance QCD corrections. Our technique
allows a clearer exploration of these effects. For the decay B d,s J /(c )e+ e one must use a
non-relativistic (NRQCD) expansion, in addition to an operator product expansion and a heavy quark
effective theory expansion. We estimate the decay rates for B d,s J /e+ e , B d,s c e+ e ,
B d,s D 0 e+ e and B d,s D 0 e+ e . 2000 Elsevier Science B.V. All rights reserved.
PACS: 13.20.He; 12.39.Hg
1. Introduction
In a recent paper [1] we considered the collection of decays B + Ds,d e+ e . The
decay rate for these is proportional to |Vub |2 . We found that over a large kinematic domain
one can reliably estimate the rate (in terms of |Vub |2 ). The process is first order weak and
first order electromagnetic, and, therefore, the amplitude involves long distance physics.
The central observation of [1] is that over a large kinematic domain the interaction is local
on the scale of strong dynamics. The amplitude can, therefore, be approximated by the
matrix elements of local operators, which can be estimated in a variety of ways and should
eventually be determined in numerical simulations of QCD on the lattice. The branching
fraction for B + Ds+ e+ e , restricted to invariant mass of the e+ e pair in excess of
()+
1 daevans@physics.ucsd.edu
2 bgrinstein@ucsd.edu
3 dnolte@ucsd.edu
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 5 9 - 0
241
Fig. 1. W-exchange quark topology diagram underlying the transition B d,s D ()0 e+ e . Emission
of a e+ e pair from any line is understood.
1.0 GeV, was estimated to be 1.9 109 . This is too small to be measured in e+ e Bfactories, but could be observable at high luminosity high energy hadronic colliders.
In this paper we consider the decays B s,d J /e+ e , B s,d c e+ e , B s,d
0
D e+ e and B s,d D 0 e+ e . These proceed via W-exchange topologies, as shown in
Fig. 1. In addition, B d,s J /e+ e and B d,s c e+ e have small contributions from
penguins, which we neglect. The goal of the paper is to show how the methods introduced
in paper [1] can be applied to the processes considered here. The kinematics of B d,s
()+
D ()0 e+ e is similar to that of B + Dd,s e+ e so one expects the methods to apply
readily. In fact, the only dynamical difference is that in B d,s D ()0 e+ e the heavy b()+ +
e e it is a heavy b-antiquark
quark decays to a heavy c-quark, whereas in B + Ds,d
that decays into a heavy c-quark. The case B d,s J /(c )e+ e is clearly different:
both quark and antiquark in the final state are heavy and they are moving together in a
bound charmonium state. As we will see the expansion that arises naturally corresponds to
NRQCD, the non-relativistic limit of heavy quarks bound by QCD into quarkonia.
()+
The processes under consideration here have advantages compared to B + Ds,d
e+ e . These processes are not suppressed by the small CKM element |Vub |2 . One might
hope that the decay rate is, therefore, substantially higher. However, the enhancement of
the rate due to bigger CKM elements is partially cancelled by small Wilson coefficients.
Therefore, all these processes have small branching fractions. While none are observable at
B-factories, some are observable at future hadronic collider experiments like LHC-B and
BTeV.
These processes are first order weak and first order electromagnetic, and, therefore, the
amplitude involves long distance physics. We will show that over a large kinematic domain
the interaction is approximated by a set of matrix elements of local operators. All these
matrix elements should eventually be determined by lattice calculations. For the processes
considered in this paper, the number of independent matrix elements is reduced by the use
of rotational, heavy quark spin and chiral symmetries.
This paper is organized as follows. In Section 2 we review the methods of Ref. [1]
that lead to an expansion in local operators. The review is done in terms of the graphs
relevant to B D 0 e+ e , which is one of the processes of interest here. In Section 3
we present a novel analysis that shows that the matrix elements of the operators in the
expansion are all related by a combination of heavy-spin, rotational and chiral symmetries.
242
We then proceed to find the short distance QCD corrections to our operator expansion
in Section 4. In Section 5 we give expressions for the differential decay rates in terms
of matrix elements of local operators. These should be considered our main results. To
get some numerical estimates of the decay rates we crudely approximate the local matrix
elements. The material in Sections 25 deals with the decays B q D 0 e+ e and B q
D 0 e+ e , and we repeat the steps applied to the processes Bq c e+ e and Bq
J /e+ e in Section 6. Our results are summarized in Section 7.
2. Operator expansion
In this section we review the method introduced in [1]. However, we will present the
method as applied to the process B d D ()0 e+ e . Therefore we will at once review the
method and perform the necessary calculation for one of the cases of interest.
The effective Hamiltonian for the weak transition in B d D ()0 e+ e , is
4GF
(2)
P T a b c
P T a u,
O8 = d
(3)
and
(x)O(0) |B + i.
(4)
hD + | d 4 x eiqx T jem
Ta
Here q denotes the momentum of the e+ e pair, jem is the electromagnetic current operator
and the operator O, defined in Eq. (2), is the long distance approximation to the W exchange graph. The full amplitude will of course also involve a similar non-local matrix
element but with the singlet operator O replaced by the octet operator O8 . For now we
concentrate on the singlet operator. None of the arguments given in this section depend on
the particular choice of the operator.
We will now argue that for heavy b- and c-quarks the non-local matrix element in Eq. (4)
is well approximated by the matrix element of a sum of local operators. The approximation
is valid provided QCD mc,b , i.e., the corrections are order QCD /mc,b . There are also
corrections of order QCD mb,c /q 2 . So our results are limited to the region were q 2 scales
243
Fig. 2. Feynman diagram representing a contribution to the Green function. The filled square
represents the four quark operator O and the
cross represents the electromagnetic current
Fig. 3. Same as Fig. 2 but with the electromagnetic current coupling to the b-quark.
Fig. 4. Same as Fig. 2 but with the electromagnetic current coupling to the u-quark.
Fig. 5. Same as Fig. 2 but with the electromagnetic current coupling to the d-quark.
like m2c,b . The region were q 2 does not scale like m2c,b is parametrically small, so the
arguments we present are theoretically sound. However, there is the practical issue of
determining a minimum q 2 for realistic calculations were our approximations can still
be trusted. We return to this practical matter below, when we attempt to estimate the rate
for this decay.
The underlying decay is represented in the quark diagrams of Figs. 25. In the heavy
quark limit, QCD mc,b , the heavy meson momentum is predominantly the heavy
quarks. This suggests the following kinematics in the quark diagrams: for the momenta of
the heavy quarks take mb v + kb and mc v 0 + kc , for the momenta of the light quarks take ku
and k and then the photons momentum is determined by conservation, q = mb v mc v 0 +
P d
ki . We can now exhibit our OPE by considering the quark Green functions in Figs. 25.
The convergence of the expansion for physical matrix elements rests on the intuitive fact
that the residual momenta ki will be of order QCD (parametrically all we need is that
these are independent of the large masses). This intuition is made explicit in Heavy Quark
Effective Theory (HQET): there are no heavy masses in the HQET Lagrangian so the only
relevant dynamical scale is QCD . Thus our expansion of a non-local product will be in
terms of local operators of the HQET.
244
Calculating the Feynman diagram of Fig. 2 with our choice of kinematics we have
iQc
/ + mc/v0
q
i
P P .
+ /kc mc
(5)
Here Qc = 2/3 is the charge of the c-quark and the tensor product corresponds to the
P
two fermion bilinears. External legs are amputated. Using q = mb v mc v 0 + ki and
expanding in ki /mc,b we obtain, to leading order
Qc
mb v/ + mc
P P .
m2b m2c
(6)
(Q)
the
(7)
Here b,c are arbitrary Dirac matrices. With c b set equal to the tensor product in (6),
c b = Qc
mb v/ + mc
P P ,
m2b m2c
(8)
(9)
The ellipses indicate terms of higher order in our expansion, and correspond to higher
derivative operators suppressed by powers of mc,b . There are also perturbative corrections
to this expression. These show up as modifications to the operator defined by setting
c b equal to (6).
e
The diagram of Fig. 3 can be analyzed in complete analogy. It leads to the operator O
with the choice
c b = Qb P P
mb + mc v/0
,
m2b m2c
(10)
q/
P
q2
(11)
q/
P .
q2
(12)
245
3. Spin symmetry
We have shown how to replace the time ordered product in Eq. (4) by a local operator.
The replacement is valid provided the invariant mass of the lepton pair is large, i.e., scales
e that replaces the time ordered product is defined by Eq. (7),
as q 2 m2c,b . The operator O
with the tensor c b defined as the sum of the contributions in Eqs. (8), (10), (11)
and (12). We now show how to relate the matrix element of this operator to the operator
with c b = P P . This operator is not only simpler, but one can estimate its
matrix elements by a variety of means, as we explain below.
e as defined in Eq. (7) for arbitrary tensor product
Consider the matrix element of O
c b between heavy meson states. We will use heavy quark spin symmetry to determine
the matrix elements of this operator between heavy meson states. Recall that the HQET
Lagrangian
(c)
(b)
(c) 0
LHQET = h (b)
v iv Dhv + hv 0 iv Dhv 0
(13)
is symmetric under the group SU (2)b SU (2)c of transformations acting on spin indices
of the heavy quark fields:
(b)
h(b)
v Sb hv ,
(c)
h(c)
v 0 Sc hv 0 .
(c)
Hv(c)
0 Sc Hv 0 ,
(15)
while an arbitrary rotation R represented by the Dirac matrix D(R) acts simultaneously on
both multiplets according to
H (Q) D(R) H (Q)D(R).
(16)
(c) (b)
(b)
eHv H (c)
(17)
H v 0 O
v 0 c b H v .
(c)
(b)
Finally, invariance under rotations implies that the remaining four indices must be
contracted. There are two possible contractions,
(b)
(b)
and Tr H v(c)
.
(18)
Tr H v(c)
0 c Tr b Hv
0 c b H v
246
We now show that the second one is excluded by chiral symmetry. The Lagrangian for a
massless quark in QCD,
/ ,
L = iD
(19)
(20)
where , the parameter of the transformation, is a real number. Under this symmetry the
transformation rule for our tensors is
b Hv(b) ei5 b Hv(b)ei5
(21)
(c)
(c)
H v 0 c ei5 H v 0 c ei5 .
(22)
and
It is seen that the first contraction of indices in (18) is invariant, but the second one is not.
We have shown that heavy quark spin symmetry, rotations and light quark chiral
symmetry combine to give
(c) (b) 1
(b)
eHv = (w)Tr H (c)
.
(23)
H v 0 O
4
v 0 c Tr b Hv
We have indicated that the invariant matrix element is a function of w = v v 0 . In
general, it is a function of v and v 0 . However, since it must be Lorenz invariant and since
v 2 = v 02 = 1, it is a function of w = v v 0 only.
The octet operator in the HQET,
a
e8 d
T a h(b)
(c)
O
v hv 0 c T u,
b
(24)
has the same spin and heavy flavor symmetry properties as its singlet counterpart.
Therefore in complete analogy we can introduce a reduced matrix element 8 :
(c) (b) 1
(b)
e8 Hv = 8 (w)Tr H (c)
.
(25)
H v 0 O
4
v 0 c Tr b Hv
The authors of Ref. [3] proposed a relation analogous to Eq. (23) for a 1B = 2 transition.
It was noted there that spin symmetry allowed more than one invariant and that, however,
all invariants lead to the same symmetry relations. One may wonder if our use of chiral
symmetry may help relate the different invariants there. We show that this is not the case.
For the 1B = 2 case the analogue of Eq. (17) is
(b)
(b)
e
(26)
b H v(b) b Hv(b) ,
H v O
1B=2 Hv
(b)
e1B=2 = d
h(vb)
h(b)
d
where O
b v (note that we define hv to create a b-antiquark).
b
Again, invariance under rotations implies that the remaining four indices must be
contracted and, again, there are two possible contractions,
(27)
Tr b H v(b) Tr b Hv(b) and Tr b H v(b) b Hv(b) .
Chiral symmetry for the antiquarks meson tensor is just as for the quarks in Eq. (21),
(28)
247
Therefore both contractions in (27) are allowed by chiral symmetry. However, it is easy to
see that for a class of operators of interest the two contractions are equivalent. If
b b = P b P
or
b b = P P b,
for any arbitrary Dirac matrix b the two contractions are related by Fierz rearrangement.
This class of operators includes the BB mixing case studied in Ref. [3].
4. QCD corrections
Consider the operator expansion in Eq. (9). We have seen that at leading order the
operator on the right hand side is given by Eqs. (7), (8). We now consider the leading-log
corrections to this relation. In the large mass limit these are formally the largest, leading
corrections to the operator expansion. A renormalization scale must be stipulated for
the evaluation of matrix elements of the composite operators on both sides of Eq. (9). It is
often convenient to evaluate the matrix elements at a low renormalization point = low .
This choice makes the matrix elements in the HQET completely independent of the large
masses of the heavy quarks. If low mc,b there are large corrections to Eq. (9) in the
form of powers of s ln(mc,b /low ). These powers of large logarithms can be summed
using renormalization group techniques. The corrections to these leading-logs are of
order 1/ ln(mc,b /low ) or s . It is important therefore to keep low small, but large enough
that perturbation theory remains valid. When we estimate decay rates below, we use low =
1.0 GeV.
To study the dependence on the renormalization point we take a logarithmic derivative
on both sides of Eq. (9). Consider first the left side. Acting with (d/d) on the charm
number current c
c gives zero, because the current is conserved. The action of (d/d)
on the composite four-quark operator is a linear combination of itself and the octet operator.
It is therefore convenient to consider instead the linear combination that appears in the
effective Hamiltonian (1):
Z
e8 + .
e + c8 O
c(x) cO(0) + c8 O8 (0) = cO
(29)
d 4 x eiqx T c
The coefficients c and c8 are such that the left hand side is -independent. This is necessary
for the physical amplitude to be independent of the arbitrary choice of renormalization
point . Therefore our task is to determine the proper -dependence for c and c8 so that
the right hand side is also independent of . Therefore, if the operators satisfy
e
e
d
O
O
(30)
e8 = O
e8 ,
d O
where is a 2 2 matrix of anomalous dimensions, then the coefficients must satisfy
d
c
c
T
=
.
(31)
d
8
8
248
Fig. 6. One loop Feynman diagrams for the calculation of the anomalous dimension matrix. The solid
diamond represents the local operators O or O8 .
p
1
ln w + w2 1 .
r(w)
2
w 1
(33)
The solution to the renormalization group equation (31) is straightforward. In terms of the
ratio of running coupling constants
s ()
(34)
z
s (0 )
and the functions
1 41 14wr(w)
,
12
b0
3 1 + 2wr(w)
,
=
4
b0
249
(35)
(36)
where the coefficient of the one loop term of the -function for QCD is b0 = 11 23 nf ,
and nf is the number of light flavors (nf = 3 in our case), we obtain
c(
0)
c()
=U
,
(37)
c8 ()
c8 (0 )
where
U =z
1
8
9z + 9z
2
3 (z z )
4
27 (z z )
.
8
1
9z + 9z
(38)
The question that remains is how to determine the coefficients c and c8 at some scale 0 .
e
But we have already determined these coefficients in Section 2. Recall that the operator O
that replaces the time ordered product is defined by Eq. (7), with the tensor c b defined
as the sum of the contributions in Eqs. (8), (10), (11) and (12) with unit coefficient. The
question can be rephrased as what is the scale 0 for which the calculation in Section 2 is
valid. What we would like to do is to determine for what choice of 0 the loop corrections
to relations like Eq. (9) will be free from large logs. The only relevant scales in the
problem are the large masses mc,b , the invariant mass of the e+ e pair, q 2 , which itself
scales like m2c,b , the small masses and residual momenta and the renormalization point 0 .
The corrections to the relations of Section 2 are guaranteed to be free from logs of the
small masses or residual momenta. But there will be logs of ratios of large masses to the
renormalization point, ln(mc,b /0 ). To avoid these one may choose 0 mc,b . For our
computations below we will use 0 4.0 GeV. If the scales mc and mb are both large but
very disparate one could review the above analysis by introducing a new renormalization
group equation to resum the logs of mc /mb . The results of this section would still resum
the logs of /mc .
We thus have that c()
(0 )
(MW )
(40)
6/23
.
(41)
For illustration we have given the leading log expression for the coefficients c(0 ) and
c8 (0 ), but in rate computations below we use the next to leading log results from [2].
We do not have at present a full next to leading log result: still missing is a computation
of the one loop corrections to the coefficients c and c8 at = 0 and of the anomalous
dimensions matrix of Eq. (30) at two loops. It is interesting to note that the coefficients
c(0 ) are significantly enhanced at next to leading log order. For the case 0 = 4.0 GeV
one has in next to leading order [2] c = 0.16, rather than the leading log result c = 0.07.
250
We emphasize that this enhancement can be systematically accounted for. The large
enhancement is not a signal of perturbation theory breaking down but rather due to the
accidental cancellation in the leading order.
5. Rates: B 0 D ()0 e+ e
We are ready to compute decay rates. Defining
Z
()0
()
0
= D
(x)Heff
(0) B 0 ,
d 4 x eiqx T jem
h
(42)
the decay rate for B 0 D ()0 e+ e is given in terms of q 2 and t (pD + pe+ )2 =
(pB pe )2 by
2
2
e
1
d
()
=
` h
(43)
,
3 q2
8
3
dq 2 dt
2 MB
e ) v(pe+ ) is the leptons electromagnetic current. A sum over final
where ` = u(p
state lepton helicities, and polarizations in the D case, is implicit.
To compute h() we need to pull together the results of the previous sections. First
the time ordered product is expanded in terms of local operators as in Eqs. (8)(12). This
0 ) and c8 (0 ) as seen
involves replacing the coefficient functions c(0 ) and c8 (0 ) by c(
e and O
e8 between
in Eq. (29). Then the matrix elements of the leading local operators O
particular states can all be expressed in terms of the reduced matrix elements and 8
defined in (23) and (25). Finally, to make all dependence on the heavy quark masses
explicit, we run down the coefficients c and c8 from the scale = 0 of order of mb,c
(which we take to be 4 GeV) to a scale = low of order of a few times QCD .
Our computation gives
(2wmb + mc )v (mb 4wmc )v 0 3(mb v 0 + mc v )
+
(44)
h =
3
(mb v mc v 0 )2
m2b m2c
and
v 0 v
mb ( + 2v v ) mc (3v v 0 + w ) 3imc
+
h =
0
2
3
(mb v mc v )
(mb v mc v 0 )2
0
.
(45)
m2b m2c
[c
+ c8 8 ]. These expressions are our central results, demonHere = GF / 2 Vcb Vud
strating that the decay rates for B 0 D ()0 e+ e can be expressed in terms of the matrix
elements and 8 . Below we make an educated guess of these matrix elements, but for
reliable results they should be determined from first principles, say, by Monte Carlo simulations of lattice QCD.
In the computation of the rate the amplitude depends on heavy quark masses mc and mb ,
while the phase space involves physical meson masses MB and MD or MD . Although it is
251
straightforward to retain the dependence on all four masses in our expressions for the decay
rates, we have chosen to express the results in terms of physical meson masses, with the
substitutions mb = MB and mc = MD or mc = MD . We are not justified in distinguishing
between quark and meson masses since the distinction enters at higher order in the 1/mc,b
expansion.
It is now a trivial exercise to compute the differential decay rate. Integrating the rate in
Eq. (43) over the variable t we obtain
2
2 G2F
d
2
=
|V
V
|
F (q).
(46)
c
+
c
cb
ud
8
8
dq 2 288MB3
q
MD () /MB . For B 0
Here F (q)
is a dimensionless function of q q 2 /m2b and m
D 0 e+ e it is given by
p
2 + q 4 2m
2 q 2 + m
4
4 1 2q 2 2m
F=
3
q 6 m(1
m
2 )2
5m
2 + 19m
4 q 2 + 30m
6 20m
4 14q 2m
2
20m
8 + 12q 6m
2 + q 2 + q 6 2q 4 + 2m
6 q 4
6m
8q 2 + 5m
10 6q 6 m
4 + 5m
2 q 8 ,
(47)
while for B 0 D 0 e+ e
F=
4(2m
2 + 1)2 (1 2q 2 2m
2 + q 4 2m
2 q 2 + m
4 )3/2
.
3q 4m(1
m
2 )2
(48)
side vacuum saturation gives (zaI fB pB / MB )(zaI fD pD / MD ). Here z is defined
in Eq. (34) and aI = 2/b0 [79] is the well known anomalous scaling power for the heavylight current in HQET. 5 Thus we obtain
p
(49)
(w) = z2aI fB fD MB MD ,
8 (w) = 0.
(50)
4 The matrix elements in B + D ()+ e+ e can be related by symmetry to the matrix element for BB
mixing, if the matrix element of the octet is negligible; see Ref. [6].
5 The two factors of zaI really correspond to distinct running, between m and
b
low for the first factor, and
between mc and low for the second. The distinction is of higher order than we have retained, if we assume that
the heavy scales mb and mc are not too disparate, that is, that s does not run much between these scales.
252
The second equation is true not just in vacuum saturation but also in the approximation that
e8 . This is not an exact
we can insert a complete set of states between the currents defining O
e8 does not equal the product of two currents.
statement because the composite operator O
But the distinction arises from their different short distance behavior. So we expect the
deviation of 8 from zero to be of order of the QCD coupling at short distances (0 )
times the unsuppressed .
Using these matrix elements we integrate the differential rate in Eq. (46) over the range
2 = 1.0 GeV as a
2
to obtain a partial decay rate. We have chosen qmin
1.0 GeV 6 q 2 6 qmax
2
2
lower limit since our OPE requires that q scale like mc,b . The corrections to the leading
terms in Eqs. (11) and (12) are of the form of an expansion in mc,b k/q 2 , where k is any
of the residual momenta and in our matrix elements is of order QCD . Parametrically, if
q 2 m2c,b , then mc,b k/q 2 QCD /mc,b 1. In addition, the region over which q 2 .
QCD mc,b where the expansion breaks down, is parametrically small. However, physical
heavy masses are not very large, and the scale mb QCD is just slightly smaller than m2c .
In order to have some non-trivial phase space we have taken q 2 & mb QCD 1.0 GeV.
The price we pay is that for the lower values of q 2 our expansion converges slowly,
mc,b k/q 2 . 1.
We find
(51)
Br B 0 D 0 e+ e q 2 >1 GeV = 1.4 108 ,
(52)
Br B 0 D 0 e+ e q 2 >1 GeV = 2.6 109 ,
where we have used |Vcb Vud | = 0.04, fD = fB MB /MD and fB = 170 MeV. It is
important to observe that the portion of phase space q 2 > 1.0 GeV is expected to give
a small fraction of the total rate since the pole at q 2 = 0 dramatically amplifies the rate
for small q 2 . The rates for Bs0 D 0 e+ e and Bs0 D 0 e+ e can be obtained to good
approximation by replacing |Vcb Vud | by |Vcb Vus |, reducing the rates by (0.22)2 0.05.
The next generation of B-physics experiments at high energy and luminosity hadron
colliders, like LHC-B and BTeV, will produce well in excess of 1011 B-mesons per year.
Our calculation includes only large invariant mass lepton pairs so detection and triggering
on the lepton pair should be straightforward. Dedicated studies must be done to determine
feasibility of detection and measurement of spectra of these decays.
6. Decays to quarkonium
6.1. Operator expansion and NRQCD
The decays Bs c e+ e and Bs J /e+ e (and obvious extensions to excited
charmonium) can be studied in a similar way. The notable difference in the operator
expansion here is that the residual momenta k of the heavy quarks in the quarkonium bound
state do scale with the large heavy mass k s mc , as opposed to the residual momenta of
the quarks in the heavy B or D mesons, k QCD . The residual momentum for the case of
quarkonia is small for a different reason: k = mc u and k 0 = 12 mc u2 are small because the
253
velocity u of the bound quarks is small [10] for heavy quarks, u s (mc ). The parameter
of the expansion is therefore mc,b k/m2c,b s (mc ).
Our best hope in making the nature of the expansion explicit is to use NRQCD [10],
the effective theory of non-relativistic quarks in QCD. As opposed to HQET, where all the
heavy mass dependence has disappeared, the Lagrangian of NRQCD still depends on the
heavy mass:
D2
.
(53)
LNRQCD = iDt
2mc
Here denotes a two component spinor field for the c-quark. A separate spinor field must
be included to describe the antiquark. We have written the Lagrangian in the rest-frame
of charmonium, but it is straightforward to boost into a moving frame. One relies on the
dynamics to generate the small parameter of the expansion. 6 For example, the two terms
in LNRQCD are of comparable magnitude if, as expected, Dt k 0 mc s2 and |D| |k|
mc s .
The operator expansion is in terms of operators with an HQET quark, a light quark and
a pair of NRQCD quarkantiquark. So instead of Eq. (7) we have
e s b h(b)
O
v c c c ,
(54)
where c and c create a charm quark and a charm antiquark, respectively. We elect
to use four component spinors throughout; the reduction to two components results from
algebraic constraints that must be imposed, just as in HQET:
1 + v/0
.
=
2
The calculation proceeds much as before. The effective Hamiltonian for the weak
transition is
4GF
0
= Vcs Vcb
c(/MW )O + c8 (/MW )O8 ,
(55)
Heff
2
where
P c
O = s P b c
(56)
P T a c.
O8 = s P T a b c
(57)
and
The operator expansion of the hadronic matrix element takes the form
Z
e8 + ,
e + c8 O
(x) cO(0) + c8 O8 (0) = cO
d 4 x eiqx T jem
(58)
6 Attempts to make the expansion in u [11] or, alternatively, in 1/c [12] explicit yield theories where the gluon
self-couplings must be perturbative. The scale of QCD must then be negligible compared with the Bohr radius of
quarkonium, QCD mc s (mc ). In our case non-perturbative gluons play a crucial role in binding the heavylight meson B.
254
Fig. 7. Feynman diagram representing a contribution to the Green function. The filled square
represents the four quark operator O and the
cross represents the electromagnetic current
Fig. 8. Same as Fig. 7 but with the electromagnetic current coupling to the c-antiquark.
a
e8 s T a h(b)
O
v c c T c .
b
(59)
The first task is to determine the tensor b c . To this order we consider Green functions
of the time ordered product in Eq. (58) with four external quarks. The in-going momenta of
the b- and s-quarks are mb v + kb and ks , respectively. The outgoing momenta of the charm
pair are mc v 0 + kc and mc v 0 + kc . As explained above, we expect kb ks QCD while
kc kc mc s (mc ). The leading term in the momentum of the electromagnetic current is
q = mb v 2mc v 0 . For the purpose of determining the expansion coefficients at tree level
we may set c = 1 and c8 = 0 and, choosing a renormalization point 0 of the order of the
large masses mc,b , we can set c = 1 and c8 = 0. There are four graphs contributing to the
tensor c b . Fig. 7 gives
c b = Qc
mb v/ mc (v/0 1)
P P ,
m2b 2mb mc w
(60)
mb v/ + mc (v/0 + 1)
P .
m2b 2mb mc w
(61)
Note that the denominator, which dictates the convergence of the expansion, scales with
m2c,b . It vanishes at w0 = mb /2mc . However, this is never in the physical region:
wmax = (m2b + 4m2c )/4mb mc = w0 (mb /4mc mc /mb ),
but mb > 2mc for the decay to be allowed.
The diagrams in Figs. 9 and 10 are just as in Figs. 3 and 5, with the replacement
q = mb v mc v 0 q = mb v 2mc v 0 .
For the first we have
c b = Qb P P
mb + 2mc v/0
,
m2b 4m2c
(62)
Fig. 9. Same as Fig. 7 but with the electromagnetic current coupling to the b-quark.
255
Fig. 10. Same as Fig. 7 but with the electromagnetic current coupling to the s-quark.
q/
P .
q2
(63)
Again we see that the expansion remains valid as long as q 2 scales with the heavy masses
(squared), and this limitation arises solely from the coupling of the photon to the light
quark.
6.2. Spin symmetry
The NRQCD Lagrangian contains separate fields for the charm quark and antiquark. The
quark Lagrangian, Eq. (53) is symmetric under spin-SU (2) transformations. The antiquark
Lagrangian is similarly invariant under a separate spin-SU (2). This case has a larger spin
symmetry than the case of decays to D-mesons. One can therefore write a trace formula
analogous to Eq. (23) without using chiral symmetry of the light quarks.
We can represent the charmonium spin multiplet (c , J /) by the 4 4 matrix
1 v/0
1 + v/0
()
c 5
.
(64)
Hv 0 =
2
2
The action of spin-SU (2) SU (2) on this is then
()
()
Hv 0 Sc Hv 0 Sc .
(65)
() e (b)
(b)
Consider the matrix element hHv 0 |O|H
v i. It must be linear in the tensors c b , Hv
()
(b)
(b)
and H v 0 . As before, acting with SU (2)b we see that b b Sb and Hv Sb Hv ,
so they enter the matrix element as b Hv(b) . Now, acting with the spin symmetries of
NRQCD, we have Eq. (65) and c Sc c Sc , so that they must enter the matrix element
()
as Tr(H v 0 c ). Finally, rotations demand that we sum over the two remaining indices,
() (b) 1
(b)
eHv = Tr H ()
.
(66)
H v 0 O
4
v 0 c Tr b Hv
() (b) 1
(b)
e8 Hv = 8 (w) Tr H ()
.
H v 0 O
4
v 0 c Tr b Hv
(67)
We have used the same symbols here for operators and reduced matrix elements as in
Sections 2 and 3, but they should be understood as distinct.
256
about large logs of the ratio mc /mb . Then one may take, say, 0 mc mb . The point
is that the coefficients on the left-hand side of (58) explicitly depend on MW /0 and the
operators implicitly depend on mc,b /0 . If we choose to do the matching at a scale 0 that
differs much from mc,b then there are implicit large corrections. Note that the right hand
side of (58) can only introduce logs of low scales over 0 , but the same infrared logs are
found on the left side of the equation.
Once the coefficients c and c8 in (58) have been determined at 0 we must ask at what
scale we should evaluate the matrix elements and how to get there. The situation is
more complicated than in the case of B 0 D 0 e+ e of Section 4 because now the matrix
element in the combined HQET/NRQCD effective theory has several scales. In NRQCD
the relevant distance scale is the inverse Bohr radius mc s (mc ) and the relevant temporal
scale is the Rydberg mc s2 (mc ). In HQET the dynamical scale is QCD . Of course QCD
also plays a dynamical role in NRQCD, but it is usually taken to be irrelevant since
one assumes QCD mc s2 (mc ) mc s (mc ). So we are faced with a multiple scales
problem. Setting equal to any one of these scales leaves us with large logs of the ratios
of to the other two. It is not known how to use the renormalization group equation to
resum these logs.
Suppose that we set mc (mc ) or mc 2 (mc ). If we then use the renormalization
group to sum powers of (mc ) ln(mc /) we will be summing powers of (mc ) ln (mc ).
Notice that these logs vanish as mc , since (mc ) 1/ ln(mc /QCD ). Contrast this
with the case QCD (or, generally, setting equal to any fixed scale as mc ).
Then (mc ) ln(mc /) 1 as mc . As a matter of principle, in the large mass limit
it is these latter logs that must be summed (they are parametrically of leading order in
the large mass expansion). Therefore we resum the leading logs with a fixed low scale
= low and choose, as before, low = 1.0 GeV in our numerical computations.
In order to use dimensional regularization and keep track of different orders in the nonrelativistic expansion we adopt the 1/c counting advocated in Ref. [12]. However, we use
a covariant gauge for our calculations. This is convenient because the Feynman diagrams
involve light and HQET quarks in addition to the NRQCD quarks. In leading order in the
1/c expansion the quark Lagrangian in (53) is replaced by
2
.
(68)
LNRQCD iDt
2mc
The only interactions are due to temporal gluon exchange. Since we work in covariant
gauge, this is not a pure Coulomb potential gluon. It is easy to see that no diagram
involving an NRQCD quark gives a divergent contribution. The self-energy diagrams for
the NRQCD quarks have an infinite piece, which however is independent of the momentum
257
e8 = O
e8 .
d O
Then the coefficients must satisfy
d
c
c
= T
,
c8
d c8
(71)
= zaI c(
c8 () = z
1
4 aI
c8 (0 ),
(72)
(73)
where z is defined in Eq. (34) and aI = 2/b0 is the well known anomalous scaling power
for the heavy-light current in HQET [79].
Contributions from higher orders in the 1/c expansion produce mixing with higher
dimension operators and are therefore excluded to the order we are working. This is easy
to see. To compensate for the powers of 1/c one must have additional velocities in the
operators. But these come from powers of /mc . The leading correction to the Lagrangian
is of order 1/c3/2 . Since two insertions are needed this gives a graph of order 1/c3 . Since
one power of c is needed to form the QCD fine-structure constant, s = gs2 /4c, the
divergent part of the graph involves p2 /m2c c2 . It is straightforward to verify this by direct
calculation.
6.4. Rates
Defining
h( ) = h |
0
d 4 x eiqx T jem
(x)Heff
(0) |Bs i,
(74)
258
h(c ) =
mb v 0 2(wmb mc )v mb v 0 + 2mc v
+
3
(mb v 2mc v 0 )2
m2b 4m2c
and
h(J /) =
(76)
2mb v v (mb 2mc w) 2mc v v 0
3
(mb v 2mc v 0 )2
+
2imc v v0
(mb v 2mc v 0 )2
8imc v v0
m2b 2mb mc w
mb + 2mc (v v 0 w ) + 2imc v v0
m2b 4m2c
.
(77)
(78)
+ c8 8 F (q).
2
3
dq
288MB
q
MJ / /MB . For Bs
Here F (q)
is a dimensionless function of q q 2 /m2b and m
J /e+ e it is given by
p
2 + q 4 2m
2 q 2 + m
4
4 1 2q 2 2m
F=
3q 6m
2 (1 m
2 )2 (1 + q 2 m
2 )2
15m
10 6m
12 + m
2 + 15m
6 + q 2 6m
4 20m
8
+ q 10 2q 6 + 6m
2 q 2 + 23m
2q 4 + 55m
2 q 8 + m
2 q 12
111m
8q 2 + 234m
6 q 4 + 104m
6q 2 + 92m
6q 6
188m
8q 4 + 58m
10 q 2 46m
4q 2 + 30m
4 q 6
124m
4q 4 + m
14 72m
8q 6 + 55m
10q 4 12m
12 q 2
+ 23m
6q 8 78m
4 q 8 + 4m
4 q 10 6m
2 q 10 48m
2q 6 ,
(79)
while for Bs c e+ e
4(1 2q 2 2m
2 + q 4 2m
2 q 2 + m
4 )3/2
.
(80)
3q 4m
2 (1 m
2 )2
For a numerical estimate we need to calculate the matrix elements and 8 . Again we
use vacuum saturation. However, now this approximation is supported by NRQCD. It is
argued in Ref. [13] that soft gluon exchange with the quarkonium is suppressed by powers
of the relative velocity u = s (mc ), and that the matrix element of the octet operator is
similarly suppressed. Therefore we take
p
(81)
(w) = zaI fB fc MB Mc ,
F=
8 (w) = 0.
(82)
259
Note that because vacuum saturation here is valid at least as a leading approximation in
a velocity expansion, the combination of coefficients in (72), (73) and matrix elements
in (81), (82) is automatically independent of the renormalization point . Spin symmetry
gives fc = fJ / . We use the measured value from the leptonic width in the tree level rate
equation,
(J / e+ e ) = 4 2
fJ2/
MJ /
(83)
(84)
(85)
where we have used |Vcb Vcs | = 0.04, and fB = 170 MeV. Again, we remind the reader that
the portion of phase space q 2 > 1.0 GeV is a small fraction of the total rate since the pole
at q 2 = 0 dramatically amplifies the rate for small q 2 . The rates for B 0 J /e+ e and
B 0 c e+ e can be obtained to good approximation by replacing |Vcb Vcs | by |Vcb Vcd |,
reducing the rates by (0.22)2 0.05. The rate (84) may seem too small to be detectable
even in the next generation of hadronic colliders. However it must be kept in mind that the
signature involves four leptons with large invariant masses (one being the J /).
7. Conclusions
We have successfully shown how to implement the OPE advertised in Ref. [1] to the
processes B d,s J /e+ e , B d,s c e+ e , B d,s D 0 e+ e and B d,s D 0 e+ e .
By the use of the OPE the long distance (first order weak and first order electromagnetic)
interaction is replaced by a sum of local operators. The application of the OPE is restricted
to a limited kinematic region.
In the processes B d,s J /e+ e and B d,s c e+ e our method leads naturally to
an NRQCD expansion for the J / and c . This illustrates that the methods of Ref. [1] are
applicable to a wider class of processes.
Furthermore we found that the number of independent matrix elements of the local
operators is severely restricted due to a combined use of heavy-spin, rotational and
chiral symmetry. The independent matrix elements could be determined, say, in lattice
simulations. Our paper shows that the processes considered can be studied in a systematic
fashion independent of any model assumptions in the kinematic regime of q 2 scaling like
m2c,b .
Using a crude estimation of the matrix elements, we found the rates of all the processes
considered to be small. We expect some of them, in particular B s D 0 e+ e , should be
accessible at planned experiments at hadron colliders, like BTeV or LHC-B.
260
Acknowledgments
We thank Mark Wise for useful discussions and conversations. This work is supported
by the Department of Energy under contract No. DOE-FG03-97ER40546.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
D.H. Evans, B. Grinstein, D.R. Nolte, Phys. Rev. Lett. 83 (1999) 4947.
G. Buchalla, A.J. Buras, M.E. Lautenbacher, Rev. Mod. Phys. 68 (1996) 1125.
B. Grinstein, E. Jenkins, A.V. Manohar, M.J. Savage, M.B. Wise, Nucl. Phys. B 380 (1992) 369.
D.J. Gross, Applications of the Renormalization Group to High-Energy Physics, in: Proceedings, Methods in Field Theory, Les Houches 1975, Amsterdam, 1976, pp. 141250.
The BaBar Physics Book, SLAC report, SLAC-R-504, October, 1998, 10121013.
D.H. Evans, B. Grinstein, D.R. Nolte, Phys. Rev. D 60 (1999) 057301.
M.A. Shifman, M.B. Voloshin, Sov. J. Nucl. Phys. 45 (1987) 292.
H.D. Politzer, M.B. Wise, Phys. Lett. 206B (1988) 681.
A.F. Falk, H. Georgi, B. Grinstein, M.B. Wise, Nucl. Phys. B 343 (1990) 1.
W.E. Caswell, G.P. Lepage, Phys. Lett. 167B (1986) 437.
M. Luke, A.V. Manohar, Phys. Rev. D 55 (1997) 4129.
B. Grinstein, I.Z. Rothstein, Phys. Rev. D 57 (1998) 78.
G.T. Bodwin, E. Braaten, G.P. Lepage, Phys. Rev. D 51 (1995) 1125; Phys. Rev. D 55 (1997)
5853 (Erratum).
Abstract
We investigate the renormalization group (RG) flow of SU(3) lattice gauge theory in a two coupling
space with couplings 11 and 12 corresponding to 1 1 and 1 2 loops, respectively. Extensive
numerical calculations of the RG flow are made in the fourth quadrant of this coupling space, i.e.,
11 > 0 and 12 < 0. Swendsens factor two blocking and the SchwingerDyson method are used
to find an effective action for the blocked gauge field. The resulting renormalization group flow
runs quickly towards an attractive stream which has an approximate line shape. This is a numerical
evidence of a renormalized trajectory which locates close to the two coupling space. A model flow
equation which incorporates a marginal coupling (asymptotic scaling term), an irrelevant coupling
and a non-perturbative attraction towards the strong coupling limit reproduces qualitatively the
observed features. We further examine the scaling properties of an action which is closer to the
attractive stream than the currently used improved actions. It is found that this action shows excellent
restoration of rotational symmetry even for coarse lattices with a 0.3 fm. 2000 Elsevier Science
B.V. All rights reserved.
PACS: 11.15.Ha; 12.38.Gc
Keywords: SU(3) lattice gauge theory; Renormalization group flow; Improved action
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 4 5 - 0
264
1. Introduction
Since Wilsons first numerical renormalization group (RG) analysis of SU(2) gauge
theory [1], there have been many Monte Carlo RG studies of non-perturbative -functions
(see Ref. [2] and references therein). In these analysis indirect information about the
-function, such as 1, has been obtained [3]. Recent progress of lattice techniques [46]
allows us to estimate directly the RG flow in multi-coupling space [7].
We study renormalization effects by means of a blocking transformation which changes
the lattice cut-off but leaves the long range contents of the system invariant. A new blocked
action S 0 as a function of blocked link variables V s is constructed from the original action
S(U ) as
Z
0
(1)
eS (V ) = DU eS(U ) V P (U ) ,
where P defines the blocking transformation. The action S 0 includes the renormalization
effects induced by blocking. In the space of coupling constants, the blocking transformation makes a transition from a point corresponding to S to a new point, S 0 . Repeating
the blocking transformation, we obtain trajectories in coupling space which define the socalled renormalization group flow.
There is a special trajectory, i.e., renormalized trajectory (RT), which starts at the ultraviolet fixed point. On the RT, the long range information corresponding to continuum
physics is preserved. Recently Hasenfratz and Niedermayer have stressed that the action
on the RT can indeed be considered as a perfect action [9]. Therefore if we find a RT
corresponding to a blocking transformation, it provides an action which gives accurate
results corresponding to the continuum limit. Even if it is an approximate one, it serves as
a well-improved action. In this sense, a pioneering work has been done by Iwasaki more
than ten years ago [8]. He estimated a RT by matching Wilson loops based on a perturbative
approximation, and proposed an improved action which we will call below Iwasaki action.
In this work, we analyze numerically the RG flow in two coupling space, (11 , 12 ), of
SU(3) lattice gauge theory and clarify the structure of the renormalization group flow. The
action is restricted to the following form:
X
X
1 13 Re Tr Pplaq + 12
1 13 Re Tr Prect .
(2)
S = 11
plaq
rect
265
as well as on some improved actions. In addition, we try to clarify the global structure of
the RG flow. An evidence that the RT sits close to the two coupling space is provided by the
fact that the flow runs quickly towards a narrow attractive stream. Characteristic features of
the flow in the strong and weak coupling regions are also found. The observed features are
reproduced by a model flow equation which incorporates a marginal coupling (asymptotic
scaling term), an irrelevant coupling and a non-perturbative attraction towards the strong
coupling limit.
Based on the flow structure, we examine the scaling properties of several actions defined
in this two coupling space. Tests are made for the rotational invariance and the scaling of
/Tc . Near the attractive stream, we find good restoration of the rotational invariance.
This paper is organized as follows. In Section 2, the basic tools for the analysis are given.
Section 3 is devoted to present simulations and numerical results of the RG flow. Scaling
tests are described in Section 4. In Section 5, a model flow equation which can reproduce
the observed RG flow is proposed. Renormalization effects beyond the two coupling space
are discussed in Section 6. A summary of results is given in Section 7.
Here c is a parameter to control the weight of the staple-like paths. A convenient notation
is also used and the sum is taken over negative as well as positive
V (n) = V (n )
direction. Q (n) is projected onto the blocked SU(3) gauge field V (n) by maximizing
Re Tr(Q (n)V (n)). 1
To determine the effective action S 0 on blocked configuration V , we use the Schwinger
Dyson method [4]. This is based on the following identity: for a link Vl0 , consider the
quantities
1 The projection from Q onto a SU(3) matrix V is not unique. In Ref. [3], we employed the polar
e/det(V
e) with V
e = Q/(Q Q)1/2 . One
decomposition, i.e., Q = V H where H is an Hermite matrix and V = V
can easily prove that both methods are equivalent for unitary matrices V s.
266
Im Tr
Vl0 Gl0
1
=
Z
0
DV Im Tr b Vl0 Gl0 eS ,
(4)
where b stands for Gell-Mann matrices. Here action is assumed to have the form,
S0 =
X X 0
l
1 13 Re Tr Vl Gl
(5)
and Gl stands for a staple coupling to the link l in a loop of type . For the present
analysis, corresponds to a plaquette and a rectangle. Eq. (4) should be invariant under
the change of variable Vl0 (1 + ib )Vl0 . Setting terms linear in to be zero, we get the
identity,
Z
h
2
i
0
(6)
DV Re Tr b Vl0 Gl0 + Im Tr b Vl0 Gl0 Im Tr b Vl0 Gl0 eS = 0,
where
Gl =
X 0
Gl .
(7)
Summing over b in the expression (6) above, we obtain the SchwingerDyson equation,
X 0 n
Re Tr Vl0 Gl0 =
Re Tr Vl0 Gl0 Vl0 Gl0 + Re Tr Gl0 Gl0
3
6
o
2
(8)
Im Tr Vl0 Gl0 Im Tr Vl0 Gl0 .
3
P
Here we have used the identity: 8b=1 Tr(b A) Tr(b B) = 2 Tr AB 23 Tr A Tr B. We apply
Eq. (8) to the blocked configurations, and calculate the expectation values h i on both
sides. Now Eq. (8) may be considered as a set of linear equations with s as unknowns.
It is noted that we may use other loop operators instead of G . However a minimal choice
is to take the same G s as the ones entering the action. In this case, the number of equations
is equal to the number of unknown couplings.
It is noted that the canonical Demon method also works for the present purpose [6]. This
method tunes the effective action so as to reproduce the mean values of the plaquette and
rectangular loops whereas the SchwingerDyson method respects wider loops which are
combination of staples such as Tr(Gl (Gl ) ). In case of a limited coupling space, they may
lead to different actions. Because the systematic errors involved in both methods are not
known in the present stage, only the results obtained via the SchwingerDyson method are
presented in this work.
Since we study the RG flow in the two coupling space, we show corresponding
SchwingerDyson equation explicitly.
!
1
0
(1)
Re Tr P
11
A11 A12
,
(9)
=
0
(2)
12
A21 A22
Re Tr P
where
A11 =
A12 =
1
16
1
16
X
6=
6=
267
i
(1) (1)
(1) (1)
(1)
(1)
(1)
Re Tr P
P Tr P
P 13 Tr P
P
Tr P
,
h
(1) (2)
(1) (2)
(1)
(2)
(2)
Re Tr P
P Tr P
P 13 Tr P
P
Tr P
i
(1) H
(1) H
(1)
H
H
P Tr P
P 13 Tr P
P
Tr P
,
+ Tr P
X h
i
(2) (1)
(2) (1)
(2)
(1)
(1)
1
Re Tr P
P Tr P
P 13 Tr P
P
Tr P
,
A21 = 16
6=
A22 =
1
16
6=
(2) (2)
(2) (2)
(2)
(2)
(2)
Re Tr P
P Tr P
P 13 Tr P
P
Tr P
i
(2) H
(2) H
(2)
H
H
P Tr P
P 13 Tr P
P
Tr P
,
+ Tr P
(10)
with
(1)
(n) = V (n)V (n + )V (n + + )V (n + ),
P
(2)
(n) = V (n)V (n + )V (n + + )V (n + + 2)V (n + 2)V (n + ),
P
H
(n) = V (n) V (n + )V (n + + )V (n + )V (n + )V (n )
P
+ V (n + )V (n + 2)V (n + 2 + )V (n + + )V (n + ) .
(11)
3. Simulation and numerical results of the RG flow
Using the techniques in the previous section, we study the coupling flow induced by the
factor two blocking. We set the blocking parameter c = 0.5.
Series of simulations on lattices of size 84 and 164 are performed. The region surveyed
in the present work is the fourth quadrant of the (11 , 12 ) plane which covers most
improved actions presently known. We use the pseudo heat-bath method to generate the
gauge fields. The blocking transformation is carried out at more than 30 points. At each
point, about 100 configurations separated by every 100 sweeps are used to determine the
0 and 0 through the SchwingerDyson method. Examples in
renormalized couplings 11
12
the determination of the coupling are shown in Tables 1 and 2. The errors are given by the
Jack-knife method and they are relatively small even at deconfined points. Analyses are
made also for data samples in which configurations are separated by 1000 sweeps. Those
data agree with that of the standard samples within error bars. An example is shown in the
last two lines of Table 1.
We check also the effect of the finite temperature phase transition on the determination
of the coupling constants. Since the blocking transformation induces renormalization effects corresponding to a change of lattice cutoff, the resulting coupling shifts are insensitive
to the phases. We examine this point by comparing results belonging to the confined phase
on the 164 lattice while remaining deconfined on the 84 lattice at several values of as
268
Table 1
0 and 0 obtained through the SchwingerDyson method.
Examples of renormalized couplings 11
12
A hundred configurations are used at each point. For () , measurements are performed every 1000
pseudo heat-bath steps, while for the rest of the data we measure every 100 steps
11
12
0
11
0
12
7.00
7.00
11.00
13.20
()13.20
0.0
0.35
0.9981
1.493
1.493
13.188(22)
8.148(29)
15.189(73)
15.445(50)
15.519(37)
1.6564(94)
1.0431(98)
2.329(19)
2.559(15)
2.583(13)
Table 2
0 and 0 on the deconfined and confined lattices. These are calculated
Renormalized couplings 11
12
with 50 configurations, and measurements are performed every 100 steps
11
6.20
12
0.0
9.8496
0.8937
9.0
1.03332
0
11
0
12
9.515(56)
9.547(14)
1.156(23)
1.1671(53)
84 deconfined
164 confined
12.368(61)
12.280(14)
1.883(16)
1.8688(41)
84 deconfined
164 confined
1.046(17)
1.0456(30)
84 deconfined
164 confined
7.211(38)
7.2086(86)
Lattice size
shown in Table 2. The finite temperature phase transition on the 84 and 164 lattices for the
plaquette action takes place at approximately 11 = 5.9 and 6.3 which roughly correspond
to a = 0.27 and a = 0.14, respectively. As seen in Table 2, both results agree within
errors.
Our results are summarized in Fig. 1, in which the coupling shift resulting after the
blocking is indicated by arrows.
As shown in the figure, several characteristics of the flow are seen. If we start from the
plaquette action (12 = 0 line), renormalization results in a negative 12 as expected by
a perturbative analysis. At 11 = 6 8, the renormalization effect is very strong making
11 twice larger and 12 negative. The resulting points are far below the line of the treelevel Symanzik action. On the other hand, in the strong coupling region below 11 < 5, the
effect of the renormalization is to reduce 11 . Therefore, the plaquette action suffers large
renormalization effects and it is far from the renormalized trajectory.
Points starting on the line defined by the tree-level Symanzik action, 12 /11 = 0.05,
are RG transformed onto ones with 12 far more negative. Although renormalization
effects are reduced, the trend is the same as that of the plaquette action. This means
that tree-level Symanzik action is still far from blocking invariant. This indicates that
269
Fig. 1. Renormalization group flow of SU(3) lattice gauge theory in the two coupling space
(11 , 12 ). There is an attractive stream to which the arrows converge. Dotted and dashed lines
correspond to (tree level) Symanzik action and Iwasaki action, respectively. The long dashed line
corresponds to the DBW2 action introduced in Section 4. On the DBW2 line, the black dot is the
point reached by twice blocking from the plaquette action at = 6.3 on a 323 64 lattice [18].
perturbative O(a 2 ) improvement is insufficient at least for the currently used range of
lattice spacings.
Now we look at flows starting from points on the line corresponding to Iwasaki action,
i.e., 12 = 0.0907311. We see that renormalization occurs approximately along the line
up to an intermediate point (11 7.3, 12 = 0.0907311). At larger 11 , however,
flows depart from the line defined by this action. Thus the present blocking transformation
renormalizes Iwasaki action further and induces more negative values of 12 above the
intermediate coupling region.
Let us turn to the global structure of the flow. Blockings are made starting from points
with 12 /11 = 0.1 0.15. For those points, the renormalization effect is relatively
small and converges to a narrow stream. This trend is manifest at strong coupling. As a
whole, there is an attractive stream which the flow approaches quickly. Furthermore, once
the flow reaches the stream, it runs along it. Therefore actions on the attractive stream are
approximately blocking invariant apart from a normalization. The shape of the attractive
stream is clearly recognized as a parabolic curve in the strong coupling region while at
points far from the origin, it is less obvious with the present data. This is a remarkable
indication that the renormalized trajectory locates close to the (11 , 12 ) coupling space.
This is an encouraging result for finding a good improved action in this two coupling space.
A closer look at the attractive flow allows to extract more information. If we start from
the Wilson action at 11 > 6.0, the first blocking leads to larger renormalization effects
as 11 increases and the resulting flow vectors have the same direction pointing towards
increasing 11 further until the flow has reached the main attractive stream. This feature
270
seems consistent with a flow induced by an irrelevant coupling. In the strong coupling
region the behavior is quite different, under the blocking the coupling moves deeper into
strong coupling, with the main stream following a parabolic behavior, as already indicated
above. In Section 5, we will try to reproduce these features by a model equation including
an irrelevant coupling, an asymptotic scaling term and a driving term derived from the area
law behavior of Wilson loops.
invariance of the heavy quark potential and independence of Tc / on the lattice spacing.
Let us define a measure of violation of rotational symmetry as
!
X [V (R) Von (R)]2 X
1
2
,
(12)
V =
V (R)2 V (R)2
V (R)2
off
off
where V (R) is the static quark potential and V (R) is its error. Von (R) is a fitting function
P
to only on-axis data. off means summations over off-axis data. This quantity is measured
on the configurations generated by the DBW2 action. For comparison, plaquette-, treelevel Symanzik and Iwasaki improved actions are also examined. A 123 24 lattice is
used for this purpose and simulations are made for lattice spacings ranging from a = 0.15
to 0.4 fm. The statistics is a hundred configurations for each data point and the error is
given by the Jack-knife method. The results are summarized in Fig. 2.
In the figure, the horizontal axis is the lattice spacing squared so as to see the expected
O(a 2) violation. As seen in the figure, both Iwasaki and DBW2 actions show excellent
restoration of rotational symmetry even at a 0.3 fm while clear a 2 violations are seen
for plaquette and tree-level Symanzik actions.
ac = a(c ),
(13)
where Nt is the temporal lattice size. In order to obtain the critical coupling c , Nt = 3, 4, 6
lattices are used and the Polyakov loop susceptibility is measured. Here, histogram method
is utilized to determine the peak of the susceptibility [19,20]. Then, c at infinite volume
limit is obtained by finite size scaling of 123 4 and 163 4 lattices. The resulting c are
given in Table 3.
271
Ns
(11 + 812 )c at V =
3
4
6
10, 12, 14
12, 16
18
0.75696(98)
0.82430(95)
0.9636(25)
Table 4
Parameters of the heavy quark potential for the DBW2 action. These simulations are carried out on
123 24 and 183 36 lattices while the values of the coupling are those determined by Tc analysis
(Table 3). Statistics is 130 configurations per data at 11 + 812 = 0.82430 and 80 configurations
per data at 11 + 812 = 0.9636
11 + 812
0.82430
0.9636
a2
0.550(17)
0.5791(96)
0.255(23)
0.357(22)
0.1555(28)
0.06996(99)
The string tension is measured on 123 24 and 183 36 lattices at the values of the
coupling determined by the Tc analysis. It is extracted from the static quark potential using
the ansatz:
(14)
V (R) = A + + R.
R
From the extracted values of at c (Nt = 3), c (Nt = 4) and c (Nt = 6) we obtain the
following values of Tc / :
272
0.6340(60) at Nt = 4,
Tc / =
0.6301(65) at Nt = 6.
(15)
In Fig. 3 we show the obtained results together with other data from plaquette (Wilson), tree-level Symanzik, tadpole improved Symanzik and Iwasaki actions taken from
Refs. [21,22].
No appreciable dependence on the lattice spacing is seen in the ratio Tc / for all the
cases including the DBW2 action.
The results of both tests is an indication that the DBW2 action is a well-improved action
as suggested by the non-perturbative RG flow analysis.
Fig. 3. Scaling behavior of Tc / . We compare the DBW2 action with results from other actions in
two-coupling space [21,22].
Fig. 4. Contours of constant aTc in the (11 , 12 )-plane. Symbols represent the phase transition
points for each Nt at the thermodynamic limit. Dotted lines connect the points with the same
aTc (= 1/Nt ) and they suggest contours of aTc = const.
273
(16)
b1
+
x
(17)
(19)
and
(E
v w)
E
d(a) =
(E
n w)
E
Z1
d 1 B p ( a) .
(20)
E = p (a).
It is noted that (E
n )
Attractive driving force in the strong coupling region comes in from the string tension.
The lowest order of strong coupling calculation for N M Wilson loops is (suppose that
NM is an even integer)
W (N M) = (11 /18)NM + (11 /18)NM2 (12 /18)P1NM
NM
+ (11 /18)NM4 (12 /18)2 P2NM + + (12 /18)NM/2 PNM/2
, (21)
274
(a)
(b)
Fig. 5. (a) Lattice spacing versus 11 . Crosses are input data from the plaquette action given
by [2325]. Filled symbols are the data of tree-level Symanzik action (squares) and DBW2 action
(diamonds) [31]. The corresponding open symbols are the results predicted by Eq. (24). (b) Flow
trajectories calculated by Eq. (24). The trajectories start at (11 , 12 ) = (5.55, 0.0), (5.70, 0.0),
(6.20, 0.0) and (6.80, 0.0). Symbols on the trajectories indicate the points corresponding to the factor
two blocking where the lattice spacing is 2n a0 (n = 1, 2, 3, . . .) with initial value a0 .
where PkNM are the tiling weights for filling the area of the loop by (NM 2k)[1 1] and
k[1 2] tiles. Then, area law for the Wilson loop leads
11 /18 = exp(a 2 ),
12 /18 = C exp(2a 2 ).
.
0 2
da 2
(22)
(23)
nE
1
d E
1 + s1
0
E
E
= 2s
)
(24)
A
B(
+
, s a .
n
ds
s
|E
n|2
0
2 + s2
Here, we keep the next order in a in the non-perturbative term as free parameters, 1 and 2 .
E are specified as vE = (cos 0 , sin 0 ) and
Parameters included are , 0 , , 1 and 2 . vE and w
v wET ) with wET = i2 w.
E nE is
w
E = (cos , sin ). Here the matrix A is A = E
v wET t /(E
0
given by nE = (1, cot( )).
By this model, we calculate lattice spacings for different actions, RG flow trajectories
and contours of constant lattice spacings. Comparisons with the present data and those
reported in Refs. [30,31] are made. Giving initial conditions for 11 , 12 and a on the
12 = 0 line (plaquette action) [2325] , those quantities are calculated and shown in Figs. 5
and 6. 2 Parameters are chosen so as to match the recently reported data of a for the
Iwasaki action [30]
2 = 420 MeV is assumed for the data in Ref. [25].
275
Fig. 6. Contours of constant lattice spacing. Filled circles show the points with a = 0.4099,
0.2702 and 0.1619 as computed by Eq. (24). The data for (aTc )1 = 4 (open squares) and 6 (open
circles) obtained by the scaling analysis (Fig. 4) are also indicated.
= 0.156,
0 = 0.205,
= 0.5,
1 = 0.1,
2 = 0.0.
(25)
As shown in the figures, lattice spacings in the 11 12 plane and the RG flow
trajectories are reasonably well reproduced. As for the flow, rapid approach to the attractive
stream is driven by the irrelevant coupling. At intermediate lattice spacings, both the nonperturbative and asymptotic scaling terms drive the trajectories. Finally, in the strong
coupling region, the non-perturbative term dominates and the parabolic behavior of the
trajectories sets on. We also note that the model describes contours of constant lattice
spacing fairly well as seen in Fig. 6.
Through the study, an understanding for basic driving mechanism of the RG flow in
11 12 plane by scaling term and an irrelevant coupling at small lattice spacing and an
attraction originated from area law in strong coupling region is suggested.
276
loops in the SchwingerDyson equation. But this effect might also be due to the truncation
of the space of couplings. See Ref. [17].
In order to examine renormalization effects outside to the two coupling space, we
perform a limited analysis in a three coupling space, (11 , 12 , twist), with a twisted loop
included in the action [28],
X
1 13 Re Tr Ptwist ,
(26)
Stwist = twist
n,6=6=
where
Ptwist = U (n)U n + U n + + U n + + +
U n + + U n + .
(27)
We will study twist in addtion to (11 , 12 ) because in Ref. [18] more general actions
were studied and it was found that the value of the twist coupling is larger that that of
the chair type loop. Moreover the action with W11 , W12 and Wtwist is often used in the
Fig. 7. RG flow of SU(3) lattice gauge theory in three coupling space (11 , 12 , twist ). Blocking is
applied for the points near the attractive stream on the (11 , 12 )-plane at twist = 0. The top figure
shows the projection of the flow onto the (11 , twist )-plane while the bottom figure is the projection
onto the (11 , 12 )-plane.
277
literatures [2,32]. We perform the blocking transformation starting from points near the
attractor in the (11 , 12 ) plane, i.e., the twist = 0 sector. The results are shown in Fig. 7.
As shown in the figure, no sizable value of twist is generated for points on the attractor.
On the other hand, if we start from the points 11 10, 12 1.0, a negative twist is
induced. This indicates that the attractor in three coupling space sits below the twist = 0
sector in this region. Although more extensive studies are necessary to clarify the exact
situation, it seems that the attractor in two coupling space gives a good starting point for
designing improved actions.
7. Summary
We investigate the renormalization group (RG) flow of SU(3) lattice gauge theory in
a two coupling space, (11 , 12 ). An extensive numerical calculation of the RG flow on
the lattice is made. Swendsens blocking followed by an effective action search using
the SchwingerDyson method is adopted to find renormalization effects. In our previous
analysis, we adopted the Demon method. Although both methods give similar results,
the SchwingerDyson algorithm is more echonomical because in case of the Demon
formula we need an additional Monte Carlo simulation for each configuration. Moreover
the SchwingerDyson method uses larger loops to determine the couplings.
Analyses are performed in the fourth quadrant of the coupling space and reveal the
presence of an attractive stream. Trajectories are first attracted towards this stream and
afterwards they move towards the origin along it. The stream converges to a parabolic
curve in the strong coupling region. These features indicate that the RT locates close to the
two coupling space and the attractive stream traces the RT.
A model flow equation which consists of asymptotic scaling, an irrelevant coupling and
a non-perturbative force corresponding to the area law can reproduce the observed features.
In this paper we have compared several actions in the two coupling space by measuring
the restoration of the rotational symmetry and the scaling of /Tc . In the regions of a
0.3, Iwasaki and DBW2 actions, which are near to RT, are superior to tree-level Symanzik
and plaquette actions. Although improveness of all actions are indistinguishable at
smaller lattice spacing within the present statistics, this indicate that non-perturbative study
of RG flows is valuable to design an improved action at intermediate lattice spacing. It is
highly desirable to pursue the same analysis for fermion actions.
The effect of truncation to a space with only two couplings is partly examined and it
is found that renormalization effects outside the two coupling space are small and the
attractive stream in this space gives hence a good starting point for improvement.
Acknowledgments
All simulations have been done on CRAY J90 at Information Processing Center,
Hiroshima University, SX-4 at RCNP, Osaka University and on VPP500 at KEK (High
Energy Accelerator Research Organization). The authors would like to acknowledge
M. Okawa for discussions on the SchwingerDyson method. They acknowledge also
278
G. Bali for performing the measurement of the heavy quark potential. H. Matsufuru would
like to thank the Japan Society for the Promotion of Science for financial support. This
work is supported by the Grant-in-Aide for Scientific Research by Monbusho, Japan (No.
11694085) and (No. 10640272). This work is supported also by the Supercomputer Project
No32 (FY1998) of High Energy Accelerator Research Organization (KEK).
References
[1] K.G. Wilson, in: G. tHooft (Ed.), Recent Developments in Gauge Theories, Plenum Press, New
York, 1980, p. 363.
[2] R. Gupta, The Renormalization group and lattice QCD, in: T. DeGrand, D. Toussaint (Eds.),
From Actions to Answers, World Scientific, 1990; in: Introduction to lattice QCD: course
lectures given at Les Houches Summer School in Theoretical Physics, Session 68: Probing
the Standard Model of Particle Interactions, Les Houches, France, 28 July5 September, 1997.
[3] QCD-TARO Collaboration, Phys. Rev. Lett. 71 (1993) 3063.
[4] A. Gonzlez-Arroyo, M. Okawa, Phys. Rev. D 35 (1987) 672; Phys. Rev. B 35 (1987) 2108.
[5] M. Hasenbusch, K. Pinn, C. Wieczerkowski, Phys. Lett. B 338 (1994) 308.
[6] T. Takaishi, Mod. Phys. Lett. A 10 (1995) 503.
[7] A. Patel, R. Gupta, Phys. Lett. B 183 (1987) 193.
[8] Y. Iwasaki, University of Tsukuba, UTHEP-118, 1983, preprint.
[9] P. Hasenfratz, F. Niedermayer, Nucl. Phys. B 414 (1994) 785.
[10] Ph. de Forcrand et al., QCD-TARO Collaboration, Nucl. Phys. B Proc. Suppl. 53 (1997) 938.
[11] R.H. Swendsen, Phys. Rev. Lett. 47 (1981) 1775.
[12] M. Creutz, Phys. Rev. Lett. 50 (1983) 1411.
[13] M. Hasenbush, K. Pinn, C. Wieczerkowski, Phys. Lett. B 338 (1994) 308.
[14] M. Creutz, Quarks, Gluons and Lattices, Cambridge Univ. Press, 1985, Chapter 10.
[15] K. Symanzik, Nucl. Phys. B 226 (1983) 187.
[16] S. Itoh, Y. Iwasaki, T. Yoshie, Phys. Rev. D 33 (1986) 1806.
[17] T. Takaishi, Ph. de Forcrand, Phys. Lett. B 428 (1998) 157.
[18] T. Takaishi, Phys. Rev. D 54 (1996) 1050.
[19] I.R. McDonald, K. Singer, Discuss. Faraday, Soc. 43 (1967) 40.
[20] A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 61 (1988) 2635; Phys. Rev. Lett. 63 (1989)
1195.
[21] Y. Iwasaki, K. Kanaya, K. Kaneko, T. Yoshi, Phys. Rev. D 56 (1997) 151.
[22] B. Beinlich, F. Karsch, E. Laermann, A. Peikert, Eur. Phys. J. C 6 (1999) 133.
[23] G.S. Bali, K. Schilling, Phys. Rev. D 46 (1992) 2636.
[24] G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Ltgemeier, B. Petersson, Nucl.
Phys. B 469 (1996) 419.
[25] S. Aoki et al., CP-PACS Collaboration, Phys. Rev. Lett. 84 (2000) 238.
[26] F. Karsch, B. Beinlich, J. Engels, R. Joswig, E. Laermann, A. Peikert, B. Petersson, Nucl. Phys.
B Proc. Suppl. 53 (1997) 413.
[27] Ph. de Forcrand et al., QCD-TARO Collaboration, Nucl. Phys. B Proc. Suppl. 63A-C (1998)
928.
[28] Ph. de Forcrand et al., QCD-TARO Collaboration, Nucl. Phys. B Proc. Suppl. 73 (1999) 924.
[29] Ph. de Forcrand et al., QCD-TARO Collaboration, Proc. of the LATTICE99, Pisa, 1999, to
appear in Nucl. Phys. B Proc. Suppl.
[30] M. Okamoto et al., CP-PACS Collaboration, Phys. Rev. D 60 (1999) 094510.
[31] A. Borii, R. Rosenfelder, Nucl. Phys. B Proc. Suppl. 63A-C (1998) 925.
[32] M. Alford, W. Dimm, G.P. Lepage, G. Hockney, P.B. Mackenzie, Nucl. Phys. B Proc. Suppl. 42
(1995) 787.
Abstract
We generalize overlap fermion by Narayanan and Neuberger by introducing a hopping parameter t.
This lattice fermion has desirable properties as the original overlap fermion. We expand Dirac
operator of this fermion in powers of t. Higher-order terms of t are long-distance terms and this
t-expansion is a kind of the hopping expansion. It is shown that the GinspargWilson relation is
satisfied at each order of t. We show that this t-expansion is useful for study of the strong-coupling
gauge theory. We apply this formalism to the lattice QCD and study its chiral phase structure at strong
coupling. We find that there are (at least) two phases one of which has desired chiral properties of
QCD. Possible phase structure of the lattice QCD with the overlap fermions is proposed. 2000
Elsevier Science B.V. All rights reserved.
PACS: 11.15.Ha; 11.15.Me; 12.38.Gc
Keywords: Lattice QCD; GinspargWilson relation; Overlap fermion; Strong coupling
1. Introduction
Formulation of lattice fermion is a long standing problem. Recently a very promising
formulation named overlap fermion was proposed by Narayanan and Neuberger [1,2] and
it has been studied intensively. However, almost all (analytical) studies on the overlap
fermion employ the weak-coupling expansion. Study on the overlap fermion interacting
with the strong-coupling gauge field is desired. Especially to clarify its phase structure is
very important, e.g., for numerical studies and in order to take the continuum limit.
In this paper we shall give a way for study of strong-coupling gauge theory of the overlap
fermion. To this end, we slightly generalize the original overlap fermion by introducing
a hopping parameter t. This lattice fermion, which we call generalized overlap (GO)
fermion, has desirable properties as the original overlap fermion. We can expand Dirac
1 E-mail: ikuo@hep1.c.u-tokyo.ac.jp
2 E-mail: nagao@hep1.c.u-tokyo.ac.jp
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 1 3 - 9
280
(m)D(m,
n)(n),
(1)
SF = a d
n,m
(2)
where r and M0 are dimensionless nonvanishing free parameters of the overlap lattice
fermion formalism. We have introduced a new parameter t. The original overlap fermion
corresponds to t = 1. For notational simplicity, we define
t
rt
1
(3)
A (dr M0 ), B , C .
a
2a
2a
It is verified that propagator at tree level U (n) = 1 has no species doublers for the
parameter region (1 t)dr < M0 < (d dt + 2t)r. This parameter region is renormalized
by the interactions. Therefore it is important to study phase structure of the system in wide
parameter region of M0 and the gauge coupling constant g 2 with fixed values of r and t, for
281
example. It is also verified by the weak-coupling expansion that the GO fermion generates
the ordinary chiral anomaly as it is desired. Actually it is verified that the t-dependence of
D(m, n) is absorbed by redefinition of the parameter M0 . In this sense the GO fermion is
not quite new.
In this paper we shall expand the GO fermion operator (2) in powers of t assuming
that t is small. As we shall show, this t-expansion is a kind of the hopping expansion and
then we expect that the t-expansion is justified and suitable for the strong-coupling gauge
theory. At strong-coupling region, movement of a single quark is suppressed by the strong
fluctuation of the gauge field, i.e., hU (m)i 0, and the number of paths in the randomwalk representation of correlation function of gauge-invariant composite fields is much
smaller than that of weak-coupling cases. 3
For notational simplicity, let us define the following quantities,
(m, n) = m+,n U (m) m,n+ U (n),
+ (m, n) = m+,n U (m) + m,n+ U (n).
In terms of the above quantities,
X
X
(m, n) B
+ (m, n),
Xmn = Amn + C
X
X
(m, n) B
+ (m, n).
X mn = Amn C
(4)
(5)
(6)
From Eq. (3), B, C = O(t) and we consider A = O(1) in later discussion. Then it is rather
straightforward to expand D(m, n) in powers of t,
p
|A|B X +
C2 X
X X mn = |A|mn
(m, n)
(m, l) (l, n)
A
2|A|
BC X
+ (m, l) (l, n) (m, l)+ (l, n) + O(t 3 ),
2|A|
C X
1
= sgn(A)mn +
(m, n)
X
|A|
X X mn
BC X
(m, l)+ (l, n) + + (m, l) (l, n)
+
2A|A|
C2 X
(m, l) (l, n) + O(t 3 ),
+
2A|A|
C X
(m, n)
aD(m, n) = 2 (A)mn +
|A|
BC X
(m, l)+ (l, n) + + (m, l) (l, n)
+
2A|A|
C2 X
(7)
(m, l) (l, n) + O(t 3 ).
+
2A|A|
3 However, we also expect that the t-expansion has a finite convergence radius for smooth configurations of
fields, for higher-order terms contain higher-powers of the difference operator (m, n) defined by Eq. (4). In the
strong-coupling phase, on the other hand, the good convergence of the t-expansion is expected after integration
over the fluctuating gauge field.
282
(8)
= (m)
(9)
(m) = 5 nm aD(m, n) (n), (m)
5,
where is an infinitesimal transformation parameter. Then in the overlap fermion
formalism, it is natural to think that the extended chiral symmetry (9) is more fundamental
than the usual chiral symmetry which is broken at finite lattice spacing.
In terms of the t-expansion, the transformation (9) is given as,
C X
(m, n) + (n),
(m) = 5 sgn(A)mn
|A|
(10)
(m)
= (m)
5.
This expression reveals the fact that the extended chiral transformation (9) has quite
different meanings depending on the sign of the parameter A. Especially for positive A,
3. Strong-coupling limit
In the previous section, we t-expanded the Dirac operator of the GO fermion. We
show that the t-expansion is suitable for the strong-coupling studies of the lattice QCD
whose action is given by,
Stot = SG + SF,M ,
1 X
Tr U U U U ,
SG = 2
g
pl
X
(m)(m),
SF,M = SF MB
(11)
where we have added the bare mass term of quarks. In this section, we shall study
chiral structure of the above system in the strong-coupling limit g 2 N , though the
strong-coupling expansion can be performed systematically. As explained above, we must
consider the cases of positive and negative values of A separately. In this section we mostly
set the lattice spacing a = 1.
3.1. Negative A
We shall consider U(N ) gauge theory for definiteness. The partition function of the
system is given by
Z
Z[J ] =
n
o
X
D D DU exp Stot +
J (n)b
m(n) ,
283
(12)
(13)
with color index a and spinor-flavor indices and . The effective action Seff (M) is
defined as
Z
(14)
Z[J ] = DM eSeff (M)+J M
where integral over color-singlet meson field M will be defined later on.
To obtain Seff (M), it is useful to notice that the following combination is invariant under
the extended chiral transformation (10),
a
= q(n)
q 1 D , q = ,
5,
2
5 q(n)
qq(m)
qq(n)
q
5 q(m)q
5 (n)
5 (m)
= (m)
(n)
n
X
C
(m)
(m, l)(l) 5 (n)
+
5
2|A|
X
(m)
(m, l)(l) (n)
o
+ (m n) + O(t 2 ).
The first step for the effective action is the one-link integral of the gauge field,
Z
W (D,D)
= dU exp Tr D U + U D .
e
(15)
(16)
For the U(1) gauge group, the above integral is easily performed as W (D, D) = DD
1
2
4 (DD) + . For the U(N ) gauge theory W (D, D) was calculated by Brezin and Gross
for large N . There are two phases in the above one-link integral, and in the strongcoupling regime, which is relevant for the present study, W (D, D) is given by the following
formula [5],
2 X
3
(c + xa )1/2
W D, D = N 2 c +
4
N a
1 X
1/2
1/2
log
(c
+
x
)
+
(c
+
x
)
,
(17)
a
b
2N 2
a,b
1
DD
N2
1 X
(c + xa )1/2 .
2N a
284
In the present study, the Dirac operator D(m, n) is given by (7), and therefore for the
expansion in powers of t, we set
a
a
= Aab + Cb
,
Db
D ab = Aab + C ab ,
C
b (n + ) a (n),
|A|
C
Aab =
b (n) a (n + ),
|A|
Aab =
(19)
where C and C are sources. The following identity is useful which is proved for an
arbitrary regular function f (x),
1
1 X
f (xa ) = f (0) Tr f () f (0)
N a
N
C
1
C (n)ba Tr a (n) b (n + ) f 0 ()
3
N |A|
C
1
C (n)ba Tr a (n + ) b (n) f 0 (0 )
+ 3
N |A|
+ O C2 ,
(20)
where
2
C
m
b(n) m
b(n + ) ,
A
2
C
0
0
m
b(n + ) m
b(n) .
= (n) =
A
= (n) =
(21)
From Eqs. (17) and (20) we obtain the one-link integral as follows [6,7],
1
1 + (1 4)1/2
1
1
1/2
W D, D = Tr (1 4) 1 + Tr log
N2
N
N
2
1
C
2
C (n)ba Tr a (n) b (n + ) 1 + (1 4)1/2
3
N |A|
1
C
2
C (n)ba Tr a (n + ) b (n) 1 + (1 40 )1/2
+ 3
N |A|
(22)
+ O C2 .
From (22), the expectation value of the gauge field is given as follows in the strongcoupling limit,
1
a
2 C
,
Tr a (n) b (n + ) 1 + (1 4)1/2
Ub (n) U =
N |A|
1
a
2 C
Tr a (n + ) b (n) 1 + (1 40 )1/2
,
(23)
Ub (n) U =
N |A|
where h iU denotes average over the gauge field U (n).
285
In Ref. [8] spontaneous symmetry breakdown of the extended chiral symmetry is argued
and an order parameter is given by
(24)
hqqi
= 1 a2 D .
5 ) and their mass m2 MB (in the
NambuGoldstone bosons appear in the channel (
flavor-nonsinglet channel). 4 The above criterion is verified by using solvable models [9].
We shall calculate the order parameter (24) in the present formalism in the strongcoupling limit. After the integral over the gauge field, the partition function is given by the
functional integral over the fermions with a new action which is a functional of the colorsinglet composite mesons m
b (n). Moreover because of the extended chiral symmetry, the
action of m
b (n) depends on the chiral invariants like (15) with the replacement of gauge
fields as in (23). Flavor-singlet extended chiral symmetry is explicitly broken by the measure of the fermion path integral. Effects of anomaly will appear in the next-leading order of
1/N and the noninvariance of the fermion measure is related with the U(1) problem [4,10].
Elementary meson fields are introduced through the identity like (up to irrelevant
constants),
Z
N
1 a
J a = det J
d d exp
N
I
N J M
= dM det M
e
,
(25)
where the integral over M is defined by the contour integral, i.e., M is polar-decomposed
H
as
M
=
RV
with
positive-definite
Hermitian
matrix
R
and
unitary
matrix
V
,
and
dM
R
dV with the Haar measure of U(Nsf ) (Nsf is the dimension of the spinor-flavor index) [7].
From (25), there appear additional terms like (N Tr log M) in the effective action. Detailed
study of the low-energy effective theory of hadrons will be given in a forthcoming paper
[10]. In this paper we shall calculate the order parameter (24). From the discussion of the
extended chiral symmetry given above and in Ref. [8], it is obvious that NambuGoldstone
pions appear if the spontaneous chiral symmetry breakdown h (1 a2 D)i 6= 0 occurs.
We assume the pattern of symmetry breaking for simplicity. By the existence of the bare
mass term of quarks,
(26)
M = v ,
where v is some constant which will be calculated from now. From (21) and (26),
2
C
0
v2 .
= =
A
(27)
(28)
286
4. Higher-order terms
(31)
in the action SF . This term is completely determined by the GW relation from the term
C X
(m)
(m, n)(n)
|A|
in the action.
It is tedious but straightforward to verify that contribution from the term (31) changes
the effective potential in (28) as
Veff qq
.
(32)
Veff
To this end we use equations like
a
C 2 a
b
(m + ) U =
Tr (m) b (m + ) g 0 ( (m))
Ub (m)Uc
NA
Tr b (m + ) c (m + + ) g 0 ( (m + )) , (33)
where
1 + (1 4x)1/2
,
g(x) = (1 4x) + 1 + log
2
1
g 0 (x) = 2 1 + (1 4x)1/2 .
1/2
(34)
like 5
1
2N
C
A
!4
m
b(m) a (m + ) b (m + + )
287
b (m + + ) a (m + ) g 0 ( (m))g 0 ( (m + )) + .
(35)
m
b(m) qq(m
+ ) (m
+ )
C X
(m + ) (m + , n)(n) + .
= b
m(m)
2|A|
After the path-integral over the gauge fields,
m
b(m) qq(m
+ ) (m
+ )
1 C 2
m
b(m) a (m + ) b (m + + )
2N A
b (m + + ) a (m + ) g 0 ( (m + )) + .
(36)
(37)
Then from Eqs. (21), (35) and (37), one can see that the contribution from the high-order
term (31) simply replaces (m) in the effective potential with
2
C
q q(m
+ )q q(m).
A
We have verified this result only at the lowest-nontrivial order, but we expect that it is
correct at all orders of t.
For completeness, we have to examine the term which comes from the M-integral and
contributes to the effective potential. We evaluate the following integral instead of Eq. (25),
Z
1
J q q .
(38)
d d exp
N
We can change the measure of the above path-integral as
(d d) (dq dq),
but there appears additional term from Jacobian,
D 1 D 2
1
.
Tr log 1 2 D = Tr
2
2 2
(39)
It is verified that at low-orders of t the above factor does not contribute. However, it is
expected that nontrivial terms which depends on the gauge fields will appear at sufficiently
high-order of t. This problem is currently under study and the results will be reported in a
future publication [10].
4.1. Positive A
In the previous section we showed that for negative A the system has desired properties
concerning the chiral symmetry and we shall call this phase QCD phase. In this section we
288
shall briefly study the case of positive A. For positive A, the t-expanded D(m, n) is given
as
C X
(40)
(m, n) + O(t 2 ).
D(m, n) = 2mn +
|A|
P
(41)
(m) = (m)5 ,
and therefore and have opposite extended-chirality with each other (at leading order
of t). This fact suggests that the phase of positive A is different from that of negative A. 6
Analysis of the effective action at the strong-coupling limit in the previous section can
be applied also for the case of positive A, and it is shown that condensation hi
has
nonvanishing value. However, in the case of positive A,
h
+
6= 0.
(42)
limit
M
M
MB 0
(43)
+ O(t).
5 =
(44)
i =
However, the strong-coupling analysis similar to that for negative A shows h
0. Next one is the nearest-neighbor quark bilinear h U i. From the analysis at the
strong-coupling limit (23), we can expect nonvanishing expectation value of the above
operators. Moreover their values depend on the direction, i.e.,
(45)
+ ) U (m)(m) 6= 0.
(m)
U (m)(m + ) = (m
As
X
C
(m)
(m, n)(n) + O(t 2 ),
1 a2 D (m) =
2|A|
1 a2 D 6= 0.
(46)
(47)
6 Strictly speaking, we cannot deny the possibility that these states are connected by a crossover rather than
a phase transition, since our analysis cannot be applied for small |A|. However, existence of a phase transition
between them is plausible. See later discussion.
289
Then from the criterion in [8] we can expect appearance of a massless particle at the
5 ), though meaning of the extended chiral symmetry is quite different from
channel (
the usual chiral symmetry in this phase.
5. Discussion
In this paper we study properties of the lattice QCD with the overlap fermions at strong
coupling. To this end, we slightly generalize the ordinary overlap fermion by introducing
the parameter t. We expand the Dirac operator of the GO fermion in powers of t, and then
apply the standard techniques of the strong-coupling expansion.
It is important and urgent to examine validity and applicability of the t-expansion, for a
drawback of the overlap fermion is its nonlocality. By numerical calculation it is verified
that for smooth configurations of the gauge field the locality is satisfied [11]. On the
other hand, we also expect that the locality is satisfied even at strong gauge coupling after
integration over the gauge field in certain parameter region of the GO fermion. Results in
the present paper support this expectation but more intensive studies are required. Tractable
models in low dimensions might be useful. Both numerical and analytical studies on them
are needed in order to argue the applicability of the t-expansion.
We find that there are (at least) two phases in the lattice QCD with the GO fermions at
strong coupling. One of them has desired properties of QCD. Though this result is obtained
by using the t-expansion, we expect that it is correct even for the ordinary overlap fermion
system since expansion parameters are B/A and C/A. Therefore for sufficiently large |A|
our results are applicable. In Fig. 1, we show a possible phase diagram of the lattice QCD
Fig. 1. Schematic phase diagram of the lattice QCD with the overlap fermions in the (1/(g 2 N), M0 )
plane. Phase A has desired properties of QCD. Extended chiral symmetry is spontaneously broken
and quasi-massless pions appear. On the other hand, the phase B is anomalous. Phase C in between is
the nonlocal phase in which the long-distance terms give important effects. The critical lines which
separate phases A, B and C may have strong dependence on the gauge-coupling constant g 2 though
they are almost vertical in the figure.
290
with the (generalized) overlap fermions. The phase A has desired chiral properties of QCD
and we call it QCD phase. On the other hand, the phase B is anomalous as we explained
in Section 3. The phase C in between cannot be studied by the present techniques, for
the t-expansion is not applicable. There it is expected that nonlocal-long-distance terms
cannot be neglected and they give substantially important effects on physical properties.
For example, the Goldstone theorem assumes the locality of the system as is well-known.
Therefore in the phase C, which we call nonlocal phase, massless pions might not appear
even if the extended chiral symmetry is spontaneously broken. It is possible that the
nonlocal phase C does not exist in certain parameter region of (r, t).
Detailed study on the QCD phase will be reported in a forthcoming paper [10].
Especially, it is interesting to see how the U(1) problem is solved and how its relates with
the noninvariance of the fermion path-integral measure under flavor-singlet extended chiral
transformation [4]. In the framework of the t-expansion, the Jacobian Tr(5 D) is easily
evaluated at low-orders of t, and it is verified that term corresponding to the anomaly
appears. However, its coefficient is not constant but depends on t. Another interesting
problem is the strong-coupling chiral gauge theory [12]. This system might be studied by
using the techniques in this paper.
R. Narayanan, H. Neuberger, Nucl. Phys. B 412 (1994) 574; Nucl. Phys. B 443 (1995) 305.
H. Neuberger, Phys. Lett. B 417 (1998) 141.
P.H. Ginsparg, K.G. Wilson, Phys. Rev. D 25 (1982) 2649.
M. Lscher, Phys. Lett. B 428 (1998) 342.
E. Brezin, D.J. Gross, Phys. Lett. B 97 (1980) 120.
I. Ichinose, Nucl. Phys. B 249 (1985) 715.
N. Kawamoto, J. Smit, Nucl. Phys. B 192 (1981) 100.
S. Chandrasekharan, Phys. Rev. D 60 (1999) 074503.
I. Ichinose, K. Nagao, hep-lat/9909035.
I. Ichinose, K. Nagao, hep-lat/0001030.
P. Hernandez, K. Jansen, M. Lscher, Nucl. Phys. B 552 (1999) 363.
M. Lscher, Nucl. Phys. B 538 (1999) 515; Nucl. Phys. B 549 (1999) 298; hep-lat/9904009.
C. Gattringer, I. Hip, hep-lat/0002002.
Abstract
In the t HooftVeltman dimensional regularization scheme it is necessary to introduce finite
counterterms to satisfy chiral Ward identities. It is a non-trivial task to evaluate these counterterms
even at two loops. We suggest the use of Wilsonian exact renormalization group techniques to reduce
the computation of these counterterms to simple master integrals. We illustrate this method by a
detailed study of a generic Yukawa model with massless fermions at two loops. 2000 Elsevier
Science B.V. All rights reserved.
PACS: 11.10.G; 11.15
Keywords: Dimensional regularization; Wilsonian renormalization group
1. Introduction
The dimensional regularization scheme devised by t Hooft and Veltman [1,2] and
later systematized by Breitenlohner and Maison [3] (BMHV) is the only dimensional
regularization scheme which is known to be consistent in presence of 5 . It gives the correct
result for the axial anomaly, at the price of breaking d-dimensional Lorentz symmetry and
chiral symmetry; while the former is easily recovered, it is a non-trivial task to satisfy the
chiral Ward identities.
More popular is the naive dimensional regularization scheme (NDR) [4] which, although
inconsistent, is much easier to use and can be handled safely in most practical situations.
For a review on the subject, see, e.g., [5].
Due to the relevance of higher loop computations in the standard model, where it is
difficult to guarantee the consistency of the naive scheme, it is worthwhile to investigate
thoroughly consistent renormalization schemes.
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 0 3 - 6
294
The difficulties encountered in computing systematically in the BMHV scheme the noninvariant counterterms can be divided in three categories:
(i) It is a complicated (but generally solvable) algebraic problem to satisfy all the
Ward identities (modulo anomaly problems), determining these counterterms as
combinations of Feynman integrals;
(ii) It is an analytically difficult problem to evaluate these counterterms, which in
general involve Feynman integrals with several masses and/or momenta; explicit
formulae with several masses at more than one loop usually range from being quite
complicated (see, e.g., [6]) to being as yet unknown;
(iii) It is burdensome to store these counterterms (which include evanescent ones) and
to compute the renormalization group beta function using them [7].
By comparison, we recall that in a vectorial theory (or in NDR, whenever possible)
step (i) is trivial in the minimal subtraction (MS) scheme, since the Ward identities are
automatically satisfied; step (ii) is much simpler, since the poles are much easier to
compute than the finite parts of the Green functions; step (iii) is almost trivial, since in
general multiplicative renormalization holds; simple formulae for the beta and gamma
renormalization group functions are available in the MS scheme [2].
The short-cuts used to get the appropriate answer for the problem at hand include:
(I) In a few cases, like in the case of an axial current insertion in a vectorial gauge
theory, it is sufficient to equate the axial vertex to the corresponding vector vertex
multiplied by 5 to be sure that the chiral Ward identities are satisfied [8];
(II) After solving the algebraic problem, avoid carrying out step (ii); this is the
philosophy of algebraic renormalization (for a review see [9]), used generally
only for demonstrative purposes, but which can also be used for making explicit
computations, as shown in some two-loop examples carried out recently [10].
To our knowledge the only paper in which the counterterms are determined systematically at one loop in the BMHV scheme in a chiral gauge theory is [11], in which the
BonneauZimmermann [12] identities are used.
There are several computations of one-loop counterterms in the BMHV scheme for
particular processes in the standard model [13,14], but no systematic one-loop treatment
has been given in such a scheme.
In this paper we describe a general method for simplifying step (ii) and partly (iii), i.e.,
we show that the counterterms can be reduced to zero-momentum Feynman integrals with
the same auxiliary mass in all propagators, and that the beta function can be computed
easily without making a direct use of the counterterms.
The auxiliary mass technique in this form has been used in the past at two or more
loops [1517] in conjunction with MS. Being the MS scheme mass-independent [18,
19], the auxiliary mass technique gives gauge-breaking terms that are polynomial in the
auxiliary mass and then can be easily treated; however in chiral theories with BMHV the
MS scheme cannot be used, and it is not easy to disentangle the gauge-breaking terms.
We will show how to do this by Wilsonian methods [20].
Wilsonian methods have been used by Polchinski [21] to simplify the proof of
renormalizability in 4 ; this proof has been further simplified and generalized to other
295
cases [2230]; in particular gauge theories have been renormalized using the effective
Ward identities introduced in [22]. In [30] mass-independent renormalization has been
studied with these Wilsonian methods; it has been also shown in this paper that the
Wilsonian effective action satisfies an effective renormalization group equation, which is
the analogous of the effective Ward identities.
In this paper we propose the exploitation in the BMHV scheme of the effective Ward
identities and effective renormalization group equation to compute the finite counterterms
and the beta function in terms of Wilsonian Green functions at zero momenta and masses,
which are easy to evaluate.
Our method is based on the use of a class of Wilsonian flows characterized by the cut-off
function
n
2
(n)
(x) =
, n = 2, 3, . . . ,
K
x + 2
to separate the propagators into hard and soft parts.
In the Feynman rules for the Wilsonian functional the usual local vertices are unchanged
whereas the propagator D is replaced by the hard propagator
(n)
D = 1 K D.
Because of the analyticity properties of this hard propagator the dimensional regularization
can be used for the ultraviolet divergences. The counterterms are defined by suitable
renormalization conditions at = R which can be chosen equal to the dimensional
regularization scale .
The renormalization program can be performed on the Wilsonian functional with the
following advantages:
(i) Since the hard propagator D is regular at zero momentum there are no
infrared problems in the evaluation of the counterterms, so that one can choose
renormalization conditions at zero momenta and masses in a mass-independent
Wilsonian renormalization scheme [30]. The counterterms turn out to be sums of
ordinary Feynman integrals at zero external momenta and with propagators having
the same effective mass = , maintaining all the computational advantages of
the auxiliary mass method. At two loops these Feynman integrals can be reduced
to a single master integral using recursion relations (for a review see [16]);
(ii) For a Wilsonian flow with n > 2 it is simple to prove that the renormalization at a
fixed implies the finiteness of the Wilsonian effective action at every value of
the Wilsonian flow, in particular at = 0;
(iii) For a Wilsonian flow with n > 2 the Ward identities and the renormalization
group equation on the usual functional generator Z = Z=0 are equivalent to the
corresponding effective Ward identities [22] and effective renormalization group
equation [30] on Z , so that choosing renormalization conditions compatible with
the effective Ward identities the validity of the Ward identities on Z follows;
furthermore the renormalization group beta functions can be expressed in terms
of Feynman integrals at zero momenta and masses.
296
= d.
(1)
(2)
where
tr I = 4,
I = I = .
(3)
In the BMHV scheme O(d) invariance is broken and one introduces O(4) O(d 4)
invariant tensors: the (d 4)-dimensional Kronecker delta and the 4-dimensional
antisymmetric tensor , satisfying
= ,
= ,
= 0.
297
(4)
(5)
1
4!
(6)
satisfying
{5 , } = {5 , } = 2 5 ,
52 = I.
(7)
In Euclidean space the reflection symmetry takes the place of hermiticity. Reflection
symmetry is an antilinear involution, under which
= ,
x0 1
= x 1 ,
x0
0 )1 ,
(x) = (x
(x)
= 1 (x 0 ),
(8)
= x
for 6= 1.
where
For a multiplet of fermionic fields, a general local marginal fermionic bilinear, involving
a scalar (A), pseudoscalar (B), vector (V ) and pseudovector (A ), all of them real fields,
can be written as
Z
H1 + H2 [ , 5 ] + H3 + iH4 5
Z
+ iH5 A + H6 5 B + iH7 V + iH8 [ , 5 ]A + iH9 V + H10 5 A , (9)
where Hi are matrices over flavour indices. Reflection invariance requires that Hi are
Hermitian.
The Green functions in d dimensions are obtained performing the Dirac algebra in the
Feynman graphs with the above rules. Actually the analytic continuation to continuous
dimensions is defined on the scalar coefficients of the Green functions expanded on a basis
for tensorial structures. For chiral theories, being O(d) broken, such a basis includes also
hatted tensors.
The poles of the Green functions for 0 are removed by local counterterms. Loop by
loop the singular part of the counterterms must subtract exactly the pole part of the Green
functions, including the hatted components; the finite parts of the counterterms instead
are not uniquely determined by the renormalization conditions which constrain only their
4-dimensional part. In fact the d 4 prescription requires, besides to take the limit 0,
to set the hatted tensors to zero.
In order to fix completely the fermionic counterterms, in the decomposition (9) we will
choose that H3 , H4 , H9 , H10 have vanishing finite term in their Laurent expansion. The
same convention is assumed for the coefficients bij , dija and dijab in the bilinear scalar
counterterms, which have the form
Z
1
i j aij + bij + Va i j cija + i j dija
2
(10)
+ Va Vb i cijab j + Va Vb i dijab j .
298
i i + cij kl i j k l ,
i cij j + c
+ /2c
S=
2
4!
(11)
ci = iyi ,
c(0)(p) = i p ,
(0)
cij kl = hij kl .
(12)
We define
yi = Si I + iPi 5 ,
Yi = Si + iPi .
(13)
The constants c, ci are matrix-valued, c = cI J , etc., where I, J are the internal indices of
the fermions I . The matrices Si , Pi are Hermitian.
We will consider a group G which is not necessarily semi-simple, with structure
constants f abc . The fields transform under linear (chiral) representations of the group
which are in general reducible:
g,
g 1 ,
i hij j ,
(14)
where
g = exp(i a t a ),
g = exp(ia ta ),
h = exp(ia a ),
(15)
and
t a = tRa PR + tLa PL = tsa + tpa 5 ,
1
1
PL = (1 5 ).
(16)
PR = (1 + 5 ),
2
2
tRa and tLa belong in general to different representations of G. The scalar fields are in a real
representation; then a are antisymmetric imaginary matrices.
i i is invariant under these transformations provided the
The Yukawa coupling y
following relations hold
yj jai + yi t a ta yi = 0,
yj jai + yi ta t a yi = 0,
(17)
(18)
or, equivalently,
Yj jai + Yi tRa tLa Yi = 0,
The tree-level action is invariant under these chiral transformations, apart from an
evanescent fermionic kinetic term. Higher corrections in the bare action will require also
non-invariant counterterms; in the next subsection we will discuss how Wilsonian methods
can be applied to determine the counterterms in order to preserve the Ward identities.
299
(19)
DS = DK .
(21)
Z[J ] = Z0 [J ] = e 2 J D
K J
Z [J ].
(24)
The Wilsonian theory at scale has a bare action differing from the one of the usual
theory at = 0 by the term
1
1
K ,
(25)
S () S() = D
2
which has ultraviolet dimension less or equal to zero, due to the above made choice of the
cut-off function K ; the renormalization of the usual theory implies the renormalization
of the Wilsonian theory and vice versa.
To impose the renormalization conditions it is useful to separate the Wilsonian 1-PI
functional generator, obtained by making a Legendre transformation on W [J ], into the
quadratic tree-level and interacting parts:
1
1
+ I [],
[] = D
2
(26)
300
In this paper the renormalization program via Wilsonian methods is applied to the
Yukawa model which is the simplest chiral theory. The symmetries are imposed choosing
renormalization conditions on the Wilsonian functional compatible with the effective Ward
identities [22,2729].
Actually the rigid Ward identities are too simple to illustrate this point, since in this
case the effective Ward identities satisfied by Z and the usual Ward identity recovered
at = 0 have the same form; however, introducing external currents one has to study the
local Ward identities, whose effective form will be used in a later section.
In the Yukawa model the bare constants in the action are chosen of the form
X
(l)
h l Ndl cA (),
cA =
l>1
Nd = (4)
/22
1+
,
2
(l)
cA () =
cA,r r .
(l)
(27)
r>0
+ irr
,
= SW
Z
(l)
(r,lr)
gA
,
(29)
gA =
r=0
(l,0)
is the l-loop graph
in which the dependence on 3 and is made explicit; gA
(r,lr)
, r = 0, . . . , l 1 is the contribution due to the r-loop graphs
contribution, while gA
(r,lr)
are independent from and .
with counterterms of overall loop order l r; gA
1 In a Wilsonian mass-independent scheme, as discussed in [30], the renormalization of the mass parameter is
treated in a way very similar to the kinetic term and therefore is included in the marginal part of the action.
301
(30)
r=1
Taking s derivatives with respect to in (29) one has still a finite expression for 0,
so that one obtains the consistency conditions
l
X
(r,lr)
r s gA,n = 0
(31)
r=1
for n = 2, . . . , l and s = 1, . . . , n 1.
For the two-loop case the consistency condition (31) for n = 2 and s = 1 is known to
hold for each Feynman graph and its counterdiagrams [35], providing a useful check.
To renormalize the theory we assign finite values to the constants gA at =
gA = rA + O().
(32)
g 1 yi g = hij yj .
(33)
Furthermore one can examine which choice for rA gives the simplest expression for the
counterterms.
For the non-chiral theory, in which the couplings Pi of Eq. (13) are vanishing and g =
g = gs , the simplest counterterms are those determined by the MS scheme, corresponding
to renormalization conditions rA (S) that can be explicitly computed.
For the chiral theory we will define the counterterms as suitable functions cA (S, P )
which, for P = 0, coincide with the corresponding functions of the non-chiral case. To
this aim loop by loop we will choose rA applying a covariantization formula to the
corresponding rA (S). A comparison of Eq. (33) with the analogous equation in the nonchiral case with the same group G:
gs1 Si gs = hij Sj
suggests the recipe for the Green functions of interest:
Si1 Si2 Si2n Si2n+1 I yi1 yi2 yi2n yi2n+1
Si1 Si2 Si2n1 Si2n yi1 yi2 yi2n1 yi2n
(34)
302
1
Tr Si1 Si2 Si2n1 Si2n Tr tr yi1 yi2 yi2n1 yi2n
4
1
= Tr Yi1 Yi2 Yi2n1 Yi2n + Yi1 Yi2 Yi2n1 Yi2n , (35)
2
where in the products the coupling constants y and y , as well as Y and Y , appear in
alternate order and Tr denotes the trace over the internal fermionic indices.
As an example let us consider the renormalization condition to be imposed on
the fermionic self-energy. In the non-chiral theory, using the minimal subtraction one
computes:
p r(S).
(36)
(p)marg = i/
Using the covariantization procedure i/
p r(S) must be replaced by i/
p r(y, y ). In order to
be compared with the standard form of Eq. (9) can be written as
1
o
in
/ r(Y, Y ) + r(Y , Y ) + [/
p , 5 ] r(Y, Y ) r(Y , Y ) , (37)
(p)marg = p
2
2
which differs from i/
p r(y, y ) by evanescent terms. Eq. (37) gives the renormalization
condition for the chiral theory. Observe that, if the theory is reflection symmetric, r(y, y )
is Hermitian.
The counterterms cA are completely determined by the pole part and by the constant
part of the corresponding vertices. They can be decomposed into two parts
NDR
+ 1cA ,
cA = cA
(38)
NDR
has the same group structure as rA and, due to the choice of renormalization
where cA
conditions, has only pole part in (see comment after Eq. (32)).
The remaining part 1cA vanishes in the non-chiral case (Pi = 0).
The non-marginal relevant terms satisfy
= 0.
(39)
ij=0 p=m=0 = 0, I=0
J
p=m=0
In dimensional regularization, being the massless tadpoles equal to zero, Eq. (39)
corresponds to have vanishing bare relevant counterterms.
3. Explicit computations
3.1. Master integrals
Using the cut-off function (20) the hard propagator has the form
r
n
p2 + M 2 X
2
.
D (p, M) = D(p, M)
2
p2 + M 2 + 2
(40)
r=1
The counterterms are computed in terms of Wilsonian Green functions at zero momenta
and masses M = 0; the corresponding Wilsonian Feynman graphs with I internal lines
P
P
have I sums nr1 =1 nrI =1 .
303
(42)
(a d/2)
.
Nd (a)
(43)
At higher loops an exact expression for these integrals is not known; they can be reduced,
using recursion relations, to a small number of master integrals which can be expanded in
; to renormalize at l-loop one must know the I (l) up to the O(1) term, the I (l1) up to
the O() term and so on (in minimal subtraction it is sufficient to know the I (l) up to the
O(1/) term, the I (l1) up to the O(1) term and so on; however, in presence of chiral
symmetries MS is not sufficient).
At two loops there is only one master integral (for a review see, e.g., [16]); it is known
how to compute the finite parts of the three-loop master integral [17].
In this paper we will restrict to two-loop computations; the recursion relation is obtained
from the identity
Z
Z
q1
dd q2
dd q1
= 0,
(44)
Nd d/2
Nd d/2 q1 (q12 + 1)a (q22 + 1)b ((q1 + q2 )2 + 1)c
which implies
(d 2b c)Ia+1,b,c c(Ia,b1,c+1 Ia1,b,c+1 )
+ 2bIa,b+1,c + cIa,b,c+1 = 0.
(45)
From this relation and the fact that Ia,b,c is totally symmetric in its indices it follows the
recursion relation
3aIa+1,b,c = c(Ia1,b,c+1 Ia,b1,c+1 )
+ b(Ia1,b+1,c Ia,b+1,c1 ) + (3a d)Ia,b,c .
(46)
Using this recursion relation one can express all Ia,b,c for a, b, c > 0 in terms of the master
integral I1,1,1 and in terms of Ia 0 ,b0 ,c0 , with one of the indices vanishing; in the latter case
the two-loop integral reduces to the product of two one-loop integrals:
Ia,b,0 = Ia Ib .
One has
(47)
304
6
9 3
+ (v 5) + O(),
2
2
Z/3
4
x
dx ln 2 sin
' 0.34391.
v 3 lim I2,2,2 = 2
0
2
3
I1,1,1 =
(48)
These recursion relations can be solved very fast by computer in the cases of interest.
Recently all the Laurent series of I1,1,1 has been computed in [36].
3.2. One-loop results
At one loop the renormalization conditions (32) on the Wilsonian vertices are
(1)
(1)rs
rij
r (1) = r(1) yi yi ,
m
= r(1)
hij rs ,
v3
ri(1) = ir(1)
yj yi yj ,
v4
v4 rs rs
Y(ij kl) + r(1)2
h(ij hkl) ,
rij(1)kl = r(1)1
(49)
where the symmetrizations (. . .) are with weight one: e.g., h(ij kl) = hij kl ; we defined
Yi1 i2 i2n1 i2n
1
Tr Yi1 Yi2 Yi2n1 Yi2n + Yi1 Yi2 Yi2n1 Yi2n .
2
(50)
NDR
defined in Eq. (38).
The same parametrization holds for the bare constants cA
The quantities rA depend in general on the choice of the Wilsonian flow; in Table 1 we
NDR
for the flow n = 2, Fig. 1 gives the corresponding graphs. To
give the values of rA and cA
compute the coefficients in Table 1 one has to make an expansion in masses and momenta
of graphs in Fig. 1, as mentioned after Eq. (28).
The remaining part of the one-loop bare constants is
4
2 2
(1)
2
p + p Tr[Pi Pj ],
1cij =
3
Table 1
One-loop coefficients
Structure
Yij
hij kl
yi yi
i yj yi yj
Y(ij kl)
rs
hrs
(ij hkl)
NDR
c(1)
#
r(1)
26
15
7
12
7
12
47
60
454
35
74
48
Graphs
Fig. 1(a)
Fig. 1(b)
Fig. 1(c)
Fig. 1(d)
48
Fig. 1(e)
Fig. 1(f)
305
i
(1)
1c(1) = p
1ci = yj 5 Pi yj ,
/ 2Pi Pi i5 {Si , Pi } ,
2
1cij(1)kl = 96 Tr S(i Sj Pk Pl) + P(i Pj Pk Pl) .
3
(51)
In [7] MS is applied in BMHV in the simplest Yukawa model with a pseudoscalar, where
there is no chiral symmetry to be maintained. Here we introduced finite counterterms,
which are exactly those needed to obtain the same renormalized Green functions as in the
MS NDR scheme. In [7] it was shown that the beta function at two loop in the MS BMHV
scheme differs from the one in MS NDR scheme by a renormalization group transformation
involving finite one-loop counterterms, which are in agreement with Eq. (51).
3.3. Two-loop results
At two loops the renormalization conditions on the Wilsonian effective action read
2 ,kl
m2kl
rijm
(52)
r = yj r1 yi yi yj + r2 yi yj yi + r3 Yij yi + r4 hij kk yi .
(53)
(54)
306
Table 2
Two-loop coefficients for the two-point bosonic functions
2
Structure
cNDR
r, rm
Graphs
hikmn hj kmn p2
1
12
7
5v
432 + 81
178v
25
9 + 81
49v
16021
8505 + 729
67
29v
108 + 324
49
144
1642
729 + 5078v
2187
4v
146
+
81
243
4421 + 121v
545
27
1
6
164
15
279
35
Fig. 2(a)
Fig. 2(b)
157
105
Fig. 2(c)
Yikj k
p2
Yij kk
p2
82 + 2
42 + 3
1 1
2
2
1
2
82 + 4
162
4 2
2
Fig. 2(a)
Fig. 2(d)
Fig. 2(b)
Fig. 2(c)
Fig. 2(e)
+ r7v4hmnpq hpqkl + r8v4 hmpkl hnpqq + r9v4 hnpqk hmpql ,
(55)
307
cNDR
yj yi yi yj
1
12 + 8
2
22
289v
2359
9720 + 5832
319v
229
324 + 486
29
40
13
120
Fig. 3(a)
Fig. 3(b)
11
10
Fig. 3(c)
6107v
2375
486 1458
13
30
73
127
30
0
Fig. 3(d)
yj yi yj yi
yj yi Yij
hij kk yj yi
3
22 + 2
Graphs
Table 4
Two-loop coefficients for the cubic vertex
Structure
Yj k yk yi yj
yk yj yi yj yk
yk (yi yj yj + yj yj yi )yk
yk (yi yj yk + yj yk yi )yj
yk yj yi yk yj
hj kll yk yi yj
hij kl yk yj yl
cNDR
r v3
4 2
2
2 1
2
1 1
2
2
2
2
1
19384 + 197v
2835
243
47 + 13v
81
486
10009 83v
22680
972
415 169v
972
1458
76 920v
729
2187
33
70
7 + 160v
81
243
647
105
107
30
647
420
47
30
34
35
Fig. 4(a)
Fig. 4(b)
17
70
Fig. 4(c)
Fig. 4(d)
Fig. 4(e)
Fig. 4(f)
Fig. 4(g)
0
1
Graphs
308
r v4
Graphs
Yninj kl
192
+ 96
10352
35
Fig. 5(a)
Ynij nkl
48
96
2 + 48
962
482 + 24
12 6
2
3
2
66416v
3782672
76545 + 6561
26272v
15088
18225 + 2187
96
Fig. 5(b)
155224v
5660407
91854 + 19683
8v
292
27 + 81
13756
105
1772
105
Fig. 5(c)
112
Fig. 5(d)
10156v
4256
243 + 729
949 + 121v
36
9
61
48
39
20
38 + 29v
9
54
104
Fig. 5(e)
107
5
7
2
23
5
Fig. 5(f)
Fig. 5(g)
Fig. 5(h)
13
Fig. 5(i)
Structure
Ynij kln
hmnij Ymkln
hmnij Ymknl
hmnij Ymp hpnkl
hmnij hmnpq hpqkl
hmnij hmpkl hnpqq
6 3
using the minimal subtraction formulas [2] and taking into account the one-loop evanescent
tensors in the bare action; after obtaining a renormalization group equation involving also
insertions of evanescent tensors, these are solved in terms of relevant couplings obtaining
the usual renormalization group equation.
To avoid making a similar subtle analysis of the bare couplings in our case, we will
obtain the beta and gamma functions working directly with the Wilsonian effective action,
which satisfies the effective renormalization group equation [30]. We will restrict to the
massless case; in [30] it is considered also the massive case.
In the compact notation of Section 2.3 the Gell-Mann and Low renormalization group
equation reads
T
+
+ J
Z[J ] = E[J ],
(56)
g
J
where Z is the renormalized functional in which the limit 0 has not yet been taken; E
is an evanescent functional, which in general is not vanishing for finite value of since the
renormalization of the theory is not strictly multiplicative, and
309
= ij kl
+ Tr Yi
+ Yi
.
g
hij kl
Yi
Y
(57)
From Eq. (24) it follows that Z [J ] satisfies the effective renormalization group
equation
T 1
T
+
+ J
h D K
(58)
Z [J ] = E [J ].
g
J
J
J
Define the functional Z [J, ] as in (22), but with
S (, ) = S () [],
1
[] T D
K .
+
+ J T
Z [J, ] = E [J ],
g
J
=0
which in terms of the 1-PI functional generator reads
[, ] = E [],
g
=0
or equivalently
+
[] = T [] + E [],
(59)
(60)
(61)
(62)
310
(63)
between two bosonic or fermionic lines. Since for n > 2 Wilsonian flows the insertion
1 [] has ultraviolet dimension less or equal to zero, it does not require renormalization,
i.e., there are not counterterms for this insertion.
Let us finally rewrite in a suitable way the gradient terms in (62); using (30) one gets
(1)
gA
(1,0)
(1)
= gA,1
= cA,1
,
(2)
gA
(2,0)
(1,1)
= 2gA,1
+ gA,1
.
(64)
These gradients have the same group structure as the renormalization constants given in
Eqs. (49), (52); the coefficients of these group structures are given in the Tables 1, 2, 3,
(2,0)
(1,1)
and gA,1
are not separately
4, 5 in the case of the n = 2 flow. Observe that, while gA,1
(2,0)
(1,1)
chiral group-covariant, the gradient term 2gA,1 + gA,1 must be covariant. This provides
a check on the computations, besides the double pole rule mentioned after (31), that we
made in a systematic way.
At one loop the Wilsonian gradients can be expressed in terms of the pole of the bare
couplings, so that the formulae for the beta and gamma functions are equivalent to the
standard formulae [2]; using
A =
h l
(l)
(4)2l A
(1) = yi yi ,
(1) = yi yi ,
2
2
1
(1)
i = 2yj yi yj + yi yj yj + yj yj yi + 2Yij yj ,
2
(1)
rs
m
ij kl = 48Y(ij kl) + 3hrs
(ij hkl) + 8Ym(i h j kl) .
ij(1) = 2Yij ,
(65)
At two loops the Wilsonian gradients are not given by the simple pole of the bare
counterterms (64); the beta and gamma functions are given in terms of these Wilsonian
gradients, the T insertions, given in the Tables 2, 3, 4, 5, and the one-loop renormalization
conditions (49).
We get
1
hikmn hj kmn 2Yikj k 3Yij kk ,
12
1
3
(2) = yj yi yi yj Yij yj yi ,
8
2
1
3
(2) = yj yi yi yj Yij yj yi ,
8
2
i(2) = 2yk yj yi (yj yk + yk yj ) 2hij kl yj yk yl 4Yj k yj yi yk
(2)
ij =
311
(66)
(1)
g gij kl
gij(1)kl
(iii) Finally, T contributes to Ynij kln through the graph in Fig. 6 and gives the
coefficient 1772/105.
These three contributions add up to the expected value 96 Ynij kln for ij(2)kl in Eq. (66).
4. External currents and local Ward identities
4.1. External currents
In order to define the currents associated to the symmetry transformations (14) we will
add in the action (11) an external gauge field Aa .
At tree level the action is now
Z
1
1
(0)
D + (D )2 + iy
i i + hij kl i j k l ,
(67)
S =
2
4!
where
D = + iAa t a ,
D i = ij + iAa ija j .
(68)
We can choose for the external gauge field the condition A a = 0, so that t a is
equivalent to ta and the fermionic gauge coupling i t a is equal to the reflectionsymmetric expression 2i ( t a + ta ).
We set the factors to one; they can be easily reintroduced by dimensional analysis.
The local infinitesimal version of transformations (14) is now
Aa = a f abc Ac b = Dab b ,
= ia ta ,
= i a t a ,
i = ia ija j ,
(69)
312
where a = 0. The tree-level action is invariant under these transformations apart from
an evanescent fermionic kinetic term.
To renormalize arbitrary products of currents we will consider in the bare action local
monomials in Aa up to fourth order.
The bare action (11) gets additional terms:
Z
1 ab a b 1 abc a b c
1
c A A + c A A A + cabcd Aa Ab Ac Ad
S (A) =
2 3! 4!
1 a
1 ab a b
a + cij
+ Aa c
Aa i j + cij
A A i j .
(70)
2
4
The marginal part SW of the Wilsonian effective action will have a similar dependence
on Aa .
The renormalization conditions on SW analogous to (32), which include the new
vertices, have to be chosen in such a way that the local chiral Ward identities hold; to this
aim the renormalization conditions are fixed compatibly with the effective Ward identities.
These identities have been introduced in [22] and further studied in [27].
An analysis similar to the one in Section 3.4 shows that the extra term of the
effective Ward identity can be represented as the insertion of non-local vertices. Under
transformation (69) one gets:
1
1
K ,
(71)
= T + O , T = D
2
where T is the generator of the 1-PI Green functions with one insertion of the operator
1
K ). Its Feynman rules are:
( 12 D
1
1
(p + q)K (p + q)t a ta SH
(q)K (q) ,
(72)
ia (p) SH
for insertion on the fermionic lines and
1
1
(p + q)K (p + q) a a DH
(q)K (q) ,
i a (p) DH
(73)
(74)
Equation (71) is the effective Ward identity [22,2729], cast in a form which we find
convenient for explicit computations.
The anomaly operator is defined by its renormalization conditions at the renormalization
scale = therefore we have to consider the marginal projection of Eq. (71) in the limit
0:
SW = T + A,
where T and A are the marginal parts of T
(75)
and
A .
313
+ i b + bij i j + bij A i j A,
(76)
2
Z
1
2
e
e
i
Tr tR AR AR + AR AR AR (R L) , (77)
A=
3
2
where
AR itRa Aa ,
AL itLa Aa .
(78)
314
i
Tr Yi Yj tLa + Yi Yj tRa (i j ),
4
121 a b
ab
Tr tR , tR Yi Yj + tLa , tLb Yi Yj + (i j )
bij =
210
16 a b
(79)
Tr tR Yi tL Yj + tLa Yi tRb Yj + (i j ) .
105
To determine the Wilsonian renormalization conditions one must find SW ; choose them
of the form
Z
1 ab a 2 b 1 ab a
(A)
a A A + a2 A Ab + a3ax f xbc Aa Ab Ac
SW =
2 1
2
1 acx xy ybd 1 abcd a b c d
aa
+
+ a
f a4 f
A A A A + iAa
4
6
1
(80)
+ Aa i aija j + Aa Ab i j aijab .
4
bija =
(A)
The effective Ward identities give the following relations between the SW coefficients
and those for T :
(a1 + a2 + b1 )ab = 0,
(a1 + a3 b4 )ab = 0,
(a3 a4 b6 )ab = 0,
(a + b)abcd = 0,
t e a a e + be = 0,
[t e , a] = 0,
(a a + ba i a a)ij = 0,
a b
b
iaik
kj + iajak ki
aijab bijba = 0.
(81)
(A)
is a, aij and a2ab . The former
Using (81), a set of independent parameters of SW
two parameters have been fixed in the previous sections, to be compatible with minimal
subtraction in the non-chiral case; making the analogous choice for the new parameter we
get
8
11
a2ab = S2 (F )ab + S2 (S)ab + zab
5
60
(82)
with zab = 0 for the minimal subtraction choice. We have checked all the relations (81) by
explicit computation of the marginal Wilsonian Green functions and the corresponding T
insertions.
Having determined the renormalization conditions, we can now compute the one-loop
bare action:
Z
1 a
1
8
ab
ab
ab
b
F
S2 (F ) S2 (S) + z F
S(1) =
4
3
3
NDR
+ S(1)
NDR
+ S(1)
+ 1S(1) ,
(83)
NDR
NDR
a Aa Aa f abc Ab Ac , S
and S(1)
are the covariantization of
where F
(1)
the kinetic terms for fermions and scalars in absence of the gauge field; 1S(1) is the nonnaive part of S(1) .
A in (77) is the AdlerBardeen anomaly [32]; for a previous derivation using Wilsonian
methods, see [38].
315
(A)
In order to compute the Bardeen anomaly in the next subsection we write 1S(1)
, the
gauge-dependent part of 1S(1) , with the variables A and V so defined:
A = V + 5 A ,
V = itsa Aa ,
A = itpa Aa .
(84)
As a consequence:
F = A A + [A , A ] = V + 5 A ,
V = V V + [V , V ] + [A , A ],
A = DV A DV A ,
(85)
where we define DV f = f + [V , f ].
A similar decomposition for the scalars fields is introduced:
Si i ,
Pi i .
(86)
(A)
(88)
Now V and A as well as and , which are respectively the vectorial and axial gauge
parameters, can be considered as independent quantities. The gauge transformation
A = + [, A ]
decomposes into vectorial and axial gauge transformations.
Under vectorial gauge transformations one has
(89)
316
V = DV ,
A = [, A ],
(90)
A = DV .
(91)
(92)
where A() is the vectorial part of the anomaly (77) which indeed can be decomposed as
A = A() + A(),
where
2
A( ) =
3
(93)
1
Tr AR AR + AR AR AR
2
1
AL AL AL AL AL .
2
(94)
Tr V V + A A
ABardeen = 4
4
12
2
(A A V + A V A + V A A )
3
8
+ A A A A .
3
(95)
One can then modify the renormalization of the product of the currents by subtracting
out all the terms depending on the Levi-Civita tensor in the finite part of the bare action (87)
and correspondingly by introducing their gauge variation in Eq. (75). Being T unchanged
the effect of subtracting Eq. (92) is to recover the vectorial gauge invariance, the subtraction
of Eq. (95) put the anomaly in the Bardeen form [39].
Observe that the -dependent part of (87) is not specific of the BMHV regularization,
being fixed its gauge variations (92), (95).
The -independent part of (87) is specific of the BMHV regularization; for instance,
it takes a different expression in [40], where a different renormalization scheme is used.
Using (87) with
i
i
A = Aa (tR tL )
V = Aa (tR + tL ),
2
2
and making the non-minimal choice
5
zab = S2 (F )ab ,
3
we find agreement with the finite counterterm computed in [11] using the Bonneau
identities [12].
317
(96)
and are shown in Table 6. In order to check the first Ward identity relation in (81) we have
also computed the graphs shown in Fig. 8 and the results are collected in Table 7. Actually
we did not consider in Figs. 7, 8 the graph with only scalar internal lines and quartic scalar
vertex: it is finite and rigid invariant, therefore it does not contribute to the bare action and
it has to match separately its own T insertion.
From the pole part of graphs in Fig. 7 we computed the gauge invariant part of S (A) at
two loops:
Z
1 a 2 ab b
(A)
F K2 F ,
=
(97)
S(2)
4
a2ab
Graphs
K1ab
K2ab
K1ab
379
674
3645 + 2187 v
68
2066
3645 2187 v
Fig. 7(a)
57469 98 v
51030
2187
9956 v
13178
+
3645
2187
160084 + 2822 v
25515
2187
98
38657
25515 + 2187 v
9674 3428 v
3645
2187
13958
195233
51030 2187 v
Fig. 7(b)
13178 9956 v
3645
2187
2822
160084
25515 2187 v
3428
9674
3645 + 2187 v
195233 + 13958 v
51030
2187
K2ab
Fig. 7(c)
Fig. 7(d)
318
(A)
1S(2)
=
Graphs
K2ab
0
Fig. 8(a)
1171 284 v
2430
729
1508 v
253
+
243
729
157 268 v
405
243
124
1751
1890 + 81 v
Fig. 8(b)
Fig. 8(c)
Fig. 8(d)
Fig. 8(e)
2
2
53
2
{A , Pi } DV {A , Pi } + 2A DV A Pi Pi
3 36
2
2
13
2
5
+
{A , Si } DV {A , Si } + A DV A Si Si
3 36
3
Z
13
22
5
4
8
1
(A Si )2 (A )2 (Si )2 +
(A Pi )2
Tr
+
4
3
9
9
3
9
4
31
4
4
+
+
(A )2 (Pi )2
(V Pi )2
3 9
9 27
13
28 29
8
(V )2 (Pi )2 + i
V Si A Pi
+
9 27
3
9
1
16 35
4
+i +
V Pi A Si + i
V A Pi Si
9 27
3 9
14
5
4
i V A Si Pi + i +
V Pi Si A .
(98)
9
9 27
1
2
Tr
We used the notation of Eqs. (84), (85); the r.h.s. of Eq. (98) is not simply quadratic in
the gauge fields because we completed it in order to have an expression gauge-invariant
319
(1)
Tabc
(2)
Graphs
6797 i 4688 iv
21870
6561
Fig. 9(a)
18119 i 12880 iv
21870
6561
0
0
2080
254
243 i + 729 iv
98
440
135 i 243 iv
4i
3
1861
1215 i + 1192
729 iv
0
55441 i
204120
109031
204120 i
Fig. 9(b)
76
2187 iv
308 iv
2187
Fig. 9(c)
Fig. 9(d)
Fig. 9(e)
98 i + 440 iv
135
243
Fig. 9(f)
Fig. 9(g)
1861 i 1192 iv
1215
729
Fig. 9(h)
under vectorial transformation. This fact has been possible because the derivatives V of
the gauge fields V appear only in the combination [ V] as expected, since dimensional
regularization and ours renormalization condition respect vectorial symmetry.
We checked the non-renormalization theorem of the anomaly only in the two-gluon
sector of Eq. (75). We have computed the part of T from which the anomaly might arise:
Z
(99)
T = a (x) Ab Ac Tabc ,
where Tabc is an invariant tensor symmetric in the last two indices. The non-renormalization
theorem requires that the completely symmetric part T(abc) , which cannot be eliminated by
a local counterterm, must vanish.
320
Fig. 9 shows the graphs which give contributions to T(abc) . Each contribution is
decomposed on a suitable basis of symmetric tensor, we have chosen a basis in which
ija never appears explicitly (symmetrization in the indices a, b, c is understood):
(1)
= Tr Yk Yk tRa tRb tRc Tr Yk Yk tLa tLb tLc ,
Tabc
(2)
(100)
Tabc = Tr Yk tLa Yk tRb tRc Tr Yk tRa Yk tLb tLc .
The results of our calculations are summarized in Table 8: as expected the sum of the
contributions of every column vanishes, so that the AdlerBardeen theorem is verified.
One could address the question whether Tabc and not only its symmetric part is actually
vanishing. For G = SU (N), using the Young tableaux one can easily prove that invariant
tensors Xabc with mixed symmetry do not exist; 2 for more general groups see [41].
5. Concluding remarks
We have shown that Wilsonian methods are useful in the BMHV dimensional
renormalization of theories with chiral symmetries; the two-loop renormalization of the
most general Yukawa theory becomes straightforward in this scheme, while in a more
conventional approach, based on the verification of usual Ward identities, it is a non-trivial
task.
In our approach one renormalizes the effective Wilsonian action along a flow on which
the subtraction integrals are easily computed using standard techniques.
There is some arbitrariness in the choice of this flow; we worked with the n = 2
flow, which is the simplest one from a computational viewpoint: it is quite obvious that
calculations with the n > 3 flows are more complicated, although the bare action is the
same. On the other hand, the n = 1 flow, which coincides with the auxiliary mass method,
does not respect renormalizability so the procedure described in this paper fails down. It is
a convenient method in non-chiral theories where minimal subtraction respects the Ward
identities.
In the future we intend to apply our formalism to chiral gauge theories.
Pi Sk Pj Sk
cij = Tr 2Si Pk Sj Pk + +
9 3
2 We thank C. Destri for explaining this point to us.
23 20
(Si Sk Pj Pk + Si Pk Pj Sk + Pi Pk Pj Pk )
9
3
4
Si Sj Pk Pk
+ 1
7
4
+
(Si Sk Pk Pj + Si Pj Pk Sk + Pi Pj Sj Sk )
9 3
13 16
Pi Pj Pj Pk ,
+
9
3
2
4
(Sj Sk Pi Pl + Sj Pk Pi Sl + Pj Sk Si Pl + Pj Pk Si Sl )
cijm ,kl = Tr 3
64 32
80 64
Pj Pk Pi Pl +
Pj Pk Pl Pi
8Pj Sk Pi Sl +
9
3
9
3
8
(Sj Sk Pl Pi + Sj Pk Pl Si + Pj Sk Sl Pi + Pj Pk Sl Si )
+ 2
2 7
Pm Pl hij mk ,
+
6
+
Pi Pi yj
c = iyj
36 3
1
1
43 10
i5 Pi yj yi Pj yi +
Tr(Pi Pj )yi .
+
8 2
54 9
The two-loop fermion-fermion-scalar counterterm is:
1ci =
7
X
n=1
ci (n),
7 2
1 2
Tr(Pj Pk )Si + i5
Tr(Sj Sk )Pi
ci (1) = iyk +
6
2
22 32
Tr(Pj Pk )Pi yj ,
+ i5
27 9
2
1 1
(Sj Si Pj + Pj Si Sj ) + i5 Sj Pi Sj
ci (2) = iyk i5
4
3 1
2 2
+ +
(Sj Pi Pj + Pj Pi Sj ) + i5
Pj Pi Pj yk ,
4
3
1
1
(Si Yj Pj + Pi Yj Sj + Sj Yj Pi + Pj Yj Si )
ci (3) = iyk i5
8 2
4
1
(Pi Pj Pj + Pj Pj Pi ) yk ,
+ i5
9 3
321
322
1 1
1 1
+
ci (4) = iyk i5
(Si Yj Pk + Pj Yk Si ) i5
(Pi Yj Sk + Sj Yk Pi )
4
4
2 2
2
+
(Pi Pj Pk + Pj Pk Pi ) yj ,
(Pi Sj Pk + Pj Sk Pi ) i5
3
1
2
1
ci (5) = iyk i5 2Sj Pi Sk Sj yi Pk Pj Pi Pk Pj yi Sk yj ,
2
3
2
ci (6) = 0,
1
ci (7) = yj Pk yl hij kl .
2
The two-loop quartic scalar counterterm is
1cij kl =
9
X
n=1
cij(2)kl (n),
3 3
8 2
SYm P 3 + i
SYm (SY P + P Y S)
c(1) = 16 Tr Ym i
3
4
8 13
+
P Ym P 3 + P Ym (SY P + P 3 + P Y S)
+
3
6
1 P Ym (SY P + P Y S)
+
3 9
P Ym (SY S P Y P ) + p.c.,
+i
4
3
Y Y Ym Y Y Y 2 Ym Y 2
c(2) = 4 Tr Ym
2
2
11 2
2
2
2
8P Ym Y + P + 8Y Y P Ym P + p.c.,
8
64 20
12
4
2
4
3 Y Ym (Y Y ) Ym + 12P Ym
+
P 4 Ym
c(3) = Tr Ym
3
32 8
3
2
2
2
2
2iSY P Pm + 2iP S P Pm P Y SYm + P Y SYm + p.c.,
48
12 Sm SP Pn + Sm P 2 Sn + Pm S 2 Pn + Pm P SSn
c(4) = Tr
128 160
2
Pm P Pn hmn ,
3
64 128
Pm P Pn P 48Sm P Sn P
c(5) = Tr
3
24
18 (Sm SPn P + Sm P Pn S + Pm SSn P + Pm P Sn S) hmn ,
323
6 7
Pm Pn hmp hnp ,
2
c(7) = c(8) = c(9) = 0,
c(6) = Tr
where the indices i, j, k, l, which are totally symmetrized, are understood. p.c. represents
the terms obtained with the substitution rule: P P , Y Y , Y Y .
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
324
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
Abstract
We use an optimized perturbation expansion called the linear -expansion to study the phase
transition in a Higgs sector with a continuous symmetry and large couplings. Our results show how
to use this non-perturbative method successfully for such problems. We also show how to simplify
the method without losing any flexibility. 2000 Elsevier Science B.V. All rights reserved.
PACS: 11.10.Gh; 02.30.Cj; 02.30.Tb
Keywords: Delta expansion; Symmetry breaking; Non-perturbative methods
Quantum Field Theory has had many successes over the last fifty years but most are
based on the use of small coupling perturbation expansions. However, there are many
interesting problems where such an expansion is not appropriate. QCD is the classic
example but phase transitions at non-zero temperatures even in models with small coupling
constants, e.g., the Electro Weak model, are now known to be in this class. It is this latter
example which motivates our work which will focus on the non-perturbative behaviour of
the Higgs sector.
Alternatives to perturbation theory, non-perturbative methods, have many limitations.
For instance it is usually very difficult to go beyond the lowest order in methods such as
large N [1] while numerical Monte Carlo codes are relatively expensive [2,3]. Another
method with a long history is the Linked Cluster Expansion [4,5] which now also exploits
large amounts of computing power [6]. In this letter we extend a non-perturbative method
called an OPE (Optimized Perturbation Expansion). In this approach one expands around a
solvable model which contains several arbitrary parameters, and thus is a variational ansatz.
The expansion is not necessarily in terms of some small parameter. One then chooses
1 t.evans@ic.ac.uk; http://euclid.tp.ph.ic.ac.uk/ time
2 m.ivin@ic.ac.uk
3 memobius@midway.uchicago.edu; present address: Enrico Fermi Institute, University of Chicago.
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 2 0 - 6
326
the variational parameters, invariably using some optimization criteria which is a highly
non-linear procedure. The optimization criteria has to ensure good results from a few
terms of the series but there has been much discussion about what is a good optimization
criteria.
As described, the OPE is a very general method so not surprisingly it has been
rediscovered several times in different guises, applied to very different applications, and
appears under many different names; it is also known as the linear -expansion [7], actionvariational approach [8], improved Gaussian approximation, variational perturbation
theory [10], method of self-similar approximation [11], or the variational cumulant
expansion [12]. For instance, the method has been to the evaluation of simple integrals
[7,13,14], solving non-linear differential equations [15], quantum mechanics [7,9,1618]
to quantum field theory both in the continuum [9,10] and on a lattice [7,8,12,1927].
To motivate this work we first note that the analysis of pure gauge theories on the lattice
using OPE used a variety of optimization schemes when fixing the variational parameters.
One can minimize the free energy [22] or demand that the series converges as fast as
possible by minimizing the high order terms with respect to the lower order terms [8]
the principle of fastest convergence. However, we believe the best results for pure gauge
models are obtained when using the principle of minimal sensitivity [28], as discussed
in [7,1921]. In this case the variational parameters, say {}, are set separately for each
physical quantity, say O, under consideration by demanding that the quantity changes as
little as possible if the variational parameters {} are varied from their optimal values {},
specifically
hOi
= 0.
(1)
i =
Combinations of these criteria are also possible [11].
Given the interest in non-perturbative effects in gauged Higgs models it is interesting
to note that the situation for the OPE in these cases [2327] is not nearly so good, and
the results are not as extensive as for the pure gauge case. The analysis of the gauged
U(1) Higgs model [23,24] fixed the variational parameters by minimizing the lowest
order calculation of the free energy, a procedure which is justified by noting the identity
hexp{O}i > exp{hOi}. However, this approach was rejected in the pure gauge analyses
and we will not pursue it here. For the more interesting gauged SU(2) Higgs model the
analysis in [26,27] was quite successful in reproducing the known Monte Carlo data but
the accuracy obtained fell short of that obtained when using similar methods for pure
gauge models. This is understandable as the analysis revealed that adding the Higgs sector
produces new problems (as indeed it does for numerical lattice Monte Carlo studies), and
the variational ansatz used was relatively simple.
Given that the Higgs sector causes new problems, it is worth considering pure Higgs
models on the lattice in the context of OPE methods, effectively the zero gauge coupling
limit of the gauged Higgs models rather than the infinite Higgs mass limit represented
by the pure gauge models. However, we have found only a discussion of the real scalar
327
model on the lattice using OPE method [12] rather than models with continuous global
symmetries relevant to realistic gauged Higgs models.
The purpose of this paper is to present results for the simplest prototype of a Higgs sector
in a gauge theory, a pair of scalar fields with an O(2) global symmetry. It is hoped that the
lessons learned in this case will improve future work on realistic problems. The model
has not been studied before using the OPE on the lattice and further the Higgs sector in
this model is different from that used in [26,27] where only the modulus of the Higgs was
required. Likewise the variational ansatz we choose here is more complicated with more
variational parameters than that used in [12] or [26,27]. We also study different schemes to
fix the variational parameters.
Our results show that the OPE can work well for problems with global continuous
symmetry and large coupling constants. We highlight the role of the symmetry in the
choice of the variational ansatz and how to use this to reduce the number of necessary
variational parameters. We also show that minimizing the free energy alone is sufficient
to locate the phase transition. Further using the minimum of the free energy to fix the
variational parameters for calculations of any quantity produced sensible results. This is
to be set against our results that show we were unable to find turning points of the form
(1) for several different types of expectation value. Thus it appears that the principle of
minimal sensitivity [28] does not work in this model in contrast with its great success in
pure gauge models [7,1921] and the rigorous results obtained with this approach to OPE
in other contexts [13,14,18].
Our starting point is the Euclidean action in four dimensions for a scalar field theory with
a global O(2) symmetry, or equivalently a global U(1) symmetry, put on a four-dimensional
hyper-cubic spacetime lattice to regulate the UV divergences
4
N X
N
X
X
(n+ n + n+ n ) +
m2 n2 + n2
S {i , i } =
n
N
X
2
n2 + n2 .
(2)
The subscript n runs over all N lattice sites and the n + subscript stands for the
nearest neighbour terms. The aim is to calculate the partition function or generating
functional, Z,
Z=
N Z
Y
di di eS({i ,i })
(3)
i=1
for large couplings . We will also be interested in various expectation values such as hi
and h 2 i.
The OPE method we will use to perform such calculations is the linear -expansion [7].
A suitable trial action S0 is added and subtracted from the real action, so that we use the
action S = S0 + (S S0 ) and = 1, i.e.,
328
Z=
Z0
n=0
(S0 S)n 0
n!
(4)
N Z
1 Y
di di ( ) exp{S0 }.
Z0
(5)
i=1
Z0 is the partition function for the action S0 . Note that is just a formal counting parameter
since it must be set to one to make contact with the physical theory. It is used to help us
truncate the infinite number of Feynman diagrams which represent the solutions to QFT,
i.e., we will work up to a finite power of . It seems that the ultimate success of any
calculation will rely on the small size of h(S0 S)n i0 . Thus we might guess that, as with
the ansatz in any variational approach, S0 needs to be as good a representation of the
correct physics as possible. However, this requires us to know the full answer before we
can make a good guess for S0 . To break this circle, we use our physical intuition to make
the best guess for the general form of S0 but at the same time we let S0 depend on some
unphysical variational parameters. When we truncate our expressions such as (4) at some
finite order in delta, our answers will depend on these parameters and we will optimize our
values for physical quantities by choosing values for these unphysical parameters using
suitable criteria. Thus we are optimizing our choice of S0 using a variational method and
it is the optimization which introduces the highly non-linear and non-perturbative element
of OPE.
In our case we choose our trial action, S0 = S0 (, ; m2 , ; j , j , k+ , k ), to be
X
1
j n + j n + k+ n2 + n2
S0 =
2
n
1
2
2
2
2 2
(6)
+ k n n n + n
2
introducing the unphysical variational parameters j , j , k+ , k . We do not introduce a
k term as we can always make an O(2) rotation of our fields to eliminate such a term.
We therefore assume that this has been done so that in general we have used up our freedom
to make such a field redefinition. Our S0 is then a reasonable guess for an action with two
scalar degrees of freedom since the form of S0 is very similar to S and their ultra local
parts are identical when j = j = k = 0 and k+ = m2 . We therefore might hope
that h(S0 S)n i0 is likely to be small, and indeed perhaps minimized for values of the
variational parameters where the ultralocal parts of S and S0 are equal. Note that our S0
is significantly different from that of [26,27] as when k 6= 0 or either j 6= 0 then this S0
is not O(2) symmetric. We shall also work with the full complex field (or its equivalent)
rather than just its modulus as used in the gauged model in [26,27].
However, there is a second criterion for the choice of S0 , namely, it must be of a form
which allows us to perform some calculations. In this case our trial action S0 consists of
329
ultra local terms. The partition function then factorizes into a product of N integrals over
the fields at each lattice site.
Let us first consider the Free energy density, F . Using as a counting parameter and
define
!
n
X
1
1
ln[Z0 ] +
Kn
(7)
F := ln Z =
N
N
n!
n=1
3
(8)
K3 = (S S0 )3 0 3 (S S0 )2 0 (S S0 ) 0 + 2 (S S0 ) 0 .
The statistical averages required to calculate the Kn can then be seen to involve integrals
over the fields at no more than (n + 1) connected lattice sites. Descriptions of the
diagrammatic method used to find all the relevant integrals are given elsewhere, for
instance, see [4,26,27].
For the free energy we choose the unphysical variational parameters, j , j , k , such
that they minimize the free energy (maximize the entropy). For the free energy calculation
only, it is also the procedure that the principle of minimal sensitivity would demand. After
fixing the parameters, F depends only on the parameters m2 and .
We also studied the expectation values hi, hi, h 2 i and h 2 i. To do this we again use
the cumulant expansion, which for the expectation value of an operator O can be expressed
as
QN R
di di OeS
(9)
hOi = Qi=1
N R
S
i=1 di di e
P
n
n
(n 1)!
n=0 h(S0 S) i0 /n!
n=1
Equating powers of produces the same type of relation between statistical averages,
involving simple integrals over n connected lattice sites, and the Ln s, similar to that for
K3 in (8). In practice we expand up to third order, L3 and K3 . hOi is then a function of the
physical parameters m2 , and the variational ones, j , j , k+ , k , but it does not depend
on N the number of lattice sites.
At this point note that there are alternatives for the optimization step used to set these
variational parameters in these expectation values. The principle of minimal sensitivity,
used with great success in many circumstances and in particular in [7,1921], is based
on (1) where {j , j , k+ , k }. We applied this method to hi, hi, h 2 i and h 2 i.
Anticipating the role that the symmetry plays (see discussion below) we also studied the
O(2) invariants hi2 + hi2 and h 2 i + h 2 i. However, we were unable to find any suitable
solutions for the variational parameters when we tried to minimize the variation in the
expectation values of the field with respect to the variational parameters. Therefore, we
reverted to the method used in [12], and in calculations of all expectation values we choose
330
unphysical variational parameters, j , j , k , which minimized the free energy for the
same set of physical parameters m2 , .
The resulting expressions for each quantity involve a sum of about 100 simple terms,
each made up of a product of various variational and physical parameters and a few of
a family of twenty six different integrals. A typical term contribution to the sum which
makes K3 in d spacetime dimensions is
1
(11)
12dj m2 + k+ k I10 I13 , (I00 )2 ,
2
where
Z
dx dy x a y b exp j x + 12 (k+ + k )x 2 + j y
Iab =
+ 12 (k+ k )y 2 x 2 y 2 .
(12)
All the integrals are of this form and so they are easily calculated numerically. The
computational time consuming part was finding the location of the minimum of the
free energy or of an expectation value to fix the variational parameters. This was done
numerically in the four-dimensional space of unphysical parameters. Though no numerical
algorithm can guarantee that it has found the global minimum, in practice it turned out that
our expressions were amenable to the simplest algorithm. Overall the computations took
no more than a couple of days on a personal computer.
It is important to check our results as the expressions are simple but long, so errors can
easily creep in. One check we used was to set in turn each of the fields to zero. The model
then reduces to that of a single real field with a 4 interaction and we find that we reproduce
the results of Wu et al. [12] who used the same approach and an S0 equivalent in this limit.
They in turn showed that these results match lattice Monte Carlo calculations extremely
well. Another check, as we shall discuss below, is to restrict the analysis to a subset of one
or two variational parameters. If we do that then the results agree with the analysis in the
full four-dimensional j, k space within numerical errors.
The results for = 25 are summarized in the figures. The numerical errors are too small
to see in the plots but the error inherent in the method can be estimated by the difference
between the results for the first and third orders. First let us look at the results for the
free energy as a function of the variational parameters. It is sufficient to study the free
energy with j = 0 and k = 0 as we will show below, so in Fig. 1 we show results for
varying j and k+ for = 25.0 and m2 = 15.0. First note that we find no minima or
any turning points in the even orders. However, Fig. 1 shows that this is due solely to
the behaviour of the k+ variational parameter which is not always included in previous
studies. In the |j | variable alone, one does seem to find clear minima as shown in Fig. 2.
On the other hand, in the full variational parameter space there are clear minima (and not
just turning points demanded by the PMS condition (1)) in the first and third order, e.g.,
for = 25, m2 = 15.0 these are at k+ 0.98 m2 and |j | 1.77. This difference in
behaviour of even and odd orders is common in OPE methods, e.g., appears in [26,27].
331
Fig. 1. Plots of the free energy from order zero (F0) to three (F3) against j = j and k = k+ for
= 25, m2 = 15, j = 0 and k = 0. Contours are evenly spaced in the interval 2.2 to 2.55,
and show that only the odd orders have minima which get shallower at higher orders.
Further, the minima in the first and third orders do not move much from the first to third
order but the minima get shallower at the higher order, i.e., the results become less sensitive
to the precise value used for the variational parameters. This is characteristic of good PMS
points.
Let us briefly consider the principle of fastest convergence of [8] in this context. This
suggests that one should fix the variational parameters by minimizing the difference
between different higher and lower order terms in the -expansion in (7). It is immediately
obvious that this principle is of no use when we have a function of more than one variational
parameter, as it provides only too few equations to constrain the several variational
parameters. However, we can arbitrarily reduce the problem to a one-dimensional one
by removing the k variational parameters, i.e., set k+ = m2 and k = 0 so that no
332
Fig. 2. Plots of the free energy from order zero (F0) to three (F3) against j = j for = 25,
k+ = m2 = 15, j = k = 0.
change in the quadratic part of the action is allowed. These values are also close to the
values actually found to minimize the free energy. We then set j = 0 by symmetry (see
below) leaving just one free parameter, the source j . We are then working with an S0
similar to that used in some of the literature. The results are plotted in Fig. 2 for example
values of = 25 and m2 = 15 where we are in the spontaneous symmetry breaking
regime. They show that even in this one-dimensional subspace, the principle of fastest
convergence is still not working well for this Higgs model. The first and third order curves
do not intersect but the second and third orders do, contradicting normal linear -expansion
behaviour where one compares every second order. However, the second and third orders
do intersect but at a value of j much bigger than the location of the minimum. In any
case the difference between the two orders changes much more slowly around the j = 0
intersection 4 suggesting that j = 0 should be chosen by this criterion. It is therefore
difficult to see any reasonable way in which the principle of fastest convergence of [8]
can be used to fix the variational parameters.
We do note that the plot in Fig. 2 does show that the difference between the the second
and third orders is still small in the region where the free energy is minimized (which is
near k+ = m2 and k = 0). Thus using the minimum free energy principle does give a
series with higher order terms giving successively smaller contributions, in the spirit of the
principle of fastest convergence.
Now let us analyze the expectation values. One can just use the values for the variational
parameters which minimize the free energy, i.e., call upon some sort of maximum entropy
4 Diagramatically, since there are no odd vertices when j = 0, odd orders give zero contributions at j = 0.
333
principle. On the other hand the very successful PMS approach [28] would demand that we
look for turning points in the expectation values with respect to the variational parameters
(1) and use these values for the particular expectation value needed. While that was
successful in gauged models [7,1921,26,27], we have not been able to find any reasonable
turning points in the expectation values of the fields or fields squared, or even to O(2)
invariant combinations of expectation values. For instance, see Fig. 3. 5
Thus a major result of this paper is the realization that the previously successful PMS
condition does not work for this pure Higgs model. However, a minimum free energy
criterion does seem to give well behaved results, i.e., weak dependence on the variational
parameters near the chosen solution, small corrections in the variational parameters and
free energy values when moving from first to third orders. There is no reason to believe
that this would not be the case in gauged models, at least for weak gauge coupling.
Now let us try to use the variational values obtained from the minimal free energy
principle to study the physics in this model for the large coupling = 25. The most
prominent feature of the complex 4 theory is the phase transition. Fig. 4 shows a plot
of the quantity hi2 + hi2 vs. m2 . We can clearly see that the curves are continuous
but that there is a sudden change in slope at m2 13.4 for the third order expansion
(and at m2 11.8 for the first order expansion). This indicates a second order phase
transition. Since we are working on a lattice and have not taken the continuum limit, the
mass parameter m is not a physical mass so the fact that the phase transition does not occur
at m2 = 0 (as it would classically) is not surprising. In any case we are working a long way
5 There are some turning points but they occur in regions where the values are changing rapidly. There was one
shallow saddle point in the hi2 + hi2 plot at = 25.0 and m2 = 15, near k = 25, |j | = 2. However, no
other expectation value at any order showed any good behaviour in this region and this suggests that this was a
feature of increasing ripples in the function as k+ increased.
334
Fig. 4. This plot shows the first and third order result of hi2 + hi2 vs. m2 for = 25. The phase
transition occurs around m2 13.4 for third order and m2 11.8 for first order.
Fig. 5. This plot shows the first and third order result of dF 2 := (h 2 i hi2 ) + (h 2 i hi2 ) vs.
m2 for = 25.
from the perturbative regime. The fluctuations about the minimum can be estimated from
the difference (h 2 i hi2 ) + (h 2 i hi2 ) and this is shown in Fig. 5. This peaks at the
phase transition as expected.
335
Fig. 6. Plot of hi vs. hi for m2 = 15.0 and m2 = 18.0, for various initial values for the
variational parameters. = 25.0. We see that there are many degenerate vacuum states for a given
m2 in the broken symmetry regime.
Another point to note is the way that non-zero results for hi and hi can be found in
the OPE. This is a key difference between this method and lattice Monte Carlo methods.
The OPE will always find one and only one vacuum solution even if there is more than
one solution. It does this by allowing the sources j to take non-zero values in which case
the classical potential is tilted favouring one vacuum solution. This contrasts with lattice
Monte Carlo methods which sample all possible vacuum states with equal likelihood.
This means OPE methods are ideal for studying non-trivial field configurations, perhaps
with defects present, by using classical sources to manipulate the effective quantum
potential.
The last plot, Fig. 6, clearly shows the role of the O(2) symmetry in this method. The
plot of hi vs. hi for m2 = 15.0 (inner curve) and m2 = 18.0 (outer curve) both show
circles, i.e., the quantity hi2 + hi2 is a constant.
It is important to note the form of the solutions found for the unphysical parameters
j , j , k+ and k . In the unbroken phase, j = j = 0. In the broken phase while |j |2 =
j2 + j2 is a non-zero constant for a given set of physical parameters m2 and , with a
different solution for the j s corresponding to each point on the circles in Fig. 6. This
immediately suggests that |j | acts as an order parameter. Thus a calculation of just the free
energy, and by implication the variational parameters, rather than of any expectation value
can be used find the location of the transition, as shown in Fig. 7.
336
Fig. 7. Plot of |j | vs. m2 for = 25.0. It shows that this variational parameter acts as an order
parameter.
Fig. 8. Plot of (1 k+ /m2 ) which minimize the third order free energy vs. m2 for = 25.0.
The solution for k+ was equal to m2 in the unbroken regime and within 10% of m2
otherwise, as shown in Fig. 8. Thus (K+ + m2 ) is acting as another order parameter. This
shows that its numerical value, chosen by minimizing the free energy, is keeping S and S0
as similar to each other as possible. However, it is not so much the numerical value of k+
but the behaviour of functions as we vary k+ which is important. As remarked above, there
is often no minimum in the k+ variable when there are minima in the other variational
parameters.
337
Finally we note that that k = 0 always minimized the free energy, within numerical
accuracy. This latter result is of particular significance with regards the symmetry of our
ansatz S0 , and we will now turn to consider this aspect.
To explain these results, in particular Fig. 6, we must consider the symmetry of the
problem. S is invariant under O(2) transformations such as
cos( ) sin( )
E
U=
,
E =
.
(13)
E E 0 = U ,
sin( ) cos( )
Our S0 of (6) is not O(2) invariant if k 6= 0 or if the j parameters are non-zero. However,
it is easy to see that
E m2 , ; jE0 = U jE, k+ , k = 0 ,
E m2 , ; jE, k+ , k = 0 = S0 E 0 = U ;
(14)
S0 ;
where jE = (j , j ). That is, there is a set of field independent transformations for j , j
and k+ which leave S0 invariant under the O(2) field transformation. Note that we must
have k = 0 for this to be true, so let us for the moment assume that this value minimizes
the free energy. 6
Using this O(2) transformation we note that the integration measure is also O(2)
invariant. It is then easy to show that at whatever order in we truncate our free energy
expression, if the set {jE, k+ , k = 0} minimizes the free energy then the set {jE0 , k+ , k
= 0} also minimizes the free energy. It then follows that a calculation of any O(2) invariant
physical object will not depend on which solution for the variational parameters we
use. In particular the free energy, h 2 i + h 2 i, and hi2 + hi2 will all give results
which are independent of the solution for the variational parameters. For objects like
hi which are not O(2) invariant, we can still relate one solution found for it, with
a particular set of variational parameters, to another solution for hi and another set
of variational parameters. The relationship is precisely the simple O(2) transformations
discussed here.
Plotting out final values for the variational parameters j and j for the different initial
conditions used in producing Fig. 6 we found that j and j also form circles in a j j
plane (for fixed m2 and ). Actually, there is a one-to-one correspondence between pairs of
j , j and hi, hi two points separated by a certain angle on the j circle correspond
to two points separated by the same angle on the expectation value circle; so to get one of
the points from the circles in Fig. 6, we fix j to a certain value, and minimize F in the
remaining 3-dimensional parameter space. Once the point is found, we fix j to a different
value, and minimize again, thereby finding a different point. This is the explicit display
that O(2) symmetry transforms both j s and s with the same rotation matrix. The fact
that we found k+ was constant for all these solutions and that k = 0 confirms the role of
the symmetry transformations (13) and (14).
The presence of the variational parameter k , which does not respect the O(2) symmetry
even if we allow field independent transformations, is a crucial distinction from the work
6 Remember in our choice of S in (6) we used the O(2) field rotation to eliminate a possible k term.
0
However, with k = 0 we regain the ability to exploit this freedom.
338
of [26,27] and [12]. It means that we have allowed our model to break the bounds of the
symmetry of S if the dynamics so choose. Thus one of the most important numerical results we have is that k = 0 to numerical accuracy (at least six significant figures). Once
this is known, only then do the symmetry arguments of the previous paragraph explain the
circles of Fig. 6.
This explicit display of the role of the symmetry in the solutions suggests an interesting
check, namely to impose the symmetry on the S0 . Thus, we ran the whole process again
holding k = 0, i.e., minimizing in a 3-dimensional parameter space. We recovered the circle of j s (at fixed m2 ). We can then take it a stage further as in addition to holding the k =
0, we can also fix one of the j s to zero, and thus minimize in a 2-dimensional parameter
space. In the unbroken symmetry regime, the j s are zero anyway, while in the broken symmetry phase there is an infinite set of equally valid solutions for the variational parameters,
so holding one of the j s at zero does not imply any loss of generality of the results.
In principle, one of the big advantages of the OPE is that one can systematically study
higher orders through a straight forward extension of the method used here. However, it is
likely that no solution for the variational parameters will be found at the next order, just
as none was found at second order. As mentioned earlier, this behaviour is common in an
OPE. Thus we would need to calculate K5 or L5 .
To summarize, the OPE method used here works well in identifying the phase transition
when there is a continuous symmetry, provided we fix the variational parameters by
minimizing the free energy and we do not use the previously successful principle of
minimal sensitivity. The explanation for the failure of the latter approach in this case after
its earlier successes seems to be our use of a variational parameters which are quadratic in
the Higgs fields. Another clear message coming from this work is that the optimal solutions
choose a trial action S0 which has the same symmetry as the full action S. This fact can be
exploited to dramatically reduce the number of variational parameters used, which in turn
will simplify and accelerate the analysis of more complicated models. Next, the source
variational parameters, j , and the combination (k+ + m2 ), act as order parameters. Thus
one can identify the transition point from free energy calculations alone which is a further
simplification when trying to find the phase diagram. We also note that this method can
be used to study just one of many equivalent vacuum solutions which distinguishes it
from lattice Monte Carlo. On the other hand, as shown here with just three orders, the
OPE has a practical and systematic way of improving its accuracy, unlike other analytic
non-perturbative methods. Thus we have demonstrated that OPE offers a practical route
to the study of models with Higgs sectors and continuous symmetries in non-perturbative
problems.
Acknowledgements
We would like to thank H.F. Jones and D. Winder for useful discussions. M.M. would
like to thank Tessella plc for their support.
339
Appendix
Fig. 9. This plot shows the first and third order result of h 2 i + h 2 i vs. m2 for = 25. It shows the
phase transition, but the kink is not as clearly visible as in other plots.
References
[1] R.J. Rivers, Path Integral Methods in Quantum Field Theory, Cambridge University Press,
Cambridge, 1987.
[2] I. Montvay, Nucl. Phys. B 26 (1992) 57.
[3] M. Lscher, hep-ph/9711205, talk given at the 18th International Symposium on LeptonPhoton
Interactions, Hamburg, 1997.
[4] M. Wortis, Linked cluster expansion, in: C. Domb, M.S. Green (Eds.), Phase Transitions and
Critical Phenomena, Vol. 3, Academic Press, London, 1974, p. 325.
[5] M. Lscher, P. Weisz, Nucl. Phys. B 300 (1989) 325.
[6] H. Meyer-Ortmanns, T. Reisz, Nucl. Phys. B 73 (1999) 892; hep-th/9809107.
[7] H.F. Jones, Nucl. Phys. B 39 (1995) 220.
[8] W. Kerler, T. Metz, Phys. Rev. D 44 (1991) 1263.
[9] A. Okapinska, Phys. Rev. D 35 (1987) 1835; hep-th/9508087.
[10] A.N. Sissakian, I.L. Solovtsov, O.P. Solovtsova, Phys. Lett. B 321 (1994) 381.
[11] V.I. Yukalov, J. Math. Phys. 33 (1992) 3994.
[12] C.M. Wu et al., Phase structure of lattice 4 theory by variational cumulant expansion, Phys.
Lett. B 216 (1989) 381.
[13] I.R.C. Buckley, A. Duncan, H.F. Jones, Phys. Rev. D 47 (1993) 2554.
[14] C.M. Bender, A. Duncan, H.F. Jones, Phys. Rev. D 49 (1994) 4219.
[15] C.M. Bender, K.A. Milton, S.S. Pinsky, L.M. Simmons Jr., J. Math. Phys. 30 (1989) 1447.
[16] W.E. Caswell, Ann. Phys. 123 (1979) 153.
[17] J. Killingbeck, J. Phys. A 14 (1981) 1005.
340
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
Abstract
We derive the invariant operators of the zero-form, the one-form, the two-form and the spinor from
which the mass spectrum of KaluzaKlein of eleven-dimensional supergravity on AdS4 N 010 can
be derived by means of harmonic analysis. We calculate their eigenvalues for all representations of
SU(3) SO(3). We show that the information contained in these operators is sufficient to reconstruct
the complete N = 3 supersymmetry content of the compactified theory. We find the N = 3 massless
graviton multiplet, the Betti multiplet and the SU(3) Killing vector multiplet. 2000 Elsevier
Science B.V. All rights reserved.
PACS: 04.60.K; 11.10.Kk; 11.15.-q; 11.30.Pb
Keywords: Harmonic analysis; KaluzaKlein; Eleven-dimensional supergravity; N = 3 supersymmetry; Anti-de
Sitter/conformal field theory
1. Introduction
In order to challenge Maldacenas conjecture [1] in all kinds of circumstances there is
a strong need for solved gauge theories on AdS manifolds. Indeed, they provide one of
the two comparison terms in the anti-de Sitter/conformal field theory correspondence [2].
A good deal of information to be checked lies already in the mass spectrum and the
supersymmetry multiplet structure of such theories.
A class of eleven-dimensional supergravity [3,4] solutions which may serve as compactified supersymmetric vacuum backgrounds is given by the FreundRubin solutions [5].
1 Present address: KMI-IRM, Ringlaam 3, 1180 Brussel, Belgium; E-mail: piet.termonia@oma.be; Tel.: +32-
342
They have the form AdS4 M, where M is a compact seven-dimensional Einstein manifold. A particular convenient number of manifolds to use for such KaluzaKlein compactifications are the G/H coset spaces. Part of their attraction comes from the fact that it is
immediate to read off their isometry groups. Consequently one knows the gauge symmetries of the fundamental particle interactions. Moreover, it is clear how to calculate the mass
spectrum of the four-dimensional theory. One can use harmonic analysis. As was shown
in [6,7] and in a series of papers [8,9], this can be done by exhibiting the group structure
of the constituents of the coset rather than laboriously solving the differential equations.
Hence the calculation of the mass spectrum gets reduced to a group-theoretical investigation of the coset and the final masses are found upon calculating the eigenvalues of some
discrete operators. This is food for computers. In this paper we outline the necessary steps
to be performed to reach this goal. Even in the compactifications where the calculations
become too gigantic the masses can still successfully be calculated in this way.
The complete list of the seven-dimensional coset spaces that may serve to compactify
eleven-dimensional supergravity to four dimensions is known [10]. Also the number of
the remaining supersymmetries and their geometries have been intensively studied in
the past. It turns out to contain some non-extremal supersymmetric cases that present
themselves as promising candidates for the anti-de Sitter/conformal field theory check.
This as opposed to KaluzaKlein on the seven-sphere which is related to the extremal
N = 8 supergravity theory [11]. There the spectrum can be derived from the short unitary
irreducible representation of Osp(8|4) with highest spin two, see [12]. From the perspective
of the three-dimensional superconformal theory this means that all the composite primary
operators have conformal weight equal to their naive dimensions. Hence no anomalous
dimensions are generated. One of such non-trivial compactifications is the one on AdS
M 111 , where M 111 is one of Wittens M pqr spaces [8,9,13]. For this manifold the complete
spectrum has been calculated and arranged in N = 2 multiplets, see [14] and [15]. This
spectrum provides some ideal material for the anti-de Sitter/conformal field theory check
which has already successfully been done [16].
In many cases of G/H compactifications the isometry group can be read off directly,
being the group G. If that is true, then the superisometry group is simply the supergroup
Osp(4|N ) G0 , where G = G0 SO(N ) and N is determined by the number of Killing
spinors that are allowed on the manifold. Yet this becomes slightly more complicated when
the normalizer N of H is non-zero. Then the isometry group becomes G N in stead of G.
This is for instance the case for eleven-dimensional supergravity on a background of
AdS4 N 010 ,
(1)
SU(3)
U (1)
(2)
with
N 010
which was introduced in the paper [17]. It has been proven to yield a N = 3 background 2
2 Actually there are many different manifolds of this form due to the present freedom in the choice of the
vielbeins on this manifolds. However, as shown in [17,18] there is a particular choice for which N = 3.
343
where the SO(3) R-symmetry group is the normalizer of the U (1) in the denominator.
Here the identification of the resulting SO(3) multiplet spectrum is not a straightforward
exercise, as can be seen in [18] if one takes the description (2) for the manifold. Still
as L. Castellani and L.J. Romans showed in [17], one can elegantly circumvent this
difficulty by taking a description which has the normalizer taken into account from the
very start. In particular, as they suggested in [17,18], one can exhibit the fact that the
N = 3 supersymmetric N 010 is equal to
SU(3) SU(2)
,
SU(2) U (1)
(3)
where the SU(2) in the denominator is diagonally embedded in SU(3) SU(2). In this
way the SO(3) SU(2) is everywhere explicitly present.
The harmonic analysis on N 010 has been done partly by Castellani in [18], using the
formulation (2). Yet due to lack of computing power his calculation did not cover the
complete spectrum. It is the scope of this paper to extend the technique of harmonic
analysis to the alternative formulation (3) and derive really the complete spectrum.
This choice is motivated by the fact that the field components will then automatically
be organized in irreducible representations of SO(3). This will make the matching of
the spectrum with N = 3 multiplets possible. We present here a sufficient part of the
mass operator eigenvalues that contains enough information to reconstruct the complete
spectrum by using the supersymmetry mass relations [8,9]. This will be used in a
subsequent paper [19] to derive the complete multiplet structure of the N = 3 N 010
FreundRubin compactifications.
The importance of the spectrum of eleven-dimensional supergravity on (1) lies in the
fact that an anti-de Sitter/conformal field theory comparison in this background is fairly
facilitated by the fact that for the four-dimensional theory the structure of the effective
Lagrangian is already known to be [20],
SU(3, 8)
.
SU(3) SU(8) U (1)
(4)
Still, as becomes clear from the work presented in this paper, the mass spectrum of the
N = 3 theory is going to be enough non-trivial to provide another powerful check of the
anti-de Sitter/conformal field theory duality.
The spectrum that we obtain here has to be organized in N = 3 multiplets. A systematic
classification of all the possible short N = 3 multiplets and their superspace structure
does not exist in the literature. Only the short vector multiplet in Table 1 has been found
explicitly in [21]. As in the case of the AdS M 111 KaluzaKlein compactification the
matching of the mass spectrum will teach us a lot if not everything about the existence
of other short multiplets. Following the arguments of [16] it turns out to be necessary to
decompose the resulting N = 3 multiplets in N = 2 multiplets. Upon doing so the states
of the resulting N = 3 multiplets can then be recognized as coming from the on-shell field
components of the superfields on the N = 2 superspace that was introduced in [22]. Their
superfield constraints are then straightforwardly read off. We postpone this issue to a future
publication.
344
Table 1
The states of the Osp(3|4) vector multiplet representation organized in representations of SO(2)
SU(2)spin and SO(3)isospin as they appear in KaluzaKlein of eleven-dimensional supergravity. The
unitary bound for the ground state is satisfied E0 = J . The names of the fields are chosen as in [8,9]
Spin
Energy
Isospin
Mass
Name
Mass
Name
J +1
J 1
16E0 (E0 1)
A/W
16E0 (E0 1)
1
2
1
2
1
2
1
2
3
2
3
2
1
2
1
2
J 1
4E0
L,T
4E0
0
0
0
0
0
J+
J+
J+
J+
J +2
J +1
J +1
J +1
E0 = J
J 2
4E0
L,T
4E0
4E0 4
L,T
4E0 + 4
J 1
4E0 4
L,T
4E0 + 4
J 2
J
J 1
J 2
J
16E0 (E0 + 1)
16E0 (E0 1)
16E0 (E0 1)
16E0 (E0 1)
16(E0 2)(E0 1)
, S/W
, S/W
16E0 (E0 + 1)
16E0 (E0 1)
16E0 (E0 1)
16E0 (E0 1)
16(E0 2)(E0 1)
The work that we present here fits in a much wider project that is currently being carried
out by the Torino group. The final scope of this AdS N 010 spectrum is to check the
anti-de Sitter/conformal field theory correspondence as it has been carried out [16] for
AdS M 111 . It will be done [23] also for eleven-dimensional supergravity on AdS Qppp .
This paper is organized as follows. We start with a short description of the geometry of
N 010 . We thereby restrict ourselves to the essentials that we will need further on. We leave
a more rigorous treatment to a future publication [24]. Then we will repeat the standard
concepts of harmonic analysis and explain how they can be applied to eleven-dimensional
supergravity on AdS4 N 010 . Using these techniques we will then compute the zero-form
operator M(0)3 , the one-form operator M(0)2 (1) , the two-form operator M(0)(1)2 and the
spinor operator M(1/2)3 , the notation of these operators being the same as in [79]. We
will then present their eigenvalues. We conclude by arguing that the information obtained
in this paper is sufficient to calculate the complete spectrum. We will do so by means of
a concrete example: the massless graviton multiplet. Moreover, in this way we will prove
that the remaining spectrum is indeed N = 3 supersymmetric. Finally, we will identify the
series of irreducible representations of SU(3) SU(2) with a common field content. For
these series we will list the eigenvalues that are present. From these eigenvalues the masses
are then obtained by applying the mass formulas from [8,9].
2. The geometry of N 010
In this section we introduce the essential concepts 3 of the geometry of N 010 . We restrict
ourselves to the elements that serve our purposes. We leave a more elaborated treatment
3 The material in this section was explained to me by Leonardo Castellani. I am indebted to him for this. He
also calculated the rescalings at an early stage of this research.
345
to [24]. We fix some freedom in the choice of the vielbeins. This will ensure that the
compactified theory on (1) will have N = 3 supersymmetry.
The N 010 coset spaces are special manifolds of the class of N pqr coset spaces,
G SU(3) U (1)
=
,
H
U (1) U (1)
(5)
where the integer numbers p, q, r specify the way in which the two U (1) generators M
and N of H are embedded in G,
2 i
rp 38 + 2i rq3 2i (3p2 + q 2 )Y ,
M =
2
RQ
1
2i q8 + 2i p 33 ,
(6)
N =
Q
with
R=
3p2 + q 2 + 2r 2 ,
Q=
3p2 + q 2 ,
(7)
where Y is the U (1) generator in SU(3) U (1) and the SU(3) generators are given
in Appendix B. We do not get into details in this paper but for a detailed description
of the definition, the geometry and the properties of these spaces we refer the reader to
the literature [17,18,24]. These spaces are seven-dimensional and can be used to make a
FreundRubin background for eleven-dimensional supergravity,
AdS N 010 .
(8)
(9)
SU(3) SU(2)
.
SU(2) U (1)
(10)
For the generators of SU(3) we take 2i and for SU(2) we take 2i , being the Gell-Mann
matrices (see Appendix B) and being the Pauli matrices. The generator of U (1) is given
by
346
T8 = 2i 8 .
(11)
a = 1, 2, 3,
H = 9, 10, 11,
(12)
where H = 9 corresponds to a = 1 and so on. We will call it SU(2)diag . For the remaining
coset generators we have
T = 2i (1 1 , 2 2 , 3 3 , 4 , 5 , 6 , 7 ).
We need the SO(7) covariant derivative on the coset space. It is defined as
D = d + B T SO(7) ,
(13)
(14)
where the one-form B is the connection of the coset space and T SO(7) are the generators
of SO(7), be it of the vector representation or the spinor representation,
scalar irrep T SO(7) = 0,
SO(7)
= [
] ,
(15)
vector irrep T
SO(7)
= 14 [ ] ,
spinor irrep T
where the matrices of the SO(7) Clifford algebra are given in the Appendix C and is
the metric of Appendix B.
As already explained, there is some freedom in the choice of the vielbeins, not all of
them leading to N = 3.
To see how this goes, let us recall that on the coset there is the invariant
L1 d L = e T + H TH ,
(16)
where L is coset representative. H is the index running on H and is the index running
on SO(7), see Appendix B for conventions. The fields e and H are the G/H vielbein
and the H connection. Using the coset vielbein e we define the coset connection B ,
de + B e = 0.
(17)
The vielbein is specified up to 7 rescalings of the coset directions, see [79,18]. We take
this into account by introducing the seven parameters r
e r e .
(18)
Then the form of the connection is obtained by solving the MaurerCartan equation, i.e.,
applying the external derivative d to (16) and using d 2 = 0,
r r
r r
1 r r
C
C
C e
B =
2 r
r
r
r
CH H ,
(19)
r
where the constants C are the structure constants of G. We take the embedding of H
into SO(7) as follows,
(20)
TH = (TH ) T SO(7) .
347
Now we specify the rescalings r . As is known [8,9], only for some well-chosen values
of these parameters does the manifold become an Einstein manifold. Moreover, as is
clear from [17,18] not all of the valuable choices for these rescalings necessarily lead
to the contemplated number of supersymmetries. To see which of them do, we look at the
curvature. The curvature on the coset space is defined by
R = d B + B B .
(21)
rA = 4 2 e
ra = 2e,
(24)
(25)
Following the conventions of the papers [8,9] we put e = 1. Then we will be able to apply
the mass relations and mass formulas that were obtained in these papers.
To conclude this section we give the explicit form of the embedding of the SU(2)diag in
SO(7) according to Eq. (20). For the vector we get
0
0
T9 =
0
0
0
0
0
0
T10 =
0
0
0
0
0
0
1
0
0
0
0
0 0
1 0
0 0
0 0
0 0
0 0
0 12
0 1 0
0 0
0
0 0
0
0 0
0
0 0
0
0 0 12
0 0
0
0
0
0
0
0
1
2
0
0
0
0
0
0
0
12
0
0
0
0
12
0
0
0
0
0
1
2
0
0
0
0
0
1
2 ,
0
0
0
0
0
0
,
1
2
0
0
ra = 2e,
ra = 10
3 e,
rA = 4 2 or
rA = 4 310 e,
(22)
(23)
that yield the Einstein curvature (25). However not all of them necessarily lead to an N = 3 theory. As we will
show in the last section of this paper, the rescaling (24) that we adopt yields N = 3 supersymmetry. We refer the
reader for a detailed study of these rescalings to a future publication [24].
348
0
1
T11 =
0
0
0
0
and,
0
0
T8 = 0
0
0
1
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 12
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
1
2
0
0
0
0
0
0
,
0
12
0
0
0
0
0
0
0
1
2
0
0
0
3
2
0 0
3
2
0
0
0
0
0 0
0 0
0
0
0
0
3
2
(26)
0
3
(27)
For the matrices TH the order of the indices is 17. For the spinor representation we have
0
0
0
T9 =
0
0
0
0
T10 =
0
0
0
0
0
0
T11 =
0
0
0
0
0
i
2
0
0
0
0
0
0
0
0
0
0
0
0
0
12
0
0
0
0
0
0
i
2
0
0
0
0
0
0
i
2
0
0
0
0
0
0
0
i
2
0
0
0
0
i
2
0
0
0
0
1
2
0
0
0
0
0
0
0
0
2i
0
0
0
0
0
0
0
0
0
12
0
0
0
0
0
0
i
2
0
0
0
0
0
0
0
0
0
0
i
2
0
0
0
0
1
2
0
0
0
0
0
0
0
0
2i
0
0
0
0
0
0
0
0
i
0
i
0
0
0
,
0
i
2
0
0
0
0
0
0
12
0
0
0
0
0
0
i
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
,
0
1
2
0
0
0
0
,
0
0
i
(28)
0
0
0
T8 =
0
0
0
0
3
0
0
2i 3
0
0
0
0
0
0
0
0
0
0
2i
0
0
0
i
2 3
0
0
0
0
i
2
0
0
0
0
0
0
0
0
3 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.
0
0
0
349
(29)
3. Harmonic analysis
In this section we provide the main ingredients of harmonic analysis that we need for the
calculation of the masses of the zero-form, the one-form, the two-form and and the spinor.
The technique of harmonic analysis has been well established, see papers [79,18], so we
restrict ourselves to the essentials. We focus on its application to our coset N 010 .
The main idea is that functions on a coset space G/H can be expanded in terms of
the components of the operators of the irreducible representations of G. In particular,
()
a complete set of functions (L(y)) on the coset G/H is given that transform in a
irreducible representation of H as follows,
()
()
(30)
L(y)h =
L(y) D () ,
where () indicates the H -irreps and the its indices. These functions depend on the
coordinates of the coset space via the coset representative L(y). Then for such functions
the expansion is given by,
X () () G
()
cG D
L(y) ,
(31)
L(y) =
()
where G are the contracted indices of G and () label all the irreducible representations of
G that contain the irreducible representation () under reduction to H as follows 5
() + () + .
H
(32)
The crucial step now is to use the fact that the covariant derivative (14) can be expressed
in terms of the coset generators plus some additional discrete operators. Indeed, there is
no need to express this covariant derivative as a differential operator. This is a generic
feature of coset manifolds [79,18]. It is extremely useful for our purposes since evaluating
the mass terms in the field equation of eleven-dimensional supergravity can then be done
without solving differential equations.
To see this, it is sufficient to realize that the harmonic expansion ultimately is an
expansion in L(y)1 . Then one uses (16) to express the derivatives in terms of the vielbein.
5 In fact () may appear multiple times in the reduction, say m times. In that case it should be understood that
the above sum contains () m times. For a more clear treatment of this see [79].
350
Then on a field Y on the coset space that sits in some representation of SO(7) the covariant
derivative becomes
r r
r r
1
C T SO(7) Y +
C T SO(7) Y. (33)
D Y = r T Y +
r
2
r
We now show how this reduction is done in the case where the coset is N 010 . In
particular we show how the representations of SU(3) SU(2) reduce to representations
of SU(2)diag U (1). We also give the constraint that is to be imposed in order to ensure
the right U (1) weight of the harmonic.
Let us look at the index structure of the objects (D () )G (L(y)) in the expansion (31).
Let us start with the indices G. Clearly, they have the following structure,
G=
k1 . . . kM2 l1 . . . lM1
m1 m2J
n1 . . . nM2
| {z } | {z }
| {z }
M2
M1
(34)
2J
being the product of a generic SU(3) Young tableau with a generic SU(2) Young tableau.
The indices run through
k1 , . . . , kM2 , l1 , . . . , lM1 , n1 , . . . , nM2 = 1, 2, 3,
m1 , . . . , m2J = 1, 2.
(35)
1 k1 . . . k2p l1 . . . l2q
3
m1 m2J
2 3 ... 3
| {z } | {z }
| {z }
2p
{z
2q
}|
M2
2J
{z
(36)
M1
(37)
and,
1 1 . . .1 ,
2 2 . . .2 ,
3 3 . . .3 .
(38)
We will use this notation henceforth. The indices k1 , . . . , k2p , l1 , . . . , l2q in the above
Young tableau now get the values 1, 2 only. The parameters p and q get half-integer values.
This yields a
pq J
(39)
351
SU(2)diag -representation in the reduction (32). From this representation one can extract the
irreducible representations by contracting pairs of the indices k, l, m with the -symbol.
Note however that it is not necessary to contract pairs of a k with an l since then one would
be over counting, as can be seen upon using the cyclic identity,
[i j ]
1 3
= 12 ij
.
3
2
(40)
Thus in the Young tableau (36) one takes only those SU(2)diag -irrepses that are obtained
by contracting pairs of ks and ms and pairs of ls and ms. The remaining indices are then
completely symmetrized.
We specify the most generic class of Young tableaux that we will need for the harmonic
analysis on N 010 and introduce the following notation for it 6 ,
(M1 , M2 , J ; p)
k(s,t,u)
1 ...k2s ,l1 ...l2t ,m1 ...m2u
= I J
2 3 3 3
3
|
|
{z
}|
2p
{z
}|
M2
{z
2(J up+s)
{z
}
}
M1
j1 j2(J u) m1 . . . m2u ,
|
{z
(42)
2J
where we have indicated the number of boxes under the Young tableaux. In the above
notation for the Young tableaux (42) we do not a priori assume symmetrization of the
indices k1 . . . k2s , l1 . . . l2t , although that is what we need in the harmonic expansion. It will
be useful for our purposes to consider the more general Young tableaux of (42). It is also
clear that in this notation,
2s 6 2p 6 M2 ,
(43)
2t 6 2(J u p + s + t) 6 M1 ,
(44)
is assumed without notice. Otherwise the Young tableaux simply do not exist. The only
free SU(2)diag -indices in the above Young tableaux are the indices k1 . . . k2s , l1 . . . l2t and
m1 . . . m2u , since the indices i1 . . . i2(J u) are contracted with the indices j1 . . . j2(J u) by
means of the s. This is the most generic way finding the embedding of a
6 We use the notation,
I J i1 j1 . . . i2J j2J .
(41)
352
st u
(45)
2 8 it has weight,
(46)
i 3 13 (M2 M1 ) + J 2p + (s + t u) .
This weight has to be matched with the U (1) weight
i
2 31
(47)
that is known from the transformation of the harmonic as under (27) and (29). Hence the
components of the SU(2)diag -representation that appear in the reduction G H will be
in a given U (1)-representation if p is constrained in terms of the G-representation labels
M1 , M2 , J and 1
(48)
p = 12 J + 13 (M2 M1 ) + (s + t u) 12 1 .
When we will refer to the U (1) weight, we will refer to the number 1 henceforth. It is
important to realize here that this constraint not only constraints p in terms of M1 , M2 , J
but it also implies that the difference M2 M1 has to be a multiple of three. Moreover as
we will see later on J has to be a positive integer. The parameter p is half-integer.
Now we know the embedding of the representations of SU(2)diag U (1) in SU(3)
SU(2) with a given SU(2)diag -spin and U (1)-weight.
In general all these Young tableaux are not independent. This can easily be seen using
the cyclic identity,
(s,t,u)
1
2
1
2
(49)
which is merely a generalization of the identity (40). This identity allows us to reduce the
number of Young tableaux that we are using in most of the cases. For instance, if we are
considering SU(3) irreducible representations that are big enough, i.e., the values M1 and
M2 exceed the numbers,
M2 > 2p + 1,
M1 > 2(J u p + s + t) + 1
(50)
0 0
(s ,t ,u1/2)
then one might make the harmonic expansion in components of the form
only. An example where this is not possible is the 0 0 1 representation of SU(2)diag
embedded in an SU(3) SU(2) with
M1 = M2 = 0,
J = 1.
(51)
353
However, there is a subset of embeddings where one can restrict oneself to the following
components,
(s,t )
2 3 3 3 3
|
{z
2s
}|
{z
{z
2(ps)
}|
}|
{z
2(J p+s)
M2
} | {z }
{z
2t
M1
j1 j2J ,
| {z }
(52)
2J
where u = 0 is understood in the notation of . In this paper, for the calculation of the
eigenvalues of the mass operators we will always assume that M1 and M2 are big enough
in order to justify an expansion in the components (52) only. In other words, here we
calculate the mass matrix for those SU(3) SU(2) representations where the harmonics
sit in the s t irreducible representation of the decomposition of the s-spin times the tspin. Afterwards we will argue the eigenvalues thus obtained, seen as functions of the
labels M1 , M2 , J , are also the functions for the eigenvalues to be found in the cases where
this assumption fails. To illustrate this we will work out (51). Yet a detailed study of the
existing eigenvalues of the mass operators in the cases of short multiplets will become
quite a subtle book-keeping exercise. Fortunately it can be implemented in some computer
programs. All this will be explained in the last section of this paper.
We now apply this to KaluzaKlein on
AdS4 N 010 .
(53)
N 010
as y. A field on (53)
We write the coordinates of AdS4 as x and the coordinates of
sits in a representation of SO(1, 3) as well as in a representation of SO(7), generically being
some multiple tensor product of the vector representation and the spinor representation. As
will be exemplified further on, these SO(7) representations decompose in representations
() of H ,
SO(7) (1 ) (n ),
hence the fields decompose in
(1 )
..
.
(n )
SU(2)diag
(54)
fragments,
(55)
(56)
354
(57)
(58)
then 2y (Y () ) (y) can be evaluated explicitly and provide the mass operators of the fourdimensional theory. The evaluation of these mass matrices and their eigenvalues is the
subject of the following sections.
To see how the covariant derivative (33) works on the components of the Young tableaux
(42), one first straightforwardly derives the following formulae,
)
(p)
( )a k(s,t
1 ...k2s ,l1 ...l2t
2s
X
(s1/2,t )
1
2
=1
2t
X
(s,t 1/2)
(a )l n k1 ...k2s ,l1 ...n ...l2t (p)
=1
(s+1/2,t +1/2)
1
2
(s+1,t )
mnk
p+
1 ...k2s ,l1 ...l2t
(s,t +1)
1
2
1
2
1
2
,
(59)
and
)
(p)
A k(s,t
1 ...k2s ,l1 ...l2t
(s+1/2,t )
2s
X
k m (A )3 m
=1
(s1/2,t )
k1 ...k ...k2s ,l1 ...l2t
1
2
1
2
(s,t +1/2)
(s+1/2,t )
+ 2(s p)(A )3 m mk1 ...k2s ,l1 ...l2t (p) k1 ...k2s ,l1 ...l2t m (p)
(s+1/2,t )
2t
X
=1
(A )l 3
1
2
(s,t +1/2)
1
2
(s,t 1/2)
(p)
k1 ...k2s ,l1 ...l ...l2t
(s,t +1/2)
(60)
where m means that k is replaced by m and n means that l is replaced by n. The hats
on the indices k and l indicate that these indices are deleted. We have suppressed the labels
M1 , M2 , J , since they do not change under the transformations. In fact, they do not change
under the covariant derivative and the evaluation of the mass operators can be done for fixed
M1 , M2 , J in the harmonic expansion (56). For the derivation of the above expressions we
355
)
used the cyclic identity (49). Mind that in the Young tableaux k(s,t
(p) we have
1 ...k2s ,l1 ...l2t
not symmetrized the indices k1 . . . k2s , l1 . . . l2t . Recall that in the notation (52) there is no
symmetrization in these indices.
Looking at the expression for the covariant derivative in Eq. (33), one sees that the
generators T are precisely given by the above formulae (59) and (60) multiplied with 2i .
Moreover, these formulae can be programmed on a computer and in this way the covariant
derivative is easily calculated.
D Y[] = 0,
D Y[ ] = 0,
D Y() = 0,
D = 0.
Table 2
The content of harmonics of eleven-dimensional supergravity: their name as in [8,9],
their SO(7) irreducible representations and their field equations
Harmonic
SO(7)-irrep
Field equation
1 = (0, 0, 0)
D D Y = M(0)3 Y
7 = (1, 0, 0)
2D D[ Y] = M(1)(0)2 Y
21 = (1, 1, 0)
3D D[ Y] = M(1)2(0) Y
35 = (1, 1, 1)
Y ()
27 = (2, 0, 0)
( 12 , 12 , 12 )
(D ) = M(0)3
( 32 , 12 , 12 )
57 = M 3 1 2
( )( )
1
D Y
= M(1)3 Y
24
(2 + 56) R Y ( ) = M(2)2(0) Y ()
(61)
356
Since we are dealing with N = 3 supersymmetry it will suffice to know the masses for
the zero-form, the one-form, the two-form and the spinor only. The rest of the multiplet
spectrum can then be determined using supersymmetry.
The decomposition of the SO(7) vector representation into irreducible representations
of SU(2)diag goes as follows
(0, 0, 0) (1, 0),
1),
(1, 0, 0) (3, 0) (2, 1) (2,
1) (3, 0) (3, 0)
(1, 1, 0) (4, 1) (4,
1) (1, 2) (1,
2) (1, 0),
(2, 1) (2,
(62)
where the irreducible representations of SU(2)diag U (1) are written as (2(s + t) + 1, 1),
1 being the U (1) weight (47). One may check now that from the formula (48) it follows
indeed that J has to be an integer number. Accordingly, an SO(7) one-form and an SO(7)
two-form can be expressed in terms of the following SU(2)diag fragments
(2)
R
C (2)
(2)
C
(2)
C
(1)
j
Ci
(2)j
C
,
(63)
C (1)i ,
(2)
(1)
Rij
Rij
O(2)
ij
(2)
C
ij k
C (2)ij k
where the fields C i1 ...in are the complex conjugate fields of Ci1 ...in and the symmetrized
indices indicate in what representation of SU(2)diag they sit. The superscript (1) and (2)
indicate that these are the fragments of the one-form and the two-form respectively. The
fields R are pseudo real, i.e.,
Ri1 ...in = i1 j1 in jn Rj1 ...jn .
(64)
For the choice of the coset generators (13) it is useful to introduce the indices (196) and
split
= ab , aA , AB ,
(65)
= { a , A },
then one can do the decomposition (63) concretely by using the components of the
matrices,
= R(0) ,
a = i(a )i j ik R(1)
jk ,
A = (A )3 i Ci(1) + (A )i 3 C (1)i ,
ab = i abc (c )i j ik R(2)
jk ,
357
aA = (a )i j (A )k 3 ik Cj(2) + (a )j i (A )3 k ik C (2)j
(2)
+ (a )i j (A )3 k il Cj kl + (a )j i (A )k 3 ik C (2)ij k ,
AB = (A )3 i (B )3 j ij C (2) + (A )i 3 (B )j 3 ij C (2)
+ i([A )i 3 (B] )3 i R(2) + ([A )i 3 (B] )3 j ik Okj .
(2)
(66)
This shows how the decomposition (32) is done. The numbers 1 in (62) are obtained by
applying the matrix T8 in (27) on the components of and . The scalar does not
transform under T8 and hence 1 = 0. One may also verify that the above decomposition
(66) is in agreement with the SU(2)diag matrices (27). The expansion in the harmonics D
then becomes 7 ,
R(0) = h(0,0) (x) D (0,0)(0)(y),
(1)
Ci
(1/2,0)
= AWc
(1/2,0)(1)
(x) Di
(0,1/2)
(y) + AWc
(0,1/2)(1)
(x) Di
(y),
g c(1/2,0)(x) D (1/2,0)(1)(y) + ij AW
g c(0,1/2)(x) D (0,1/2)(1)(y),
C (1)i = ij AW
j
j
(1/2,1/2)
(1/2,1/2)(0)
(1)
(1,0)(0)
(1,0)
(y) + AWr
(x) Dij
(y)
Rij = AWr (x) Dij
(0,1)(0)
(y),
+ AWr(0,1)(x) Dij
R(2) = Zr(0,0)(x) D (0,0)(0)(y),
C (2) = Zc(0,0)(x) D (0,0)(2)(y),
ec(0,0)(x) D (0,0)(2)(y),
C (2) = Z
(1/2,0)
(1/2,0)(1)
(0,1/2)
(0,1/2)(1)
(2)
(x) Di
(y) + Zc
(x) Di
(y),
Ci = Zc
(1/2,0)(1)
(0,1/2)(1)
(2)i
ij e(1/2,0)
ij e(0,1/2)
= Zc
(x) Dj
(y) + Zc
(x) Dj
(y),
C
(1/2,1/2)
(1/2,1/2)(0)
(2)
(1,0)(0)
(y) + Zr
(x) Dij
(y)
Rij = Zr(1,0)(x) Dij
(0,1)(0)
(y),
+ Zr(0,1)(x) Dij
(1/2,1/2)
(1/2,1/2)(0)
(2)
(1,0)(0)
(y) + Zo
(x) Dij
(y)
Oij = Zo(1,0)(x) Dij
(0,1)(0)
(0,1)
(y),
+ Zo (x) Dij
(3/2,0)
(3/2,0)(1)
(1,1/2)
(1,1/2)(1)
(2)
(x) Dij k
(y) + Zc
(x) Dij k
(y)
Cij k = Zc
(1/2,1)
(1/2,1)(1)
(0,3/2)
(0,3/2)(1)
(x) Dij k
(y) + Zc
(x) Dij k
(y),
+ Zc
(3/2,0)
(3/2,0)(1)
(1,1/2)
ec
ec
(x) Dmnp
(y) + im j n kp Z
(x)
C (2)ij k = im j n kp Z
(1,1/2)(1)
(1/2,1)(1)
im j n kp e(1/2,1)
(y) + Zc
(x) Dmnp
(y)
Dmnp
(0,3/2)
(0,3/2)(1)
ec
(x) Dmnp
(y),
(67)
+ im j n kp Z
(s,t )(1)
where the index structure on the harmonics Di1 ...i2(s+t) is given by the symmetrized Young
tableau
7 We expand the complex conjugates in the conjugate representations. Upon doing so there appear some
irrelevant signs in the front of the harmonics. We have absorbed these signs in the x fields.
358
(68)
with p constrained as in (48). We see that we have summed in the harmonic expansion a
multiple of times over the same irreducible representations of G when a given irreducible
representation of SU(2)diag appears multiple times in the reduction G H . The names of
the x-fields refer to names that were used in the papers [8,9]. For instance the x-field h(0,0)
gives the masses for the spin-2 fields h , the x-fields AW will contribute to the vectors
A and W , and Z will contribute to the vector Z. So now, for a given representation of G,
labeled by M1 , M2 , J we have the following basis of four-dimensional fields
Z (0,3/2)
c
(1/2,1)
h h(0,0) (x),
AW (0,1)
r
(1/2,1/2)
AWr
AWr(1,0)
(0,1/2)
AW
AWc(1/2,0) (x),
AWc
(0,1/2)
AW
gc
(1/2,0)
gc
AW
Zc
(1,1/2)
Zc
(3/2,0)
Zc
(0,3/2)
Z
ec
(1/2,1)
Z
e
c
(1,1/2)
Z
e
c
e(3/2,0)
Zc
Zo(0,1)
(1/2,1/2)
Zo
(1,0)
(x). (69)
Z Zo
Z (0,1)
r
Z (1/2,1/2)
r
(1,0)
Zr
(0,1/2)
Zc
(1/2,0)
Zc
(0,1/2)
Z
ec
(1/2,0)
Z
e
c
Z (0,0)
c
e(0,0)
Zc
(0,0)
Zr
In order to determine the mass spectrum we have to apply the invariant Laplace Beltrami
equations of Table 2 to the harmonics in (67) which will then determine the masses of
the four-dimensional x fields by means of the Eq. (58). Clearly, since we have a onedimensional, a seven-dimensional and a twentyone-dimensional basis in (69) for the zeroform, the one-form and the two-form respectively the matrices M(0)3 , M(1)(0)2 and M(1)2 (0)
will be 1 1, 7 7 and 21 21. Notice that for the two-form we will already need to handle
a 21 21 matrix. Had we not restricted ourselves to representations made with s-spin and
t-spin only, then we would have had to handle a 41 41 matrix for the two-form. Moreover
we would be over counting in the harmonic expansion. We then calculate the eigenvalues
of these operators and the masses are then found by using the mass formulas of [8,9].
In order to obtain the complete multiplet spectrum by the method of paper [15], it is most
convenient to have the eigenvalues of the spinor also. The spinor is also decomposed in
irreducible representations of SU(2)diag as follows,
359
(70)
Concretely, when using the SO(7) -matrices for the Clifford algebra as we are advocating
in the appendix, then one can see from (28) that its components are
(71)
=
i .
ij
Moreover the Majorana condition
C = ,
(72)
with the conjugation matrix as in the appendix, is translated into (pseudo) reality conditions
on its components,
= ,
i = ij j ,
i = ij j ,
ij = ik j l kl .
(73)
The numbers 1 are read off from (29). Then similarly as for the forms, the harmonic
expansion acquires the form,
= (0,0)(x) D (0,0)(0)(y),
(1/2,0)
(x) Di
(1/2,0)
(x) Di
i =
i =
(1/2,0)(1)
(1/2,0)(1)
(1,0)(0)
ij = (1,0)(x) Dij
(0,1/2)
(y) +
(0,1/2)
(y) +
(0,1/2)(1)
(x) Di
(0,1/2)(1)
(x) Di
(1/2,1/2)(0)
+ (1/2,1/2)(x) Dij
(y),
(y),
(0,1)(0)
+ (0,1)(x) Dij
So we conclude that for the x-fields of the spinor we need the basis
(0,1)
(1/2,1/2)
(1,0)
(0,1/2)
(1/2,0) (x)
(0,1/2)
(1/2,0)
. (74)
(75)
(0,0)
for which we should find an 8 8 matrix M(2)2 (0) .
At this stage we have all the ingredients to calculate the matrices M(0)3 , M(0)2 (1) , M(0)(1)2
and M(1/2)3 . As explained before, it is possible to restrict oneself on the terms in the
expansion for a given irreducible representation of G. Such representation is characterized
by the labels M1 , M2 , J , see (34). To do the calculation, we need to perform the following
steps:
360
1. Invert (66). We know how the mass operators work on the the fields . They are
given in Table 2. To see how they work on the fragments we invert (66).
2. For all of the fragments in (63) and (71) draw the Young tableaux of D(s,t )(1). Check
whether it exists. Solve the constraint (48) and then check the bounds (43) and (44).
3. Apply the operators of Table 2, by evaluating the covariant derivative (33) on the
Young tableaux of the fragments. Recall that these are not differential operators.
All one has to do is to evaluate the operators T and do the multiplication with the
structure constants. To this end one can use the formulae (59) and (60).
All the above steps can be programmed in say M ATHEMATICA. We assume that the bounds
(43) and (44) are satisfied and calculate the four contemplated operators. Because of this
assumption these matrices are the matrices for the long N = 3 multiplets. Indeed, since
all the possible fragments are present that implies that all the field components of the
multiplets are present. We list the matrices in Appendix A.
We finish this section by presenting the eigenvectors of the operators M(0)3 , M(0)2(1) ,
M(0)(1)2 and M(1/2)3 , which is what we ultimately need for the masses. We use the notation
16
2(M12 + M22 + M1 M2 + 3M1 + 3M2 ) 3J (J + 1) .
(76)
3
We write the subscript 0 to remember that this is the eigenvalue of the zero-form operator.
(f )
We denote the eigenvalues by , where f = 0, 1, 2, s refers to the zero-form, the oneform, the two-form and the spinor respectively and enumerates its different eigenvalues.
Then,
The zero-form eigenvalues:
H0
(0) = H0 .
(77)
p
(1)
1 = H0 32J 8 4 H0 32J + 4,
p
(1)
2 = H0 32J 8 + 4 H0 32J + 4,
p
(1)
3 = H0 + 24 4 H0 + 36,
p
(1)
4 = H0 + 24 + 4 H0 + 36,
p
(1)
5 = H0 + 32J + 24 4 H0 + 32J + 36,
p
(1)
6 = H0 + 32J + 24 + 4 H0 + 32J + 36,
(1)
7
= H0 .
(78)
(79)
(80)
(81)
(82)
(83)
(84)
1 = H0 + 64J,
(85)
(2)
2
(2)
3
(2)
4
= H0 + 32J + 32,
(86)
= H0 + 32J + 32,
(87)
= H0 + 32,
(88)
361
(2)
5 = H0 + 32,
(89)
(2)
6
(2)
7
(2)
8
(2)
9
(2)
10
(2)
11
(2)
12
(2)
13
(2)
14
(2)
15
(2)
16
(2)
17
(2)
18
(2)
19
(2)
20
(2)
21
= H0 + 32,
(90)
= H0 32J,
(91)
= H0 32J,
(92)
= H0 64J 64,
p
= H0 + 48 + 8 H0 + 36,
p
= H0 + 48 8 H0 + 36,
p
= H0 + 32J + 48 + 8 H0 + 32J + 36,
p
= H0 + 32J + 48 8 H0 + 32J + 36,
p
= H0 32J + 16 + 8 H0 32J + 4,
p
= H0 32J + 16 8 H0 32J + 4,
p
= H0 32J 8 4 H0 32J + 4,
p
= H0 32J 8 + 4 H0 32J + 4,
p
= H0 + 24 4 H0 + 36,
p
= H0 + 24 + 4 H0 + 36,
p
= H0 + 32J + 24 4 H0 + 32J + 36,
p
= H0 + 32J + 24 + 4 H0 + 32J + 36.
(93)
(94)
(95)
(96)
(97)
(98)
(99)
(100)
(101)
(102)
(103)
(104)
(105)
H0 32J + 4,
(106)
H0 32J + 4,
(107)
H0 + 36,
(108)
H0 + 36,
(109)
H0 + 32J + 36,
(110)
H0 + 32J + 36,
p
= 10 H0 + 36,
p
= 10 + H0 + 36.
(111)
(s)
2 = 6 +
(s)
3
(s)
4
(s)
5
(s)
6
(s)
7
(s)
8
= 6
= 6 +
= 6
= 6 +
(1)
p
p
p
p
p
(2)
(2)
(112)
(113)
362
(114)
To see which of the fragments survive the bounds we now apply the two Table 3 and 4 as if
they were two sieves. We now illustrate how to use these table by some concrete examples,
1. Let us consider the G-irrep with
M1 = M2 = 0,
J = 0.
(115)
Sifting the fragments with Table 3 we see that none of the fragments of the
decomposition survive the bounds except for 1(1/2,1,0), 1(0,1/2,0), 0(1/2,1/2,0), 0(0,0,0),
1(1/2,0,0), 1(1,1/2,0). Then sifting the fragments with Table 4, we see that these
fragments only
0(0,0,0)
(116)
survive. This implies that in the harmonic expansion (67) only the harmonic D (0,0)(0)
appears.
363
Table 3
The two lower bounds as in (43) and (44). The rows represent the lower bound in (43) and the
columns represent the lower bound in (44)
M2 M1
2
3
J>
M2 M1
1
3
M2 M1
3
M2 M1
+1
3
M2 M1
+2
3
3
M1 M2
2
3
1(0, 2 ,0)
M1 M2
1
3
2(0,0,0)
1
(0, 21 ,0)
1
( ,1,0)
1 2
(0,1, 21 )
(0, 21 ,1)
(0,0, 23 )
0(0,1,0)
(0, 23 ,0)
1
M1 M2
3
(1, 21 ,0)
1
1( 2 ,0,0)
0(0,0,0)
1 1
1(0,0, 2 )
1 1 1
( , , )
1 2 2 2
(0, 21 , 21 )
( , ,0)
0 2 2
(0, 21 ,0)
(0,1, 21 )
( ,1,0)
1 2
3
M1 M2
+1
3
( ,0,0)
1 2
0(1,0,0)
1
( ,0,0)
1 2
(1, 21 ,0)
M1 M2
+2
3
( ,0,0)
1 2
(1,0, 21 )
1
( ,0, )
0 2 2
(0,0, 21 )
2(0,0,0)
( , , )
1 2 2 2
(1,0, 21 )
( ,0,1)
1 2
0(0,0,1)
(0, 21 ,1)
1 1 1
( ,0,1)
1 2
(0,0, 23 )
2. Let us consider
M1 = M2 = 0,
J = 1.
(117)
Then from Table 4 we see that only 1(0,0,3/2), 0(0,0,1), 1(0,0,3/2) satisfy the upper
bounds. From Table 3 we see that of these only
0(0,0,1)
satisfies the lower bounds.
3. Let us also consider the G-irrep with
(118)
364
Table 4
The two upper bounds as in (43) and (44). The rows represent the upper bound in (43) and the
columns represent the upper bound in (44)
2M1 +M2
2
3
J6
2M1 +M2
1
3
2M1 +M2
3
2M1 +M2
+1
3
2M1 +M2
+2
3
2M2 +M1
2
3
1( 2 ,0,0)
1
(1, 21 ,0)
1
( ,1,0)
1 2
1
2M2 +M1
1
3
(0, 23 ,0)
(1,0, 21 )
( ,0,0)
1 2
(1, 21 ,0)
( ,1,0)
1 2
(0, 3 ,0)
1 2
1
0(1,0,0)
1 1
( , ,0)
0 2 2
0(0,1,0)
1 1 1
( , , )
1 2 2 2
1
( ,0,0)
1 2
2(0,0,0)
(0,1, 21 )
1
(0, 1 ,0)
1 2
(1,0, 21 )
2M2 +M1
3
1 1 1
( , , )
1 2 2 2
1
( ,0,0)
1 2
(0,1, 21 )
( ,0, )
0 2 2
( ,0,1)
1 2
0(0,0,0)
(0, 21 , 21 )
(0, 21 ,1)
(0,0, 21 )
1(0, 2 ,0)
1
2M2 +M1
+1
3
( ,0,1)
1 2
2(0,0,0)
(0, 21 ,1)
0(0,0,1)
(0,0, 23 )
1(0,0, 2 )
(0,0, 23 )
2M2 +M1
+2
3
M1 = M2 = 1,
J = 0.
(119)
(120)
365
identify the series of G-irrepses that have the same content. Before doing so, let us restrict
our attention to some of the above examples.
By means of the first example we show that the spectrum that we have obtained now
is really the complete spectrum in spite of the fact that we have only been considering
harmonics with s and t indices. This we argued was valid for certain SU(3) SU(2)
representations where M1 and M2 are big enough. Here we show that the matrices that
we have calculated in the previous section can actually be used to obtain the eigenvalues
for all the other cases where M1 and M2 are not big enough. We will do this by means
of a concrete and particular relevant example: the massless graviton multiplet. Moreover,
the subsequent material will also serve as a proof of the fact that the compactification on
AdS N 010 formulated as SU(3) SU(2)/SU(2) U (1) and with the rescalings chosen
as in (24) has indeed N = 3 supersymmetry.
Clearly, the massless graviton multiplet contains the graviton field which is to sit in the
representation M1 = M2 = J = 0. Applying the mass formula
m2h = M(0)2 ,
(121)
we find that it has mass zero. The gravitini and the graviphoton correspond to case we
already mentioned, M1 = M2 = 0 and J = 1,
1
(122)
This is a case where M1 and M2 are not big enough to express everything in terms of
SU(3) SU(2) Young tableaux with s-spin and t-spin only, as we have assumed in all the
previous discussion. We illustrate that the eigenvalues of the previous section can still be
used to derive the masses of the graviphoton and the gravitino.
Let us first consider the one-form. We show that it contains the graviphoton in its
harmonic expansion. It sits in in the fundamental of SO(3). Using the information in (118)
1) do not appear.
we see that in the reduction G H , the representations (2, 0) and (2,
Indeed, one can not put one SU(2) index i in (122). The fragment (3, 1) appears as (118),
(1)
namely for the fragment Rij there is 1 i j . This can be seen as the Young tableau
ij(0,0,1)(0, 0, 1; 0),
(123)
where p = 1 is given by (48). Let us now formally apply the generalized cyclic identity
(49) to this. We write,
(1/2,1/2)
(124)
Clearly none of the above terms exists as a Young tableaux, whereas (123) does. In order
to obtain the mass matrices for the one-form we should in principle derive the formulae of
the type (59) and (60) for the case u 6= 0. We argue now that such work can be avoided by
using the information that we have already obtained.
Since all these formulas are linear one does not have to derive these formulae on the
Young tableau (123), but one introduces the objects even if they do not correspond to
366
an irreducible representation of SU(3) SU(2) with the definition that they transform as
in the transformation rules (59) and (60). Then one uses the identity between (123) and
(124) and applies a and A to (124). Even though the separate terms in the calculation do
not represent irreducible representations of SU(3) SU(2) the sum of all of them on the
other hand will. Concretely, following this line of argument, the harmonic expansion for
the one-form can be written as,
(1/2,1/2)
(0,0,1)
Dij(1,0) + 2 Dij
Dij(0,1) ,
(125)
R(1)
ij = AW
(s,t )
where the Dij are defined similarly as the -objects upon applying the cyclic identity
two times as in (124).
In order to exhibit the results for the matrices that we have already obtained in the
previous section, we consider M(0)2(1) for
M1 = M2 = 0,
J = 1.
(126)
(1)
Ci
AWr(0,1)
(1/2,1/2)
AW = AWr
(127)
(1,0)
AWr
only. Hence, it is a 3 3 matrix
0
24
0
M(0)2 (1) = 96 48 96 .
0
24
0
(128)
It is useful to realize that we can make change of basis in the harmonic expansion (67)
AW 0 = (P 1 )T AW ,
T 1
P Dij ,
R(1)
ij = AW P
1 2 1
P = 1
0 1 .
1 4 1
(129)
(130)
Then the first term in the harmonic expansion (129) gets the form (125). The matrix (128)
in the basis AW 0 gets the form,
48 0
0
0
1 T
T
0 .
(131)
M(0)2 (1) = (P ) M(0)2 (1) P = 0 0
0 0 96
So we see that the eigenvalue of M(0)2 (1) for component AW (0,0,1) in (125) of the
graviphoton is given by the first entry of the above matrix,
M(0)2 (1) = 48.
Mind that the matrix P diagonalizes the matrix M(0)2 (1) .
(132)
367
As is known from [8,9], the mass of the vector A is given by the following mass formula,
q
(133)
m2A = M(0)2(1) + 48 12 M(0)2 (1) + 16.
So we find a massless vector indeed.
So far we have found only the vectors and the graviton of the massless graviton multiplet.
Still it is crucial to check that we have the right number of gravitini in this multiplet. That
will prove that we have indeed N = 3. Hence, let us now consider the spinor. We are
interested in the representation (122). Again there is only a contribution for the fragment
(3, 0) in the decomposition (70) and the treatment is the same as for the (3, 1) of the oneform. Indeed, there only appears the fragment ij in the expansion,
(1,0)
ij = (0,0,1)(x) Dij
(1/2,1/2)
+ 2Dij
(0,1)
Dij
Consequently, from the 8 8 matrix M(1/2)3 there is only the upper 3 3 matrix
4 2 0
M(1/2)3 = 8 8 8
0
(134)
(135)
2 4
0 0
0
0
0 4
(136)
M(1/2)
0 .
3 =
0 0 12
Only the first zero entry of this matrix is physically relevant. Using the fact,
m = M(1/2)3 ,
(137)
we conclude that there are three massless spin- 23 fields in the fundamental of SO(3). They
provide the gravitini of the N = 3 massless graviton multiplet.
So we have found the massless graviton, gravitini and graviphoton of the N = 3
(2, 3( 32 ), 3(1), 12 ) graviton multiplet.
To complete the treatment of (126), we also consider the eigenvalues of the two-form for
this representation of G. As in the case of the one-form, the only fragments that contribute
are the ones with u-spin 1, i.e., for the representation (126) the 21 21 matrix works on
the basis,
Zo(0,1)
(1/2,1/2)
Zo
Zo(1,0)
(138)
Zr(0,1)
(1/2,1/2)
Z
r
(1,0)
Zr
only. Hence, it gets the form of a 6 6 matrix,
368
32 16
0
32
0
0
64
0
64
0
32
0
0
16 32
0
0
32
.
M(0)(1)2 =
16
0
0
32 24
0
0
16
0
96 16 96
0
0
16
0
24 32
(139)
Following the same line of reasoning as for the one-form and the spinor, we introduce the
diagonalization matrix
2
4
2
1 2 1
1
2
1
1 2 1
2
0
2
1
0 1
.
(140)
P =
2
0
2
1 0 1
1 + 3 4 1 + 3
4 1
1 + 3 1
1 3 1
4 1
1 3 4 1 + 3
Thus diagonalizing we find,
0
1 T
M(0)(1)
) M(0)(1)2 P T
2 = (P
48
0
0
=
0
0
0
0
96
0
0
0
0
0
0
0
0
16 (2 + 2 )
0
0
16 (2 + 2 )
0
0
0
0
0
0
0
0
0
0
. (141)
0
0
16 (3 + 3 )
0
0
16(3 + 3 )
16 3 +
3
(142)
as being unphysical. Only the first two eigenvectors in P correspond to the two 0(0,0,1)
that we have found in the decomposition G H in (62). So we see that the first
eigenvector (corresponding to the first row of the matrix P ) is the longitudinal unphysical
mode that corresponds to the one-form eigenvector that we obtained above. It has the
same eigenvector M(0)(1)2 = 48. Furthermore, there is the eigenvalue M(0)(1)2 = 96 whose
eigenvector is an extra massive vector Z with mass
m2Z = M(0)(1)2 = 96.
(143)
369
To this end we have to consider the one-form. Acting on the fragments with (120) it it
becomes the 3 3 matrix,
80 16 i 2 16 i 2
,
24 i 2
96
0
(144)
0
96
24 i 2
which has eigenvalues
M(0)2 (1) = 48, 96, 128.
(145)
The first eigenvalue 48 according to (133), gives rise to a massless vector A. This is the
Killing vector of the SU(3) isometry.
6. The series
In the previous section we explained how the masses of the N = 3 theory can be
calculated from the matrices of Appendix A when M1 , M2 , J are given. Yet for a
systematic study of the complete spectrum we need to do this for all the irreducible
representations of G. In order to do so it is useful to arrange the representations that are
allowed by the bounds (43) and (44) and the constraint (48), into series that give rise to the
same content of fragments. This can be done most easily by using the Tables 3 and 4.
First of all, notice that it is sufficient to consider the cases where
M2 > M1 .
(146)
Indeed, the irreducible representations with M2 < M1 are the conjugate representations
of SU(3) and hence correspond to the complex conjugate multiplets in the N = 3 theory.
Reflecting on the two Tables 3 and 4 we conclude that we should make a distinction among
the following series R and En ,
(
J > 13 (M2 M1 ) + 2,
(147)
R : (M1 > 4);
J 6 13 (2M1 + M2 ) 2,
(
J > 13 (M2 M1 ) + 2,
(148)
E1 : (M1 > 3);
J = 13 (2M1 + M2 ) 1,
(
J > 13 (M2 M1 ) + 2,
(149)
E2 : (M1 > 2);
J = 13 (2M1 + M2 ),
(
J > 13 (M2 M1 ) + 2,
(150)
E3 : (M1 > 1);
J = 13 (2M1 + M2 ) + 1,
(
J > 13 (M2 M1 ) + 2,
(151)
E4 : (M1 > 0);
J = 13 (2M1 + M2 ) + 2,
(
J = 13 (M2 M1 ) + 1,
(152)
E5 : (M1 > 3);
J 6 13 (2M1 + M2 ) 2,
370
(
E6 : (M1 = 2);
(
E7 : (M1 = 1);
(
E8 : (M1 = 0);
(
E9 : (M1 > 2);
J = 13 (M2 M1 ) + 1,
J = 13 (2M1 + M2 ) 1,
J = 13 (M2 M1 ) + 1,
J = 13 (2M1 + M2 ),
J = 13 (M2 M1 ) + 1,
J = 13 (2M1 + M2 ) + 1,
J = 13 (M2 M1 ),
J 6 13 (2M1 + M2 ) 2,
(
J = 13 (M2 M1 ),
(
E11 : (M1 = 0);
(
E12 : (M1 > 1);
(
E13 : (M1 = 0);
(
E14 : (M1 > 0);
J = 13 (2M1 + M2 ) 1,
J = 13 (M2 M1 ),
J = 13 (2M1 + M2 ),
J = 13 (M2 M1 ) 1,
J 6 13 (2M1 + M2 ) 2,
J = 13 (M2 M1 ) 1,
J = 13 (2M1 + M2 ) 1,
J = 13 (M2 M1 ) 2,
J 6 13 (2M1 + M2 ) 2.
(153)
(154)
(155)
(156)
(157)
(158)
(159)
(160)
(161)
The series R is the regular series and contains the long multiplets. For the calculation of
the masses of the components, one takes the matrices in the Appendix A. The series E are
the exceptional series. One may verify that in these exceptional series some columns of
the Tables 3 and 4 are absent. In order to see which rows survive the bounds it is useful to
make a distinction between the cases
M2 = M1 + 3j,
(162)
for j a non-negative integer. Hence, the series (148)(161) have a different content
j
depending on the parameter j . We will denote them as En . One may verify that it is
sufficient to make a distinction between
j = 0,
j = 1,
j > 2.
>
(163)
371
2. Consider each of the fragments 1(s,t,u) in F with non-zero u-spin. If both the
fragments 1(s,t+1/2,u1/2) and 1(s+1/2,t,u1/2) are present as well then eliminate
1(s,t,u) . This has to be done because of the cyclic identity (49). Otherwise we are
overly expanding. Call the remaining set of fragments Fc .
3. Consider the fragments of Fc with non-zero u-spin that are left. They can not be
eliminated. In fact they span a basis for the harmonics. However, in the spirit of
the previous section, they have to be expressed in terms of the objects (that do
not correspond to Young tableaux). All of this has to be done as explained in the
example of (117). We recall that this is to avoid unnecessary work. This determines
an unnormalized row vector of the conjugation matrix P . See for examples (124)
and (125), where the row vector is
( 1
2 1 ) .
(164)
4. For all the remaining fragments with zero u-spin, add the unit vector to P .
5. Fill the rest of P in such a way as to get an invertible matrix. The choice of the vectors
is completely free as long as the matrix P becomes invertible.
6. Apply P as in (131).
7. Delete the unphysical row and columns of the resulting matrix. The rows and columns
that have to be deleted are the ones for which we had to insert the vectors in step 5.
8. Calculate the eigenvalues of this matrix. They have to be in the lists (77), (78)(84),
(85)(105) and (106)(113). Since we are doing this for specific values of M1 , M2 , J
we have to plug in these values also in these lists.
Remark that after step 2, we have found a complete basis of fragments. Given some values
for M1 , M2 , J , all the above steps can be implemented in a M ATHEMATICA program. So
it is sufficient to choose one representative M1 , M2 , J for the series in (148)(161) and let
the computer perform these steps.
We now list the results of this. For the zero-form, the one-form, the two-form and the
spinor we list which of the eigenvalues are present and we give the masses of the fields
(see [79,15] for conventions concerning names)
h, , A, W, Z, L , , S
(165)
>
>
>
>
>
>
>
E1 , E2 , E5 , E6 , E7 , E9 , E10 , E11 .
(166)
From the zero-form eigenvalue (0) we can obtain the mass of the spin-2 field h and the
spin-0 fields and S [79,15]:
372
m2h = (0) ,
p
m2 = (0) + 176 + 24 (0) + 36,
p
m2S = (0) + 176 24 (0) + 36.
(167)
So for each G-irrep in the series (166) there is a field h, a fields and a field S with
masses as given in (167).
6.2. The one-form
We list which of the eigenvalues are present for the one-form. We use the notation of
(78)(84). In the following tables we suppress the superscript (1).
For the series R all seven eigenvalues are present:
R
1 , 2 , 3 , 4 , 5 , 6 , 7
(168)
1 , 2 , 3 , 4 , 5 , 6 , 7
E20
E30
E40
E50
E60
E70
E80
E90
2
E10
0
E11
0
E12
0
E13
0
E14
4 , 5 , 6 , 7
6
none
1 , 2 , 3 , 4 , 5 , 6 , 7
1 , 2 , 3 , 4 , 5 , 6 , 7
4 , 5 , 6 , 7
6
(169)
1 , 2 , 7
1 , 2 , 7
none
empty
empty
empty
0 , E 0 , E 0 . Indeed, if
Mind that there are no values for M1 , M2 , J that match the series E12
13
14
j = 0 then (159), (160) and (161) would imply that J = 1 or J = 2. We have indicated
this in the table with empty. For the series E40 there exist values of M1 , M2 , J . However,
no eigenvalues (1) are present. We have indicated that with none.
For the exceptional series with j = 1:
E11
1 , 2 , 3 , 4 , 5 , 6 , 7
E21
E31
E41
E51
E61
E71
E81
E91
1
E10
1
E11
1
E12
1
E13
1
E14
3 , 4 , 5 , 6 , 7
373
5 , 6
none
1 , 2 , 3 , 4 , 5 , 6 , 7
1 , 2 , 3 , 4 , 5 , 6 , 7
3 , 4 , 5 , 6 , 7
5 , 6
(170)
1 , 2 , 3 , 4 , 7
1 , 2 , 3 , 4 , 7
3 , 4 , 7
1 , 2
1 , 2
empty
E1
>
E2
>
E3
>
E4
>
E5
>
E6
>
E7
>
E8
>
E9
>
E10
>
E11
>
E12
>
E13
>
E14
1 , 2 , 3 , 4 , 5 , 6 , 7
3 , 4 , 5 , 6 , 7
5 , 6
none
1 , 2 , 3 , 4 , 5 , 6 , 7
1 , 2 , 3 , 4 , 5 , 6 , 7
3 , 4 , 5 , 6 , 7
5 , 6
(171)
1 , 2 , 3 , 4 , 7
1 , 2 , 3 , 4 , 7
3 , 4 , 7
1 , 2
1 , 2
none
From the one-form eigenvalue (1) we can obtain the masses of the vectors A and W ,
p
m2A = (1) + 48 12 (1) + 16,
374
p
m2W = (1) + 48 + 12 (1) + 16.
(172)
So for each entry in the above tables there is both a vector A and a vector W whose mass
can be obtained from by the formulas (172).
6.3. The two-form
We list which of the eigenvalues are present for the two-form. We use the notation of
(85)(105). Mind that there are the multiplicities of the eigenvalues
(2)
(2)
2 = 3 ,
(2)
(2)
(2)
4 = 5 = 6 ,
(2)
(2)
7 = 8 .
(173)
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21
(174)
The exceptional series E with j = 0 contain:
E10
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 10 , 11 , 12 , 13 , 14 , 16 , 17 , 18 , 19 , 20 , 21
E20
E30
E40
E50
E60
E70
E80
E90
0
E10
0
E11
0
E12
0
E13
0
E14
1 , 2 , 3 , 4 , 5,14 , 10 , 12 , 13 , 19 , 20 , 21
1 , 2,10 , 12 , 21
none
1,2 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21
1,2 , 4 , 5 , 6 , 7 , 8 , 10 , 11 , 12 , 13 , 14 , 16 , 17 , 18 , 19 , 20 , 21
1,2,19 , 1,3,19, 4 , 5,14 , 10 , 12 , 13 , 20 , 21
12 , 21
1,7 , 2,4 , 9 , 14 , 15 , 16 , 17
1,7 , 2,4 , 3,5 , 14 , 16
1,7,11,13,15,17,18,20
empty
empty
empty
(175)
375
E11
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21
E21
E31
E41
E51
E61
E71
E81
E91
1
E10
1
E11
1
E12
1
E13
1
E14
1 , 2 , 3 , 4 , 5 , 6 , 10 , 11 , 12 , 13 , 18 , 19 , 20 , 21
1 , 2 , 3 , 12 , 20 , 21
1
2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21
1,14 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 10 , 11 , 12 , 13 , 15 , 16 , 17 , 18 , 19 , 20 , 21
2 , 3 , 4 , 5 , 6 , 10 , 11 , 12 , 13 , 18 , 19 , 20 , 21
2 , 3 , 4,20 , 12 , 21
4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 14 , 15 , 16 , 17 , 18 , 19
4 , 5 , 6 , 7 , 8 , 10 , 11 , 14 , 15 , 16 , 17 , 18 , 19
4 , 5 , 6 , 7,18 , 10 , 11 , 19
1,7 , 9 , 14 , 15 , 16 , 17
1,7 , 9,16 , 14 , 15 , 17
empty
(176)
E1
>
E2
>
E3
>
E4
>
E5
>
E6
>
E7
>
E8
>
E9
>
E10
>
E11
>
E12
>
E13
>
E14
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21
1 , 2 , 3 , 4 , 5 , 6 , 10 , 11 , 12 , 13 , 18 , 19 , 20 , 21
1 , 2 , 3 , 12 , 13 , 20 , 21
1
2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21
2 , 3 , 4 , 5 , 6 , 7 , 8 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21
2 , 3 , 4 , 5 , 6 , 10 , 11 , 12 , 13 , 18 , 19 , 20 , 21
2 , 3 , 12 , 13 , 20 , 21
4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 14 , 15 , 16 , 17 , 18 , 19
4 , 5 , 6 , 7 , 8 , 10 , 11 , 14 , 15 , 16 , 17 , 18 , 19
4 , 5 , 6 , 10 , 11 , 18 , 19
7 , 8 , 9 , 14 , 15 , 16 , 17
7 , 8 , 14 , 15 , 16 , 17
9
(177)
376
Mind that besides the multiplicities in (173), there may be occasional multiplicities that
arise for certain values of M1 , M2 , J . If that occurs, then we have indicated it in the above
tables by writing the eigenvalues with multiple indices. For instance, 1,2 in E50 means
(2)
(2)
(2)
that for the values M1 = M2 > 3, J = 1 we have 1 = 2 = 3 in (105).
From this one can obtain the masses of the vector field Z,
m2Z = (2) .
(178)
So for each entry in the above tables there is a vector Z with mass (178).
6.4. The spinor
We list which of the eigenvalues are present for the spinor. We use the notation of (106)
(113). In the following tables we will suppress the superscript (s).
For the series R we find all of the eight eigenvalues:
R
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
(179)
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
E20
E30
E40
E50
E60
E70
E80
E90
0
E10
0
E11
0
E12
0
E13
0
E14
3 , 4 , 5 , 6 , 7
6
none
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
3 , 4 , 5 , 6 , 7
6
1 , 2 , 7 , 8
1 , 2 , 7 , 8
7
empty
empty
empty
(180)
E11
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
E21
E31
E41
E51
E61
E71
E81
E91
1
E10
1
E11
1
E12
1
E13
1
E14
3 , 4 , 5 , 6 , 7 , 8
377
5 , 6
none
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
3 , 4 , 5 , 6 , 7 , 8
5 , 6
(181)
1 , 2 , 3 , 4 , 7 , 8
1 , 2 , 3 , 4 , 7 , 8
3 , 4 , 7 , 8
1 , 2
1 , 2
empty
E1
>
E2
>
E3
>
E4
>
E5
>
E6
>
E7
>
E8
>
E9
>
E10
>
E11
>
E12
>
E13
>
E14
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
3 , 4 , 5 , 6 , 7 , 8
5 , 6
none
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
3 , 4 , 5 , 6 , 7 , 8
5 , 6
(182)
1 , 2 , 3 , 4 , 7 , 8
1 , 2 , 3 , 4 , 7 , 8
3 , 4 , 7 , 8
1 , 2
1 , 2
none
From the this one can obtain the masses of the spin- 23 field and the spin- 21 field L ,
mL = (s) + 16 .
(183)
m = (s) ,
378
So for each entry in the above tables there is the field and the field L with masses (183).
(184)
Acknowledgements
This work fits in a much wider project at the university of Torino which is mentored
by Pietro Fr. I am indebted to him for providing this background. I have received the
essential knowledge about the geometry of the spaces N 010 from Leonardo Castellani. I
also benefited from useful discussions with Leonardo Gualtieri and Davide Fabbri at an
early stage of this work. I cordially thank all of them. I also thank Lorenzo Magnea for
letting me terrorize his computer. I thank Leonardo Gualtieri for carefully reading through
the manuscript.
M(0)3 = H0 =
379
+ 2(M12 + M22 + M1 M2 + 3M1 + 3M2 ) 3J (J + 1) .
16
3
(185)
AWr(0,1)
AWr(1,0)
AWr
32 + H0 16 M1 + 16 M2
8 (3 J + M1 M2 )
16 (3 + 3 J M1 + M2 )
16 + H0
16 (3 + 3 J + M1 M2 )
16 i
2 (3 + 3 J M1 + M2 )
3
8 (3 J M1 + M2 )
8i
2 (6 + M1 + 2 M2 )
3
8i
2 (3 J M1 + M2 )
3
8i
2 (3 J + M1 M2 )
3
8i
2 (6 + 2 M1 + M2 )
3
H0 + 16 (2 + M1 M2 )
16 i
2 (3 + 2 M1 + M2 )
3
0
16 i
3
0
16 i
3
(0,1/2)
AWc
2 (3 J + M1 M2 )
16 i
2 (2 M1 + M2 )
3
16 i
3
0
16
3 (M1
16
3 (3 J
M2 )
M1 + M2 )
0
0
16 i
2 (3 + 3 J + M1 M2 )
16 i
2 (3 + 2 M1 + M2 )
3
16
3 (3 + 3 J
H0 +
16 i
3
H0
M2 )
0
0
g c(1/2,0)
AW
2 (3 + M1 + 2 M2 )
2 (3 + 3 J M1 + M2 )
+ M1 M2 )
16
3 (3 + M1
g c(0,1/2)
AW
16 i
3
2 (3 + 3 J + M1 M2 )
(1/2,0)
AWc
H0
2 (3 + M1 + 2 M2 )
0
16 i
3
2 (M1 + 2 M2 )
16 i
2 (3 J M1 + M2 )
3
16
3 (3 + M1
16
3 (3 J + M1 M2 )
H0 + 16
3 (M1 M2 )
16
3 (3 + 3 J
M2 )
M1 + M2 )
(186)
380
(1/2,1/2)
2
3
(3 + M1 M2 )
2
3
4
3 2 (3 + 2 M1 + M2 )
4
3
(3 J M1 + M2 )
2 (3 J + M1 M2 )
2
3 2 (6 + 2 M1 + M2 )
2
2 (6 + M1 + 2 M2 )
3
2
2 (3 J M1 + M2 )
3
3
2 (3 + 3 J M1 + M2 )
0
(3 + 3 J M1 + M2 )
2
3
(M1 M2 )
(0,1/2)
2
3
(3 + 3 J + M1 M2 )
4
3 2 (M1 + 2 M2 )
4
2 (3 J M1 + M2 )
3
4
(3 J + M1 M2 )
3
(6 + M1 M2 )
4
3
(3 + 3 J M1 + M2 )
(3 + M1 M2 )
2
3 2 (3 + 3 J M1 + M2 )
2
3 2 (6 + M1 + 2 M2 )
(0,1/2)
4
3
2 (3 + M1 + 2 M2 )
(1/2,0)
2 (3 + M1 + 2 M2 )
4
2 (3 + 3 J M1 + M2 )
3
4
3
4
3
(3 + M1 M2 )
4
3 2 (3 + 3 J + M1 M2 )
4
3
(3 + 3 J + M1 M2 )
4
3
2
3
(3 J + M1 M2 )
(3 + 3 J M1 + M2 )
(1,0)
2 (3 J + M1 M2 )
4
2 (2 M1 + M2 )
3
(1/2,0)
(0,0)
4
3 2 (3 + 3 J + M1 M2 )
4
2 (3 + 2 M1 + M2 )
3
4
3
(3 J + M1 M2 )
8
3
(M1 M2 )
(3 J M1 + M2 )
4
2 (3 J + M1 M2 )
3
4
3
(6 + M1 M2 )
2 (2 M1 + M2 )
4
3 2 (M1 + 2 M2 )
4
2 (3 J M1 + M2 )
3
2 (3 + 3 J + M1 M2 )
16
0
4
3
(3 J M1 + M2 )
2
3 2 (6 + 2 M1 + M2 )
4
3
(3 + M1 M2 )
(3 + 3 J + M1 M2 )
4
3
4
3
2
3
381
4
3
(187)
The matrix M(0)(1)2 of the two-form is
(0,3/2)
(1/2,1)
Zc
H0
64
3
64
3
(M1 M2 )
(3 + 3 J M1 + M2 )
0
32
3
64
9
(3 J + M1 M2 )
H0
128
9
64
9
(9 + M1 M2 )
(3 J M1 + M2 )
0
128
9
(3 + 3 J + M1 M2 )
H0 +
64
9
64
9
(6 + M1 M2 )
(3 + 3 J M1 + M2 )
32
9
2 (3 J + M1 M2 )
64
9
2 (9 + 2 M1 + M2 )
2 (6 + 2 M1 + M2 )
0
0
16
9
32
9
2 (9 + 2 M1 + M2 )
(3 + 3 J M1 + M2 )
0
0
64
9
32
9
0
32
9
2 (3 + 3 J + M1 M2 )
16
9
(M1 M2 )
2 (12 + 2 M1 + M2 )
0
0
32
27
2 (3 + 3 J + M1 M2 )
2 (12 + 2 M1 + M2 )
2 (3 J + M1 M2 )
0
16
9
Zc
2 (6 + 2 M1 + M2 )
16
3
(1,1/2)
Zc
0
16
27
(3 + 3 J + M1 M2 )
382
(3 J M1 + M2 )
0
0
0
32
27
(3 + M1 M2 )
0
0
0
0
0
0
(188)
(3/2,0)
Zc
0
0
64
3
(6 + 3 J + M1 M2 )
H0 +
64
3
(3 + M1 M2 )
0
H0
64
3
0
0
0
0
(3 + M1 M2 )
(6 + 3 J M1 + M2 )
32
2 (6 + 3 J M1 + M2 )
3
0
0
16
3
2 (6 + 3 J
16
2 (6 + 3 J + M1 M2 )
3
0
64
9
(3 + 3 J + M1 M2 )
H0
128
9
2 (6 + 3 J + M1 M2 )
16
9
ec(1/2,1)
Z
0
32
3
64
3
ec(0,3/2)
Z
(6 + M1 M2 )
(3 + 3 J M1 + M2 )
0
32
2 (M1 + 2 (6 + M2 ))
64
2 (3 + 3 J M1 + M2 )
9
0
M1 + M2 ) 16
2 (M1 + 2 (6 + M2 ))
9
32
2 (3 + 3 J M1 + M2 )
9
9
0
16
9
64
9
(6 + 3 J M1 + M2 )
0
32
27
16
27
(3 + M1 M2 )
(3 + 3 J M1 + M2 )
0
0
0
0
0
0
(6 + 3 J + M1 M2 )
0
0
0
(189)
128
9
H0 +
ec(3/2,1)
Z
0
0
0
0
(3 J + M1 M2 )
64
9
64
9
ec(1,1/2)
Z
(9 + M1 M2 )
8
3
8
3
(3 J M1 + M2 )
2 (M1 + 2 M2 )
2 (3 + 3 J M1 + M2 )
0
0
8
3
2 (3 + 3 J + M1 M2 )
8
3 2 (6 + 2 M1 + M2 )
(3 + 3 J + M1 M2 )
H0 +
64
Zo(0,1)
0
64
3
383
2 (9 + M1 + 2 M2 )
64
3
(M1 M2 )
H0
32
3
32
3
(6 + M1 M2 )
(3 + 3 J M1 + M2 )
32
2 (3 J M1 + M2 )
9
32
2 (6 + M1 + 2 M2 )
3
32
2 (9 + M1 + 2 M2)
9
16
2 (3 J M1 + M2 )
9
16
16
27
16
3
(3 J + M1 M2 )
32
27
(M1 M2 )
2 (6 + M1 + 2 M2 )
0
16
9
(3 + 3 J + M1 M2 )
2 (3 + 2 M1 + M2 )
0
2 (3 + 3 J M1 + M2 )
(190)
(1/2,1/2)
Zo(1,0)
Zo
2 (3 + M1 + 2 M2 )
8
2 (3 J M1 + M2 )
3
Zr(0,1)
8
3
8
3
0
8
3
2 (6 + M1 + 2 M2 )
8
3
2 (M1 + 2 M2 )
2 (3 + 3 J M1 + M2 )
0
384
2 (3 + 3 J M1 + M2 )
0
8
3
8
3
2 (3 + 2 M1 + M2 )
0
8
3
0
16
3
2 (3 + 3 J + M1 M2 )
8
3 2 (2 M1 + M2 )
(3 J + M1 M2 )
32 + H0
16
3
8
3
2 (3 J + M1 M2 )
(3 J M1 + M2 )
2 (3 + 3 J + M1 M2 )
8
2 (6 + 2 M1 + M2 )
3
0
0
32
0
32
3
(3 + 3 J + M1 M2 )
H0 +
32
3
(6 + M1 M2 )
H0 16 (4 + M1 M2 )
16
16 (3 + 3 J M1 + M2 )
16
2 (3 J + M1 M2 )
4
2 (6 + 2 M1 + M2 )
9
4
2 (6 + M1 + 2 M2 )
9
4
2 (3 J M1 + M2 )
9
4
9
8
9
32
9
2 (3 + 3 J + M1 M2 )
0
8
9
2 (3 + M1 + 2 M2 )
2 (3 + 2 M1 + M2 )
0
32
9
2 (3 + 3 J M1 + M2 )
0
0
(191)
(1/2,1/2)
Zr
0
8
3
(0,1/2)
Zr(1,0)
Zc
0
0
8
2 (6 + M1 + 2 M2 )
3
8
3 2 (3 + 3 J M1 + M2 )
2 (3 + M1 + 2 M2 )
8
3 2 (3 J M1 + M2 )
0
0
8
3
(3 + 3 J + M1 M2 )
2 (3 J + M1 M2 )
8
2 (3 + 2 M1 + M2 )
3
8
3
16
3
0
8
3
2 (3 + 3 J + M1 M2 )
8
2 (2 M1 + M2 )
3
8
3
385
(3 + M1 M2 )
(3 + 3 J M1 + M2 )
0
16
3
2 (3 + M1 + 2 M2 )
16
2 (3 + 3 J M1 + M2 )
3
32
32
8 (3 J + M1 M2 )
16 + H0
16 (3 + 3 J + M1 M2 )
32
2 (3 + M1 + 2 M2)
3
32
2 (3 + 3 J M1 + M2 )
3
8 (3 J M1 + M2 )
16
2 (3 J + M1 M2 )
9
16
2 (6 + 2 M1 + M2 )
9
16
2 (6 + M1 + 2 M2 )
9
16
2 (3 J M1 + M2 )
9
H0 + 16 (4 + M1 M2 )
32
2 (3 + 3 J + M1 M2 )
9
H0
16
9
80
9
(33 + 5 M1 5 M2 )
(3 + 3 J M1 + M2 )
32
2 (3 + M1 + 2 M2 )
9
0
0
8i
8 2 (2 M1 + M2 )
2 (3 + 3 J M1 + M2 )
(192)
ec(0,1/2)
Z
(1/2,0)
Zc
8
3
8
3
(3 J + M1 M2 )
16
3
0
8
3
ec(1/2,0)
Z
(M1 M2 )
0
8
3
(3 J M1 + M2 )
(3 + 3 J + M1 M2 )
16
3
8
3
(3 + M1 M2 )
(3 + 3 J M1 + M2 )
(3 J + M1 M2 )
16
3
(M1 M2 )
(3 J M1 + M2 )
8
3
386
16
3
2 (M1 + 2 M2 )
16
3
16
3
2 (3 J + M1 M2 )
16
3
2 (2 M1 + M2 )
2 (3 J M1 + M2 )
32
3
32
3
H0 +
2 (M1 + 2 M2 )
32
3
2 (2 M1 + M2 )
(18 + 5 M1 5 M2 )
H0
0
0
0
8 2 (3 J + M1 M2 )
8 i 2 (6 + M1 + 2 M2 )
2 (3 + 3 J + M1 M2 )
16
3
2 (3 + 2 M1 + M2 )
0
32
3
16
9
80
9
2 (3 + 3 J + M1 M2 )
32
3
(3 J + M1 M2 )
16
9
2 (3 J + M1 M2 )
2 (3 J M1 + M2 )
80
9
16
3
0
32
3
2 (3 + 2 M1 + M2 )
0
80
9
(18 + 5 M1 5 M2 )
(3 J M1 + M2 )
(3 + 3 J + M1 M2 )
H0 +
16
9(33 + 5 M1
5 M2 )
8 2 (3 J M1 + M2 )
8 2 (M1 + 2 M2 )
8 i 2 (6 + 2 M1 + M2 )
8 i 2 (3 + 3 J + M1 M2 )
(193)
Zc(0,0)
ec(0,0)
Z
Zr(0,0)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
16
3
2 (6 + M1 + 2 M2 )
8i
3
2 (3 J + M1 M2 )
16
2 (3 + 3 J + M1 M2 )
3
16
2 (6 + 2 M1 + M2 )
3
2 (3 + 3 J M1 + M2 )
8 i
3
8 i
2 (2 M1 + M2 )
2 (M1 + 2 M2 )
3
8i
2 (3 J M1 + M2 )
3
0
H0
0
0
0
H0
H0
0
0
387
(194)
Appendix B. Conventions
The indices , are the SO(7) Lorentz indices, we have negative definite metric
= ( ).
(195)
A, B, C = 4, 5, 6, 7.
(196)
The indices on H :
m, n, p, q = 9, 10, 11,
N = 8.
(197)
0 1 0
0 i 0
1 0 0
2 = i 0 0 ,
3 = 0 1 0 ,
1 = 1 0 0 ,
0 0 0
0 0 0
0 0 0
0 0 1
0 0 i
0 0 0
5 = 0 0 0 ,
6 = 0 0 1 ,
4 = 0 0 0 ,
1 0 0
i 0 0
0 1 0
0 0 0
1 0 0
8 = 1 0 1 0 .
(198)
7 = 0 0 i ,
3
0 i 0
0 0 2
The SU(2) generators are: 2i a
0 1
0 i
1 =
, 2 =
,
1 0
i 0
3 =
0
.
1
1
0
(199)
f147 = 12 ,
f345 = 12 ,
f367 = 12 ,
f156 = 12 ,
f246 = 12 ,
f458 =
f678 =
3
2 ,
f257 = 12 ,
3
2 .
(200)
388
0
0
1 = 0
0
i
0
i
0
0
0
0
3 =
0
i
0
0
0
i
0
0
0
0
0
0
i
0
0
0
0
0
0
0
0
0
0
i
0
0
0
0 0
i 0
0 i
0 0
0 0
0 0
0 0
0 0
0
0
0
i
0
0
0
0
1
2
0
0
0
0
0
1 0
0
0
1 0
5 = 0
0
0
0 1 0
0
0 12
0
0
0
0 12
1
0
0
0
0
0
7 = 1 0
0
0
0 1
2
0
0
i
2
0
0
0
i
0
0
0
0
0
0
0
0
0
i
2
0
0
0
0
0
i
0
0
0
1
2
0
0
0
0
0
1
2
0
0
0
0
0
i
0
0
0
0
0
0
0
0
0
1
i
2
0
0
0
0
0
i
0
i
0
0
0
0
0
i
2
0
0
0
2 = 0
0
1
0
0
0
0
,
0
0
i
i
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0 1 0
0
0 12
1 0
0
0
0
0
0
1
0
1 0
,
0 1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
i
4 = 0
0
0
0
i
0
6 = i
0
0
0 0 0 12
1 0 0 0
0 0 0 0
0 0 1 0
0 1 0 0
0 0 0 0
1
0 0 0
2
0 0 0 0
0
0
0
0
0
i
0
0
i
2
0
0
0
0
0
i
2
i
2
i
2
0
0
0
0
0
0
0
0
0
0
i
2
i
2
0
0
0
0
0
0
0
0
i
0
0
0
0
0
0
0
i
0
0
0
0
0
i
0
0
i
2
0
0
0
0
0
i
2
0 12
0
0
0
0
0
0
,
0
0
1 0
0 12
1
0
0
i
0
0
0
0
0
0
0
0
i
i
0
0
0
0
0 0
0 i
0 0
i 0
0 i
0 0
0 0
0 0
0
0
0
,
i
0
0
0
0
0
,
0
0
0
0
0
0
.
0
0
0
0
1 0 0
0 0 0
0 0 0
0 0 1
C=
0 1 0
0 0 0
0 0 0
0 0 0
0 0 0
0 1 0
1 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 1
0
0
0
0
0
0
1
0
0
0
0
0
.
0
0
0
(201)
References
[1] J. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor.
Math. Phys. 2 (1998) 231; hep-th/9711200.
389
Abstract
Soliton time delays and the semiclassical limit for soliton S-matrices are calculated for non-simply
laced affine Toda field theories. The phase shift is written as a sum over bilinears on the soliton
conserved charges. The results apply to any two solitons of any affine Toda field theory. As a byproduct, a general expression for the number of bound states and the values of the coupling in which
the soliton S-matrix can be diagonal are obtained. In order to arrive at these results, a vertex operator
is constructed, in the principal gradation, for non-simply laced affine Lie algebras, extending the
previous constructions for simply laced and twisted affine Lie algebras. 2000 Elsevier Science
B.V. All rights reserved.
1. Introduction
In 1 + 1 dimensions, a well-known class of integrable theories are the Affine Toda Field
Theories (ATFTs). For each affine Lie algebra g,
we can associated an ATFT. For simplicity
just the untwisted algebras will be considered. (The twisted cases were considered in [4].)
If the coupling is imaginary, there exist degenerate vacua and solitons interpolating these
vacua. An N -soliton solution can be written as [1]
ej =
hj |g(t)|j i
h0 |g(t)|0 imj
(1)
with
g(t) =
N
Y
bi(k) ()
eQi(k) Wi(k) F
k=1
E-mail: kneipp@cbpf.br
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 0 4 - 8
(2)
391
(3)
,
(4)
where k is the rapidity of the kth soliton, xk0 its position at t = 0, k is a phase and i(k)
is the mass particle of species i(k). Let g be the underlying Lie algebra. For each dot
of the Dynkin diagram of g a soliton species can be associated. Each species may have
various different topological charges. For example, sine-Gordon, which is associated with
g = a1(1) , has one soliton species with two topological charges: the well-known soliton and
anti-soliton solutions. The mass for each species is [1]
Mi =
4hi
,
i2 | 2 |
(5)
where h is the Coxeter number of g. From (1), we can equivalently write the N -soliton
solution as
=
r
1X v
j lnhj |g(t)|j i,
j =0
where 0 is the negative of the hightest root. Written in this form, it becomes clear that
hj |g(t)|j i are the -functions which appear in the Hirota method, used in [79].
Quite a lot of work has been done in order to obtain the S-matrix for the particles [1020]
and for the solitons [2127] of ATFT. As usual, the proposed S-matrix must be constructed
in agreement with the S-matrix axioms. However, in order to confirm that the proposed Smatrix is associated with a given theory, one must check if it has the correct semiclassical
limit. In this paper we obtain the semiclassical limit for the soliton S-matrices of nonsimply laced ATFT, extending the results in [5]. This semiclassical result provides a test
for proposals of exact soliton S-matrices for non-simply laced ATFT. That comparison will
be done in a future publication.
bi ( ) has a vertex operator construction in the
The highest non-vanishing power of F
principal gradation which was first proven for level one representations [2,28,29] and then
extended for any representation of simply laced [3] and twisted [4] affine Lie algebras.
From this construction, it was shown [5] that any given soliton regains its original shape,
after having collisions with other solitons, as expected by a soliton solution. The only effect
is a time delay which was calculated. From the time delay, the semiclassical limit for the
transmission amplitude of the simply laced soliton S-matrix was obtained [5].
In the present paper, these results are extended for the remaining case of non-simply
laced affine Lie algebras obtaining a semiclassical expression which holds for any affine
Lie algebra. The paper is divided as follows: in Section 2, after a brief review, it is shown
bi ( ) also has a vertex
that the non-simply laced case, the last non-vanishing power of F
construction. Then, the asymptotic behavior of the solitons is analyzed in Section 3 and the
time delay resulting from the collision of two or more solitons is calculated in Section 4.
From the time delay, the semiclassical limit of the soliton S-matrix which holds for any
ATFT is calculated, in Section 5. The phase shift is written in terms of the soliton conserved
charges. In Section 6 a general expression for the number of bound states in the direct and
392
cross channel of the S-matrix is obtained along with the values of the coupling in which
the S-matrix can be purely elastic. Our conclusions are stated in Section 7.
bi ,
zN F
N
(8)
N=
z being a complex variable. The numbers ai are normalization constants which can be fixed
bi (z)|0 i = 1, implying that [4]
by imposing h0 |F
bi (z)|j = e2ii j .
(9)
j |F
From the commutation relation (7), it follows directly that
bi (z).
bi (z) = i q [M] zM F
bM , F
E
(10)
For a simply-laced g,
in the representation with highest weight j of level mj , the highest
bi (z) has a vertex operator construction [2,3]
nonvanishing power of F
bi mj
bi (z) F (z) = e2ii j Yi Y+i
V
mj !
where
Yi
= exp
X i q([M])zM
EM
M
(11)
!
ME
(12)
bi (z)V
bk (w) := e2i(i +k )j Yi Yk Y+i Y+k
:V
p
h
Y
zk (hii )hki
Xhiihki (zi , zk ) =
.
1 p
zi
393
(13)
p=1
Let us now prove for non-simply laced affine Lie algebras, the highest non-vanishing power
bi (z) also admits a vertex operator construction. As is well-known [6], all non-simply
of F
laced Lie algebras can be obtained from a simply-laced Lie algebra g as a fixed subalgebra
g under an outer automorphism of g. The Dynkin diagram of g , 1(g ), is obtained
by identifying the vertices on each separate orbit of . The set of vertices in the orbit
containing the vertex i will be denoted by hii and pi stands for the number of vertices.
Denoting by i and i , the simple roots and fundamental weights of g, the g simple roots
are [30]
1 X
i ,
(14)
hii =
pi
ihii
(15)
ihii
have the correct inner products with the simple roots. From (14), it follows that
X
2
v
=
p
=
i .
i
hii
2
hii
ihii
(16)
Turning to the affine Lie algebras, one can once more obtain all the non-simply laced
g as fixed subalgebras of simply-laced g by an outer automorphism . One notes that the
exponents of g are a subset of the exponents of g and it was proven [2] that for this subset
of exponents,
bM.
bM = E
(17)
E
It was also shown that
b (i) .
bNi = F
F
N
Therefore, the generators of g which belong to the fixed subalgebra g are
bM ,
if M is an exponent of g ,
E
X
hii
i
b , N Z.
b :=
F
F
N
(18)
(19)
ihii
(20)
Using this relation, (14) and the fact that all the roots in the orbit hii the have same color,
it follows that for the generators (19),
hii
bhii = hii q [M] F
b
bM , F
(21)
E
N
M+N .
394
Then,
bi (z)
F
(22)
are in g and
bhii (z),
bhii (z) = hii q [M] zM F
bM , F
E
(23)
bhii (z) :=
F
ihii
bM g .
for E
As it has been explained in [4], under a folding procedure, the inequivalent fundamental
representations of g with highest weights j , (j ) , 2 (j ) , . . . become identified as a
single fundamental representation of g whose highest weight is denoted by hj i and with
level mhj i = mj = m (j ) = . . . .
Recall that as g is simply laced, in the g representation with highest weight j , the
bi (z) has the vertex operator construction (11). This
highest non-vanishing power of F
remains true in the hj i representation of g as does (F i (z))mhji +1 = 0. Since [2]
i
b (i) (z) = 0,
b (z), F
(24)
F
it follows that in the hj i representation,
bhii (z) pi mhji +1 = 0,
F
pi mhji
Y (F
bhii
bi (z))mhji
bhii (z) (F (z))
=
.
V
(pi mhj i )!
mhj i !
(25)
ihii
But each factor in the above product can be written as a vertex operator. Performing the
normal ordering (12), using (15) and (16), it follows that
v
bhii (z) = bhii mj e2ihii hji Yhii Y+hii
(26)
V
where
hii
Y
v q([M])zM
X hii
= exp
bhii =
ME
!
bM ,
E
(27)
(28)
06p<q6pi
can be obtained by explicit computation for the order three automorphism of d4 and for
(1)
, by using the fact that i q([M]) are proportional to the
the degenerate exponents of d2n
components of the eigenvectors of the Cartan matrix, which are well-known. Therefore,
395
the above vertex operator construction makes sense in the non-simply laced algebra g ,
since the sum is restricted to its exponents.
With the above result, we conclude that for any affine Lie algebra, in a level mj
bi (z) has a vertex operator given by
representation, the highest non-vanishing power of F
(26), with pi = 1 and bi = 1 for the simply laced and twisted algebras.
Using the definition (25), it is straightforward to check that, in the hj i representation,
bhii (z) = m v q [M] zM V
bhii (z).
b ,V
(30)
E
M
hj i hii
The vertex operator (26) indeed satisfies this commutation relation. It is interesting to note
bhii (z) in the hj i representation satisfy the same commutation relation
bM /mhj i and V
that E
hii
b
b
as EM and F (z) satisfy for the dual algebra (g )v .
Using (6), the normal ordering of two vertex operators can be performed, in the hj i
representation, giving
bhki (zk ) = Xhiihki (zi , zk )mhji : V
bhii (zi )V
bhki (zk ) :
bhii (zi )V
V
where
!
X v q([N]) v q([N]) zk N
i
k
.
N
zi
(31)
(32)
NE
Similarly to the simply laced case [2], Xhiihki can be also written as
Xhiihki (zi , zk ) =
h
Y
(1 p
p=1
v ) v
zk p (hii
hki ,
)
zi
(33)
where = exp(2i/ h). Moreover, from the commutation relation (23) we find that
hii
bhki (zk ) = F
bhki (zk )Y+hii (zi )X1/pk (zi , zk ),
Y+ (zi )F
hiihki
(34)
(35)
which will be very useful in the next sections. Using the fact that
h
X
p kv = 0
p=1
and
h
X
p p k = h k + R (g) ,
p=1
396
(1+c(k))
2h
(36)
where k is the rapidity of the kth soliton and c(k) is the color associated to the k dot of
the Dynkin diagram. Then, Xik can be expressed as a function of ik i k :
Xik (ik ) =
h
Y
eik e 2h (4p+c(i)c(k))
p v v
i
p=1
i
v
m
V (z)j = b j e2ii j .
j b
i
(37)
bi ( )), Ai
Then, the one-soliton solution created by the group element 1 g(t) = exp(Ai F
Qi Wi , will take the form
v
m
1 + + bi j e2ii j (Ai )mj pi
,
(38)
ej =
[1 + + bi (Ai )pi ]mj
where the dots indicate intermediate powers of Ai . These intermediate powers, whose
coefficients we have not calculated, do not affect the asymptotic limits x + or x
, which are equivalent to Ai or Ai 0, respectively. Thus:
2ij v
i ,
x ,
ej = e
1,
x .
In particular, this shows that, asymptotically, does approach one of the degenerate vacua.
bi ( ) will
We can also conclude that the topological charge for the soliton created by F
satisfy
2i v
i + R (gv ) .
Qtop (+) () =
Using the normal ordering (31), we obtain that the two-soliton solution created by
bi (zi ) exp Ak F
bk (zk ),
g(t) = exp Ai F
takes the form
ej =
v v
p p
1 + + (bi bk )mj e2i(i +k )j (Ai i Ak k )mj Xik (ik )mj
,
p p
[1 + + bi bk Ai i Ak k Xik (ik )]mj
with the asymptotic limits exp[2i(v + v ) ] and 1, confirming the expected result
i
that the solution interpolates degenerate vacua, and with the topological charge satisfying
2i v
i + kv + R (gv ) .
Qtop =
397
g(t) = eAi(2) (2 )F
(39)
where we consider that 1 > 2 . In order to obtain the time delay, we shall be tracking
each soliton in time. By tracking a soliton i we mean remain in its vicinity, which is near
x = xi0 + vi t. Following this tracking procedure for soliton 1, which is described in detail
in Section 4 of [5], and using Eqs. (34) and (35), it follows that in the past t , in the
vicinity of soliton 1,
A Fbi(1) ( )
1
j e i(1)
j
j
(40)
e
mj ,
i(1) ( )
b
F
A
1
0 e i(1)
0
which corresponds to an one-soliton solution of species i(1), velocity v1 , initial position
x10 and phase 1 . On the other hand, in the future t , in the vicinity of soliton 1,
ej e
2iv
i(2) j
bi(1)
0 |eXi(1)i(2) (12 )Ai(1) F (1 ) 0 j
(41)
Again, we recognize once more a one-soliton of species i(1), velocity v1 and phase 1 .
1/pi(1)
, which is real and positive, changes the modulus of Ai(1) and
However, the factor Xi(1)i(2)
0
hence x1 (see (4)). The effect is
i(1) cosh x x10 v1 t i(1) cosh x x10 v1 t +
1
ln Xi(1)i(2)(12 ). (42)
pi(1)
So, soliton 1 regains its original shape after the collision, and the only effect of the collision
is that solution (40) differs from (41) by a translation in spacetime. More precisely, the
lateral displacement of soliton 1, due to its collision with soliton 2, 112 x, satisfies
E1 112 x =
Mi(1)
2h
ln Xi(1)i(2)(12 ) = 2 ln Xi(1)i(2)(12 ),
pi(1) i(1)
| |
(43)
where E1 = Mi(1) cosh 1 is the energy of soliton 1 and (16) has been used as well as the
mass formula (5). The time delay 112 t of soliton 1 with momentum P1 , is obtained from
P1 112 t = E1 112 x.
Following [5], one can repeat the procedure by tracking the slower soliton 2, from which
it follows that
E2 121 x =
2h
ln Xi(1)i(2)(12 ) = E1 112 x,
| 2 |
(44)
398
as expected, where 121 x is the lateral displacement of soliton 2 due to its collision with
soliton 1. Therefore, we can combine both results as
Pi 1ik t = sign(i k )
2h
ln Xik (ik ).
| 2 |
(45)
It is not difficult to extend this procedure from the collision of two solitons to the
collision of any number of solitons. Like [5], it results for the mth soliton that
!
X
X
2h
ln Xi(m)i(k) (mk )
ln Xi(m)i(k) (mk ) ,
Em 1m x = 2
| | v >v
v <v
k
where 1m x is the total spatial displacement for the mth soliton. The left summation are the
contributions from the solitons faster than m and the right summation are the contributions
from the slower solitons.
Z
d ln Xik ().
(47)
Note that since the time delay does not depend on the soliton topological charges, then
only the integration constant ia,kb (0) may depend on them.
()
()
()
Let us denote by xi and yi = i2 xi /2 the components of the left and right eigenvectors of the Cartan matrix of g associated to the common eigenvalue 4 sin2 (/2h) (
being a exponent of g ) and satisfying
xi()yi
= ,
()
The vectors xi
()
c(i)xi
xi()yj() = ij .
399
(48)
satisfy 2
(h)
= xi
(49)
()
and the inner product iv q() can be written in terms of xi as [15]
i
iv q() = i 2h e 2h (1+c(i)) xi().
(50)
Using this result, (36) and the exponential form (32) for the Xik () function, the above
integral can be written as
4h2 X (N) (N) (eN 1)
xi xk
,
(51)
ia,kb ( ) = ia,kb (0) + 2
| |
N2
NE
(+nh)
xi
()
[c(i)]n xi .
m=1
where are the exponents of the underlying Lie algebra. If we put = 0, we recognize the
term inside the brackets as the series expansion on simple fractions of the sin2 function
[32], resulting that
X x (N) xk (N) 2 X x ()x ()
2 v v
i
i
k
=
=
k ,
N2
h2 4 sin2
h2 i
2h
N>0
(53)
where the last equality can be checked directly just using the fact that x () are eigenvectors
of the Cartan matrix.
The infinite series (52) can be written as a finite sum of polylogarithm functions,
Lim (y)
y n /nm ,
|y| < 1,
n=1
p=1
n=1
X x nh+
ip
1 X i p
e h Lim e h x =
.
h
(nh + )m
Then,
4 2 v v
k
| 2 | i
h
4h2 X () () e
1 X
p
ip
2
h
Li
+ 2
xi xk
+
cos
e
.
2
| |
2
2h
2h
p=1
(54)
400
Alternatively, the phase ia,kb ( ) can be written in terms of the infinite soliton conserved
bi ( )
charges. Indeed, the conserved charges for the one-soliton solution created by exp QF
are [33]
N
N
(N)
PiN (i ) = 2 iv q [N] ziN = 2 2h x i eNi , N > 0.
| |
| |
Then,
ia,kb ( ) = ia,kb (0)
X P N (i )P N (k )
4 2 v v
k
i
2
2h|
|
k
| 2 | i
2N N 2
(55)
NE
for i > k . This shows an intimate relationship between soliton S-matrix elements and
soliton conserved charges of ATFTs. Indeed, as pointed out in [11,34], the phase shift of
any purely elastic exact S-matrix for an integrable theory could be written as a sum over
bilinears on the quantum conserved charges. In the next section, the values of the coupling
in which the soliton S-matrix can be purely elastic are obtained. So for these values of the
coupling, one could expect that an expression like this for the phase shift would be exact,
with PiN being the quantum conserved charges and making use of the substitution (58),
given in the next section. For example, for sine-Gordon, the integral representation of exact
phase shift of the solitonsoliton transmission amplitude [21] is
1
ss ( )
=
2i
h
1
=
2i
" Z
dt 2i t sinh(1 )t
e
t
sinh t cosh t
dt 2i t
e
t
#
dt 2i t
e tanh t coth t ,
t
where
1
h|
h|
2|
2|
1
.
=
4
4
(56)
2k
X
(2k 1) X e
k
2e(2k1)
1
ss ( )
cot
tan
=
1
(2k 1)
2
k
h
k=1
k=1
where = sgn Re . One clearly sees that for = 1/n, n = 2, 3, . . . , when the exact
reflection amplitude vanishes, the last summation vanishes and the exact phase shift takes
a form similar 4 to (51) or (55), with the first term of the series expansion of the cotangent
given by the semiclassical term. From this expression of the phase shift one gets for free
the sine Gordon soliton quantum charges.
It is interesting to note that the ATFT particle S-matrix, which is purely elastic for any
value of the coupling, has a phase shift [19] similar to (55) but with the conserved charges
being proportional to the right eingenvector yi() instead of xi() . Therefore, for the values of
4 Remembering that the exponents of a 1 are the odd integers.
1
401
the coupling in which the soliton S-matrix is purely elastic, there is an intriguing similarity
between the g-ATFT
soliton S-matrix and the g v -ATFT particle S-matrix. Clearly for each
particle species there is a soliton species with many possible topological charges.
4 2 v
i kv + kv
2
| |
nik ( 2 ) h ,
(57)
where kb means the antisoliton of the soliton of species k and topological charge b.
Following the super-Levinson theorem [31], the largest integer smaller than nik ( 2 ) gives
the number of bound states in
both, the direct and the cross channels. Note that for sine(1)
Gordon (g = a1 ) 1 = 1/ 2, and we recover the well-known result 5 that the number
of bound states is the largest integer smaller than n11 = 4/(h | 2 |), in the semiclassical
approximation. In order to obtain the exact result, one should replace [35] h | 2 |/(4) by
given in (56). In view of the arguments in [37], one would expect that a similar substitution
1
h|
h|
h | 2 |
2| h
2|
(58)
4
4 hv
4
in (57) would give the exact number of bound states.
These bound states should correspond to poles in the exact S-matrix associated not just
to breathers but also for the fusing of soliton solutions, which appear as poles of Xik ( )
[5] and are governed by Doreys fusing rule [13]. Some of the bound states were analysed
[24] for some particular algebras and particular colliding soliton species.
By direct inspection, one can conclude that
vk = kv + R (g ).
(59)
402
purely elastic [36] in the semiclassical approximation. Now we shall show that same is true
for the other ATFTs.
In two dimensions, a two-state S-matrix must satisfy the unitarity and crossing relations
(61)
S ( ) = S 1 ( ) ,
S (i ) = S ( ).
(62)
S (i + ) = S 1 ( ) .
(63)
In the semiclassical approximation, we must use the crossing relation involving the analytic
continuation + i rather than i , since the semiclassical approximation
breaks down on the imaginary axis, as was pointed out by Coleman [38].
Considering that the S-matrix is purely elastic, conditions (61) and (63) become
S ( )S ( ) = 1,
S (i
(64)
1
+ ) = S
( ),
(65)
(66)
where N must be an integer number and gives the number of bound states in the direct
channel [31].
Let us see the constraint the crossing relation imposes on the phase shift of ATFT. Before
doing this, recall the fact that [15] c(k) = (1)h c(k) and [13] x () = (1)+1xk() . Then,
k
(67)
Using (59) and the fact that ia,kb (0) and ia,kb (0) must satisfy (66), it implies that
the above equation can only be consistent if the coupling satisfies the condition (60).
Therefore, only for these values of the coupling the unitarity and crossing relations for
purely elastic S-matrix is fulfilled. So, the S-matrix can only be purely elastic for these
values in the semiclassical approximation. Once more, this result holds at the semiclassical
approximation and in order to obtain the exact result one should probably perform a
403
substitution like (58). Comparing relation (67) with the super-Levinson relation (57)
implies that ia,kb () + ia,kb () = 0 mod(2 h ). It would be interesting if one could
calculate the constants ia,kb (0), which gives the number of bound states in the direct
channel [31].
7. Conclusions
In this paper, we have extended our vertex operator construction for non-simply laced
affine Lie algebras from which we obtained the time delay resulting from the collision
of two (or more) solitons. From the time delay, we obtained the semiclassical limit for
the transmission amplitude of the two soliton S-matrix which holds for any ATFT. The
semiclassical phase shift appeared as a sum over bilinears on the soliton conserved charges.
Using the super-Levison theorem, an universal expression for the number of bound states
in the direct and cross channels of the S-matrix was obtained. We also obtained the values
of the coupling in which the S-matrix can be purely elastic. For these values, one would
expect that the form of the exact phase shift would be like (55), with PNi being the quantum
soliton conserved charges.
Acknowledgements
I would like to thank D. Olive and R. Paunov for discussions and for reading the
manuscript, L.A. Ferreira for invitation to visit IFT-UNESP where part of this work was
conceived and FAPERJ for financial support.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
D.I. Olive, N. Turok, J.W.R. Underwood, Nucl. Phys. B 401 (1993) 663.
D.I. Olive, N. Turok, J.W.R. Underwood, Nucl. Phys. B 409 (1993) 509.
M.A.C. Kneipp, D.I. Olive, Nucl. Phys. B 408 (1993) 565.
M.A.C. Kneipp, D.I. Olive, Commun. Math. Phys. 177 (1996) 561.
A. Fring, P.R. Johnson, M.A.C. Kneipp, D.I. Olive, Nucl. Phys. B 430 (1994) 597.
D.I. Olive, N. Turok, Nucl. Phys. B 215 (1983) 470.
T.J. Hollowood, Nucl. Phys. B 384 (1992) 523.
H. Aratyn, C.P. Constantinidis, L.A. Ferreira, J.F. Gomes, A.H. Zimerman, Nucl. Phys. B 406
(1993) 727.
N.J. MacKay, W.A. McGhee, Int. J. Mod. Phys. A 8 (1993) 2791.
A.E. Arinshtein, V.A. Fateev, A.B. Zamolodchikov, Phys. Lett. B 87 (1979) 389.
H.W. Branden, E. Corrigan, P.E. Dorey, R. Sasaki, Nucl. Phys. B 338 (1990) 689.
P. Christe, G. Mussardo, Int. J. Mod. Phys. A 5 (1990) 4581; Nucl. Phys. B 330 (1990) 465.
P. Dorey, Nucl. Phys. B 358 (1991) 654.
P. Dorey, Nucl. Phys. B 374 (1992) 741.
A. Fring, D.I. Olive, Nucl. Phys. B 379 (1992) 429.
G.W. Delius, M.T. Grisaru, D. Zanon, Nucl. Phys. B 382 (1992) 365.
E. Corrigan, P.E. Dorey, R. Sasaki, Nucl. Phys. B 408 (1993) 579.
404
Abstract
R
In the absence of Gribov complications, the modified gauge fixing in gauge theory DA
R
R
R
g
{exp[SY M (A ) f (A )dx]/ Dg exp[ f (A )dx]} for example, f (A ) = (1/2)(A )2 , is
g
R
(A ) R
identical to the conventional FaddeevPopov formula DA {(D fA
)/ Dg (D f (Ag ) )}
exp[SY M (A )] if one takes into account the variation of the gauge field along the entire gauge
orbit. Despite of its quite different appearance,the modified formula defines a local and BRST
invariant theory and thus ensures unitarity at least in perturbation theory. In the presence of
Gribov complications, as is expected in non-perturbative YangMills theory, the modified formula
is equivalent to the conventional formula but not identical to it: both of the definitions give rise to
non-local theory in general and thus the unitarity is not obvious. Implications of the present analysis
on the lattice regularization are briefly discussed. 2000 Elsevier Science B.V. All rights reserved.
1. Introduction
The standard gauge fixing procedure of general gauge theory formulated by Faddeev
and Popov [1] provides a convenient framework for perturbation theory. The BRST
symmetry appearing there [2] controls the SlavnovTaylor identities [3,4] and ensures
the renormalizability and unitarity. This formulation however suffers from Gribov
complications [5] in the non-perturbative level. The lattice formulation of gauge theory
is known to introduce further complications which may partly be the artifacts of
lattice regularization. A naive modification of BRST invariant formulation of continuum
theory [6,7] does not quite resolve the basic issue of lattice regularization, as is illustrated
by the no-go theorem of Neuberger [8] about the lattice implementation of BRST
symmetry. Recently the issue related to this last point has been partly resolved by Testa [9].
A possible generalization of the FaddeevPopov formula, which may provide an alternative approach to the Gribov-type complications, has been proposed by Zwanziger [10],
and Parrinello and Jona-Lasinio [11]. It is known that their modified formula reduces to
the conventional FaddeevPopov formula in a specific limit of the gauge fixing parameter [1012]. In the present note, we show explicitly that the modified formula is reduced
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 0 2 - 4
406
to the FaddeevPopov formula in the absence of Gribov complications if one takes into
account the motion of the gauge variable along the entire gauge orbit, without taking a specific limit of the gauge fixing parameter. We thus see that the modified formula, despite of
its quite different appearance, does not go beyond the FaddeevPopov prescription. In particular, the locality and unitarity is ensured by both prescriptions in the absence of Gribov
complications, such as in perturbation theory. In the presence of Gribov complications, as
is expected in non-perturbative formulation, both of the original FaddeevPopov formula
and the modified formula, which are equivalent but not identical to each other, give rise to
non-local action: the validity of unitarity is thus not obvious.
In contrast, a modified BRST formula [6,7] ensures unitarity if one assumes the
asymptotic condition such as the LSZ condition [13], but the full justification of the
modified BRST formulation in the non-perturbative level is absent at this moment.
(2.1)
Dh e g(A )dx ,
= DA e
where we suppress the non-Abelian index, and and h stand for the gauge parameters;
Ah
stands for the field variable obtained from A by a gauge transformation specified
by h. We also defined
(2.2)
g A f A f A0 ,
(2.3)
Namely, 0 (x) is the value of the gauge orbit parameter (x) which gives rise to the
minimum value of f (A ) at each point of the spacetime. We may generally assume
f (A ) > 0 (or more generally, bounded from below) to ensure the convergence of the
path integral in the denominator in (2.1). By definition we have
Z
Z
Z
f (A )
f (A )
(x) = dx D (x) = 0
(2.4)
f (A )dx = dx
(x)
A (x)
for any choice of (x) at (x) = 0 (x), namely, 0 (x) is implicitly defined by
f (A 0 )
= 0.
A0 (x)
(2.5)
We understand that the absence of Gribov complications implies that Eq. (2.5) has a unique
solution 0 (x) for each gauge orbit. In this case, Eq. (2.3) also has a unique solution 0 (x):
407
D(g(A
) ) eS0 (A ) dx
Z = DA R
R
0 0 2 dx
h
D0 Dh(g(A
) )e
(R
R 2 )
Z
D(g(A
)) eS0 (A ) dx
,
(2.6)
= DA R
R
0 2 dx
h
D0 Dh(g(A
)) e
where we defined
p
g A = g(A ), g A > 0,
p
g A = g(A ), g A < 0,
(2.7)
with (g(A
))2 = g(A ). In the second expression of (2.6) (x) is a generic gauge
orbit parameter in the infinitesimal neighborhood of 0 (x), and h(x) is a generic gauge
parameter in the infinitesimal neighborhood of the unit element. Here we used the fact that
we can bring the relation
(2.8)
g A 2 = 0
to
0
g A = 0
(2.9)
by choosing a suitable gauge parameter 0 (x) for an arbitrary (x) in the absence of
Gribov complications. In fact, 0 (x) = 0 (x) in the absence of Gribov complications. This
statement is established by noting that we can compensate any variation of (x) by a
suitable change of the gauge orbit parameter (x) as
f (A )
g(A )
(x) =
(x) = 2 (x)
(x)
(x)
because
f (A )
6= 0
(x)
(2.10)
(2.11)
408
We thus finally write the partition function (2.6) after the path integration over and 0
as
Z
Z=
Z
n
.
o S (A )
)
e 0 .
Dh g Ah
DA g(A
(2.12)
This is thepstandard FaddeevPopov formula for gauge theory with a specific gauge fixing
g(A
) = g(A ) = 0.
To write the above path integral formula (2.12) in a more manageable manner, we use a
representation of the -function
g
A0 +
R
R
1
2
lim0 D e 2 dx
(2.13)
f (A )
(x)
=g
+ dx
(x) =0
Z
f (A )
D (A )
(y)
+ dx dy (x)
(y)
A (x) =0
Z
f (A )
D (A )
(y),
= dx dy (x)
(y)
A (x) =0
where we used
g A 0 = 0,
A 0
f (A )
dx
(x)
(2.14)
(x) = 0.
(2.15)
=0
=
R
R
1
lim0 D exp 2
dx2
1
(2.16)
=
,
det O
where we defined
f (A )
D (A )
(2.17)
O(x, y)
(y)
A (x) =0
(2.18)
det O = det O.
Note that we may assume that the operator O(x, y) is proportional to (x y) and
a
positive definite operator in the absence of Gribov complications: we can thus define O
409
in (2.18) as an operator whose eigenvalues are given by the square root of the eigenvalues
of O and still proportional to (x y).
We also have
1 R
lim0 exp 2
dx dy (x)O(x, y)(y)
g A0 + =
R
R
1
lim0 D exp 2
dx2
0 +
0 +
1 R
dx dy D f (A0 + ) (y)O(x, y)1D f (A0 + ) (x)
lim0 exp 2
A
A
=
R
R
1
lim0 D exp 2
dx2
+
)
f (A 0
1
,
(2.19)
= D
0 +
O
A
where we used the fact that
Z
+
)
f (A 0
f (A 0 )
(x) = D
(x) dy O(x, y)(y)
D
+
A0
A0
Z
= dy O(x, y)(y).
(2.20)
Oe 0
det
Z = DA D
A
O
Z
R
f (A )
[iB 1 D ( A )+c Oc]dxS0 (A
)
= DA DB Dc Dc e
Z
=
R
f (A )
[iBD ( A )+cOc]dxS
0 (A )
DA DB Dc Dc e
,
(2.21)
where c and c stand for the FaddeevPopov ghosts, and B is the NakanishiLautrup field.
In the last expression we re-defined the auxiliary variables as
c c O,
(2.22)
B B O,
which leaves the path integral measure invariant.
We have thus established the equality
Z
n
.Z
o
R
R
h
S0 (A ) f (A )dx
Dh e f (A )dx
DA e
Z
R
(A )
)+cOc]dxS
[iBD ( fA
0 (A )
= DA DB Dc Dc e
(2.23)
f (x0 ) = 0 and f (x0 ) > 0. We then have f (x) = f 00 (x0 )/2 (x x0 ) and (f(x)) ( f 00 (x0 )(x x0 ))
p
R
R
f 0 (x)). Consequently, (f(x))/ dx (f(x)) = ( 001
f 0 (x))/ dx ( f 00 (x0 )(x x0 )) =
( 001
f (x0 )
( 001
f (x0 )
f 0 (x))/( 001
f (x0 )
) = (f 0 (x))f 00 (x0 ).
f (x0 )
410
The final path integral formula (2.23) is local and invariant under the BRST transformation with a Grassmann parameter
i abc b c
f c c ,
ca = B a ,
B a = 0 (2.24)
2
for a rather general class of gauge fixing function f (A ). We here write the gauge index
explicitly. This BRST symmetry itself holds even before the field re-definition in (2.21),
since the field re-definition (2.22) is consistent with BRST symmetry. Eq. (2.23) is the
main result of the present note.
So far we assumed the absence of Gribov complications, and thus the arguments are
applicable to perturbation theory. In the presence of Gribov complications, as is expected in
non-perturbative formulation, the analysis becomes more complicated. We here understand
the Gribov complications as simply meaning the appearance of multiple solutions of
Z
(2.25)
f A dx = 0,
Aa = i(D c)a ,
ca =
at (x) = k (x), k = 1, 2, . . . , n. We also assume that these k (x) are globally defined in
the entire spacetime.
In the presence of Gribov complications, we cannot make a definite statement. In the
following, we briefly sketch how one can transform (2.1) to an expression which is as local
as possible; this may be relevant for the analysis of the issues related to unitarity. In the
modified path integral formula [10,11], the local minimum solutions of (2.25) correspond
to the so-called Gribov copies. The local maximum solutions of (2.25) correspond to the socalled Gribov horizons, namely, the obstruction in the analysis of (2.10). We can then still
arrive at the second expression in (2.6), but now 0 (x) stands for one of those k (x) which
give the local minima of f (A ). For these solutions, the operator in (2.17) is considered
as a positive operator. We thus obtain
Z
n
.Z
o
R
R
h
) f (A
)dx
S0 (A
DA e
Dh e f (A )dx
Z
n
.Z
o
R
R
h
Dh e g(A )dx
= DA eS0 (A ) g(A )dx
X Z
XZ
k
k
k
hk
eS0 (A )
DA g A
Dh g A
=
k
X
k
1
1
f (A k )
=
DAk D
eS0 (A )
k
A
Ok
det Ok
k
k
Z
X
X
p
k
1
f (A k )
=
O
det
eS0 (A ) , (2.26)
DAk D
k
k
A
det Ok
k
k
XZ
where the integration variable Ak and the variable inside the action and the -functional
constraint stand for a generic gauge field in the infinitesimal neighborhood of the local
minimum solutions of (2.25). 2
2 We here take a view that we should sum over all the Gribov copies, as is the case in the modified BRST
formulation [14,15]. It is argued in [10,11] that the integrated functional has an absolute minimum solution for
411
On the other hand, the conventional FaddeevPopov formula gives (by remembering
that the -function is by definition positive)
Z
Z
f (A )
f (Ah
)
Dh D
eS0 (A )
Z = DA D
h
A
A
X
Z
f (A )
eS0 (A ) ,
= DA D
(2.27)
A
| det Ok |
k
f (A )
f (A )
D
det
eS0 (A )
DA D
A
A
Z
R
f (A )
f (A )
D ( A )c]dxS0 (A
[iBD ( A )+c
)
= DA DB Dc Dc e
,
(2.28)
where the integrand is no more positive definite in
the presence of Gribov copies, as we
f (A
)
do not take the absolute value of det{ [D ( A )]}. In (2.28) we integrate over all the
gauge field configurations. The BRST symmetry combined with the asymptotic condition
such as the LSZ prescription defines a unitary theory. We emphasize that the Lagrangian
for the FaddeevPopov ghosts in (2.28) is not degenerate in general with respect to the
f (A )
degenerate solutions of D ( A ) = 0; consequently, the asymptotic condition such as
the LSZ condition may well pick up a unique asymptotic field A despite the presence of
the Gribov copies. (In pure YangMills theory without the Higgs mechanism, we expect
the gluon confinement and thus the asymptotic condition may be replaced by the use of a
Wilson loop, for example.)
3. Abelian example
An example of Abelian gauge theory may be illustrative, since we can then work out
everything explicitly. Note that there is no Gribov complications in the Abelian theory at
R
2
Min f (A
)dx if one chooses a suitable f (A ) such as f (A ) = (1/2)(A ) . This suggests that, if one could
argue that the Gribov horizon is overcome in the analysis of (2.10) in the
modified
FaddeevPopov formula, one
R
would have a chance to achieve a relaxation to the absolute minimum of f (A
)dx in the path integral. In such a
case (and if the absolute minimum is unique), the Gribov complications would largely be resolved in the modified
FaddeevPopov formula.
412
least in a continuum formulation. As a simple and useful example, we choose the gauge
fixing function [10,11]
1
f (A) A A
2
and
D
f
A
(3.1)
= A .
(3.2)
Dh e dx 2 (A )
Z = DA e
Z
R
)c] dx
.
= DA DB Dc Dc eS0 (A )+ [iB A +c(
(3.3)
2
dx 12 (Ah
)2
= Dh e dx 2 (A + h)
Dh e
Z
R
1
2
2
dx 12 [(A
1
) 2( A ) B+B ]
p
= DB
e
det
R
R
1
2 1
1
dx 12 (A
) + 2 A A dx
p
=
e
,
det
p
where we defined h = B. Thus
Z
p
S (A ) 1 R A 1 A dx
Z = DA det e 0 2
R
R 2
Z
1
1
S0 (A
) 2 B dx+ [iB A +c c] dx
= DA DB Dc Dc e
,
(3.4)
(3.5)
c = 0,
c = B,
B = 0,
(3.6)
with a Grassmann parameter . Note the appearance of the imaginary factor i in the term
iB( )1/2 A in (3.5).
When one defines
R
R
Z
S0 (A ) 2 B 2 dx+ [iB 1
A
+c c] dx
(3.7)
Z() = DA DB Dc Dc e
one can show that
Z( ) =
Z
DA DB Dc Dc
e
S0 (A
) 2
R
B 2 dx+ [iB
1
A
+c
c] dx
Z
= Z() +
e
DA DB Dc Dc
S0 (A
) 2
R
B 2 dx+ [iB
413
B 2 dx
1
A +c
c] dx
(3.8)
On the other hand, the BRST invariance of the path integral measure and the effective
action in the exponential factor gives rise to
R
R 2
Z
Z
S0 (A
) 2 B dx+ [iB A +c c] dx
dx e
DA DB Dc Dc (cB)
Z
Z
0
0
0
0
= DA DB Dc Dc (c0 B 0 )dx
R
R 02
1
0
0
0 c0 ] dx
S0 (A0
) 2 B dx+ [iB A +c
e
Z
Z
= DA DB Dc Dc
cB
+ BRST (cB)
dx
R
R 2
S0 (A
) 2 B dx+ [iB A +c c] dx
,
(3.9)
e
0
0
where the BRST transformed variables are defined by A0
= A + i c, B = B, c =
0
c + B, c = c. Namely, the BRST exact quantity vanishes as
Z
Z
BRST (cB)
dx
DA DB Dc Dc
R
R 2
S0 (A
) 2 B dx+ [iB A +c c] dx
e
Z
Z
= DA DB Dc Dc B 2 dx
R
R
S0 (A ) 2 B 2 dx+ [iB 1
A
+c c] dx
= 0.
(3.10)
e
(3.11)
Dh e 2 (A ) dx
Z = DA eS0 (A ) 2 (A ) dx
R
Z
1
S0 (A
)+ [iB A +c c] dx
= DA DB Dc Dc e
Z
R
)c] dx
= DA DB Dc Dc eS0 (A )+ [iB A +c(
(3.12)
(3.13)
which is consistent with BRST symmetry and leaves the path integral measure invariant.
We thus established the desired result (3.3).
An alternative way to arrive at the result (3.12) from (3.5) is to rewrite the expression
(3.5) as
414
Z
DA DB D Dc Dc
R
!
1
p
A
R
eS0 (A ) 2 (B +2iB)dx+ c c dx
!
Z
1
= DA D Dc Dc p
A
R
R 2
1
eS0 (A ) 2 dx+ c c dx .
(3.14)
We next note that we can compensate any variation of by a suitable change of gauge
parameter inside the -function as
1
p
= .
(3.15)
By a repeated application of infinitesimal gauge transformations combined with the
invariance of the path integral measure under these gauge transformations, we can re-write
the formula (3.14) as
!
Z
R
R 2
1
1
A eS0 (A ) 2 dx+ c c dx
DA D Dc Dc p
!
Z
R
= DA Dc Dc p
A eS0 (A )+ c c dx
R
Z
1
S0 (A
)+ [iB A +c c] dx
= DA DB Dc Dc e
Z
R
)c] dx
= DA DB Dc Dc eS0 (A )+ [iB A +c(
(3.16)
after the field re-definition of auxiliary variables B and c in (3.13). This procedure is the
one we used for the non-Abelian case.
4. Conclusion
We have shown explicitly the equivalence of the modified path integral formula [10,11]
to the conventional FaddeevPopov formula [1] without taking any limit of the gauge fixing
parameter, if the Gribov complications are absent. In the presence of Gribov complications,
these two formulas are equivalent but not identical to each other, and both give rise to nonlocal theory in general and thus the unitarity is not obvious.
From a view point of non-perturbative definition of gauge theory, a BRST invariant
formulation of lattice gauge theory is important [16]. The Neubergers stricture in the
BRST invariant lattice formulation [8] corresponds tothe appearance of an even number of
f (A )
gauge variables, and thus (when we use the continuum notation in (2.28))
Z
f (A )
f (A )
(4.1)
D
DA D
det
eS0 (A ) = 0.
415
The recent analysis by Testa [9] suggests that the BRST invariant ansatz [6,7] can be
implemented for Abelian theory and also for the Abelian projection of non-Abelian theory
by a suitable generalization of the -function on the lattice. It remains an interesting
problem to extend the analysis of Testa to fully non-Abelian lattice theory which may
eventually overcome the Neubergers stricture.
As for the practical implications of the present continuum analysis on the lattice
simulation, it may be important to remember that the gauge fixing by adding an (effective)
mass term to the lattice action [16] may not go beyond the conventional FaddeevPopov
procedure if one accounts the variation of the gauge variable along the entire gauge orbit.
This property will be important when one analyzes the issues related to unitarity. It should
also be emphasized that both of the original FaddeevPopov formula and the modified one
give rise to a positive definite integrand in the path integral even on the lattice [16].
The present note was motivated by the discussions at the RIKEN-BNL Workshop, May
2529, 99 and at the NATO Advanced Research Workshop at Dubna, October 59, 99.
One of the authors (K.F.) thanks all the participants of those workshops for stimulating
discussions.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
Abstract
We find further evidence for the conjecture relating large N ChernSimons theory on S 3 with
topological string on the resolved conifold geometry by showing that the Wilson loop observable of
a simple knot on S 3 (for any representation) agrees to all orders in N with the corresponding quantity
on the topological string side. For a general knot, we find a reformulation of the knot invariant in
terms of new integral invariants, which capture the spectrum (and spin) of M2 branes ending on M5
branes embedded in the resolved conifold geometry. We also find an intriguing link between knot
invariants and superpotential terms generated by worldsheet instantons in N = 1 supersymmetric
theories in 4 dimensions. 2000 Elsevier Science B.V. All rights reserved.
1. Introduction
Large N gauge theories have been conjectured by t Hooft to be related to string theories.
A particularly simple example of gauge theories is the ChernSimons theory, solved by
Witten. It is thus natural to ask about the large N limit of ChernSimons theory and look
for an appropriate stringy description. Some aspects of large N limit of ChernSimons
theory were studied some time ago in [1,2].
It was conjectured recently [3] that at least for some manifolds (including S 3 ) the large
N limit does give rise to a topological string theory on a particular CalabiYau background.
This conjecture was checked at the level of the partition function on both sides; The Chern
Simons answer was already well known, and the topological string partition function
was recently computed in two different ways (one by mathematicians, and one by using
physical reasoning about the structure of BPS states).
It is natural to extend the conjecture to the observables of ChernSimons theory,
which are Wilson loop operators. Namely we should consider product of Wilson loop
On leave from University of California, Berkeley; hooguri@lbl.gov
1 vafa@string.harvard.edu
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 1 8 - 8
420
observables for any choice of representation on each knot. We show how this question
can be formulated in the present context and explicitly check the map for the case of
the simple circle in S 3 (unknot). The computation on the ChernSimons side is well
known. On the topological string side, we end up with topological string amplitudes on
Riemann surfaces with boundaries. Mathematically these have not been studied, however
by connecting the partition function of topological strings to target space quantities we
compute them in terms of spectrum of M2 branes ending on M5 branes embedded in the
CalabiYau threefold. The target space interpretation is also related to generation of N = 1
superpotential terms in four dimensions (which we relate it analogously to the spectrum of
domain walls).
For a general knot finding the explicit form of the M5 brane embedded in the CalabiYau
is not trivial, though physically we argue it should be possible. In this way we reformulate
knot invariants in terms of new invariants capturing the spectrum of M2 branes ending on
M5 branes.
The organization of this paper is as follows: In Section 2 we review the large N
conjecture for ChernSimons theory. In Section 3 we show how the Wilson loop observable
for arbitrary knot and representation can be formulated in this set up, and apply the gauge
theory/geometry correspondence for the case of the simple knot. In Section 4 we show how
the results anticipated from the Wilson loop observables can be directly obtained using the
spectrum of M2 branes ending on M5 branes (or D2 branes ending on D4 branes). We
also point out connections with generation of superpotential terms with theories with 4
supercharges. In Section 5 we present some concluding remarks and suggestions for future
work.
2
,
k+N
t=
2 iN
,
k+N
(2.1)
421
where is the string coupling constant and t is the Khler modulus of the blown-up S 2 .
The coupling constant gCS of the ChernSimons theory, after taking into account the finite
2 . Therefore the Khler moduli t given by (2.1) is
renormalization, is related to as = gCS
2
i times the t Hooft coupling gCS N of the ChernSimons theory. The geometric motivation
of the conjecture is based on starting with the topological strings on conifold geometry
T S 3 and putting many branes on S 3 , for which we get a large N limit of ChernSimons
on S 3 supported on the brane. The conjecture states that in the large N limit the branes
disappear but lead instead to the resolution of the conifold geometry where an S 2 has
blown up. In fact this conjecture parallels the motivation for the AdS/CFT correspondence
conjecture: As noted in [3], since the open topological string theory couples to closed
topological string theory through a gravitational ChernSimons action [4], putting 3-branes
on S 3 deforms the gravitational background so as to produce a blown up S 2 . In fact the
volume of the S 2 was computed in this way.
The conjecture has been checked as follows: Start with the vacuum amplitude Z(S 3 ) of
the ChernSimons gauge theory on S 3 (with the normalization Z(S 2 S 1 ) = 1);
r
Ns
N1
ei 8 (N1)N k + N Y
s
3
.
(2.2)
2 sin
Z(S ) =
(k + N)N/2
N
k+N
s=1
log Z(S 3 )
is given by
The large-N expansion of
"
#
X
3
2g2
Fg (t) ,
Z(S ) = exp
(2.3)
g=0
X
1
i 2
i
t i m+
t 2 + t 3 +
n3 ent ,
F0 = (3) +
6
4
12
n=1
1
1
log 1 et ,
F1 = t +
24
12
with m being some integer, and for g > 2,
"
#
X
(1)g1
1
(1)g1
Bg
2 (2g 2)
n2g3 ent .
Fg =
2g(2g 2)
(2)2g2
(2g 3)!
(2.4)
(2.5)
n=1
Here Bg is the Bernoulli number, which is related to the Euler characteristic of the moduli
space Mg of genus-g Riemann surfaces as
g =
(1)g1
Bg .
2g(2g 2)
(2.6)
By using this and the formula for the Chern-class of the Hodge bundle over the moduli
space
Z
(1)g1
3
cg1
=
2 (2g 2)g ,
(2.7)
(2)2g2
Mg
422
X
g
3
cg1
n2g3 ent .
Fg =
(2g 3)!
(2.8)
n=1
Mg
It turns out that the expressions (2.4) and (2.8) for Fg are exactly those of the gloop topological string amplitude on the resolved conifold. The constant map from the
1
t in F1 [8], and
worldsheet to the target space gives rise to the term 12i t 3 in F0 [6] 2 , and 24
R
3
c
in
F
with
g
>
2
[9].
Regarding
worldsheet
instantons,
since
the
only
non-trivial
g
Mg g1
2-cycle in the target space is the blown-up S 2 , their contributions are from multi-coverings
of the Riemann surface onto the S 2 . For g = 0, 1 and 2, the expressions in the instanton
terms in (2.4) and (2.5) agree with the results of [6,8], and [9], respectively. More recently,
the contribution of of the worldsheet instantons are evaluated for all g in [5], in complete
agreement with (2.8). These expressions can also be derived, as was done in [10], from
the target space view point by identifying what the topological strings compute in type
IIA compactification on the corresponding CalabiYau threefold. It turned out that the full
structure of Fg is encoded in terms of the spectrum of wrapped D2 branes on the Calabi
Yau. This will be reviewed later in this paper.
In this paper, we provide further evidence for the conjecture. We will show that the
Wilson loop expectation value of the ChernSimons theory also has a natural interpretation
in terms of the topological string on the resolved conifold geometry.
2.2. Conifold transition
As noted above, the geometric insight that led to the conjecture is the fact that one
can view the ChernSimons theory as the open topological string theory. Consider the
cotangent space T S 3 of S 3 as the target space of the topological string. It was shown in [4]
that, if we wrap N D-brane on the base S 3 of the cotangent space, the open topological
string theory on the D-brane is equivalent to the ChernSimons theory with the gauge
group SU (N). The cotangent space has the canonical symplectic form
=
3
X
dq i dpi
(q S 3 , p Tq S 3 ),
(2.9)
i=1
and the base S 3 is a Lagrangian submanifold. Therefore the open string on the D-brane
allows the topological twist of the A-type.
At this point, it would be useful to review basic facts about the conifold transition. The
space T S 3 can also be regarded as a deformed conifold geometry,
4
X
y2 = a 2
(y C 4 ),
(2.10)
=1
2 The coefficients of 1 and t in F have analogous interpretations [7], and they also agree with the Chern
0
Simons prediction [3].
423
x p = 0.
(2.11)
The first equation suggests that the base S 3 of radius a is located at p = 0 and the second
equation shows that p are coordinates on the cotangent space at x S 3 . As a 0, the
S 3 shrinks to a point and a singularity appears. It is known as the conifold singularity.
In addition to the deformation by a, the conifold singularity
X
y2 = 0
(2.12)
can be smoothened out by the blow-up. It is described as follows. By introducing two pair
of complex coordinates (u, u)
and (v, v)
by
u = y1 + iy2 ,
u = y3 iy4 ,
v = y3 + iy4 ,
v = y1 iy2 ,
(2.13)
(2.14)
v = zv.
(2.15)
424
Fig. 1. The conifold singularity can be either deformed to T S 3 or resolved by the S 2 blow-up.
The conjecture in [3] states that the open topological string theory on the N D-branes on S 3 of
the deformed conifold is equivalent to the closed topological string theory on the resolved conifold
geometry.
of T S 3 . We can probe the dynamics on these D-branes by introducing another set of Dbranes. First we define a Lagrangian 3-cycle associated to the knot q(s) S 3 as follows 3 .
At each point q(s) on the loop, we consider 2-dimensional subspace of Tq S 3 orthogonal
to dq/ds. By going around the loop, we can define the 3-cycle,
dq i
= 0, 0 6 s < 2 .
(3.1)
C = q(s), p pi
ds
The topology of C is R 2 S 1 . The symplectic form vanishes on C, so it is a Lagrangian
submanifold. 4 The 3-cycle intersects with the base S 3 along the loop q(s). See Fig. 2.
Now let us wrap M D-branes on C. We then have the SU (M) ChernSimons theory on C
as well as the SU (N) ChernSimons theory on S 3 . In addition, we also have a new sector of
open string with one end on C and the other on S 3 . One can easily quantize the topological
string in this sector and obtain a complex scalar field living on the intersection, namely the
loop q(s), which transforms according to the bi-fundamental of SU (N) SU (M). To see
that there is one complex scalar field of this type, we note that, in the relevant open string
sector, i.e., the Ramond sector, there are two states one with N = 2 U (1) charge 1/2 (a
scalar) and the other with +1/2 (a 1-form). The physical states of the topological string
3 We thank C. Taubes for discussion on these Lagrangian cycles.
4 The cycle C defined here is Lagrangian but is not necessarily special Lagrangian. In order for to make the
425
Fig. 2. For each loop q(s) S 3 , one can define a unique Lagrangian 3-cycle which extends in the
cotangent direction and intersects S 3 on the loop q(s).
come from the sector with U (1) charge 1/2, and that turns out to correspond to the scalar
living on the loop q(s). The action for the scalar field is Gaussian, and integrating it out
gives the determinant,
dq i
d
+ (Ai Ai )
,
(3.2)
Z = exp log det
ds
ds
where A and A are the ChernSimons gauge fields on S 3 and C, respectively. We can
evaluate the determinant by diagonalizing A and A (this is allowed since we are dealing
with the one-dimensional problem along the intersection). By using the formula
X
d
+ i =
log(n + ) = log sin( ) + const,
(3.3)
log det
ds
n=
we find
Z(U, V ) = exp tr log U 1/2 V 1/2 U 1/2 V 1/2
= exp tr log 1 U V 1
#
"
X1
n
n
tr U tr V
,
= exp
n
(3.4)
n=1
where U and V are path-ordered exponentials of the gauge fields along the loop,
I
I
U = P exp A SU (N),
V = P exp A SU (M),
and we used det U = det V = 1.
We are interested in taking N to infinity for a fixed M. We view the M branes on C as
a probe. In this context, integrating out the gauge fields A will leave us with an effective
theory on the probe brane, which is an SU (M) ChernSimons on C plus some corrections.
Let us define
426
Z
1
exp F (t, V ) = R
[DA]
[DA] exp(SCS (A; S 3 ))
#
"
X
1
3
n
n
tr U tr V
exp SCS (A; S ) +
n
n=1
= Z(U, V ) S 3
(3.5)
which can be viewed as the generating functional for all the observables of the Chern
Simons gauge theory on S 3 associated to the unknot. Then we obtain an effective theory
on the C brane, which is the deformation of the ChernSimons theory as
C) + F (t, V ).
S = SCS (A;
(3.6)
TrRj U
S3
S0j
,
S00
(3.7)
where Sij is the modular transformation matrix for the characters of the SU (N) current
algebra at level k. If we know hTrRj U i for all the representations, we can compute any
product of traces of U n in the fundamental representation by using the Frobenius relation,
X
(Y ; , n1 , . . . , nh ) TrR(Y ) U,
(3.8)
tr U n1 tr U nh =
Y
where W is the finite Weyl group of SU (N), (w) = 1 is the parity of the element w
W, j is the weight vector for the representation j , and is a half of the sum of positive
roots. Therefore S0j /S00 takes the form of the character of the finite dimensional group
SU (N), namely
(3.10)
TrRj U S 3 = TrRj U0 ,
where U0 is a fixed element of SU (N) which, in the fundamental representation, takes the
form
i(N1)
k+N
U0 =
(N3) i
k+N
0
0
0
0
0
0
..
.
0
0
0
0
i(1N)
e k+N
427
(3.11)
Since U0 in (3.10) is the same for any Rj , to evaluate correlation functions of the Wilson
loops, we can simply replace U by the c-number matrix U0 as
(3.12)
tr U n1 tr U n2 tr U nh = tr U0n1 tr U0n2 tr U0nh ,
and
tr U0n =
sin
sin
nN
k+N
n
k+N
= i
ent /2 ent /2
,
2 sin(n/2)
where and t are as defined in (2.1). Substituting this back into (3.4), we find
"
#
nt /2
X
e
ent /2
n
tr V
Z(U, V ) S 3 = exp i
.
2n sin(n/2)
(3.13)
(3.14)
n=1
As we will see, this is exactly the form we expect for the topological string on the resolved
conifold geometry.
3.3. Conifold transition of the Wilson loop
In the case of the unknot, it is straightforward to identify the effect of the conifold
transition on the Lagrangian submanifold C. Let us start with T S 3 expressed as (2.10),
and consider the following anti-holomorphic involution on it.
y1,2 = y1,2 ,
y3,4 = y3,4.
(3.15)
Since the symplectic form changes its sign under the involution, its fixed point set is
automatically a Lagrangian submanifold of T S 3 . This will be our C. If we write y =
x + ip , the invariant locus of the action (3.15) is
p1,2 = 0,
x 3,4 = 0
(3.16)
(3.17)
Therefore C intersects with S 3 along the equator of S 3 , i.e., the loop q(s) is the unknot.
The loop q(s) in this case is identified with
q(s):
(x 1 )2 + (x 2 )2 = a 2 ,
x 3 = x 4 = 0.
To define C after the conifold transition, we continue to identify it with the invariant
locus of the anti-holomorphic involution. To describe this explicitly let us use the
coordinates (u, v, z) or (u,
v,
z1 ) defined by (2.13) and (2.15). In these coordinates, the
Z2 invariant set C is characterized by
u = v,
v = u,
(3.18)
428
Fig. 3. After the conifold transition, the Lagrangian 3-cycle touches the base S 2 along the equator
|z| = 1 and extends in the fiber directions following the constraint u + zv = 0. The worldsheet
instanton can either wrap the northern hemisphere, as shown in the figure, ending on the equator,
or wrap the southern hemisphere.
(3.19)
(3.20)
Because of (3.19), z is pure phase. Therefore one may view that C is a line bundle over the
equator |z| = 1 of S 2 (the fiber being the subspace of O(1)+O(1) given by u+zv = 0).
In particular, C intersects with the base S 2 along |z| = 1. See Fig. 3. Since the intersection
is one-dimensional, C remains a Lagrangian submanifold even after the S 2 is blown up and
the symplectic form is modified.
According to the conjecture of [3], topological string with N D-branes wrapping on the
base S 3 of T S 3 is equivalent to topological string on the resolved conifold without Dbranes. Here we are adding M D-branes on C on one side, and have traced it over to the
other side. On the T S 3 side, the effective theory on the M probe branes was the Chern
Simons action plus some corrections (3.6). So the conjecture gives the falsifiable prediction
that, after the conifold transition, we should also see the effective theory on the brane to be
a deformed version of the ChernSimons theory (3.6). Indeed it has been shown in [4] that
when there are holomorphic maps from Riemann surfaces with boundaries to the target
space, with boundaries lying on the D-brane, the ChernSimons action gets deformed. In
the original geometry of T S 3 , there are no such maps. However we got the deformation
by integrating the gauge theory on S 3 and the scalar field living on the knot. At large N ,
we have made a transition to a new geometry without any other sectors. But now, there are
non-trivial holomorphic maps that can end on C! Since C intersects with the base S 2 of the
429
resolved conifold along |z| = 1, there are holomorphic maps from Riemann surface with a
boundary with the image having the topology of disc. It is shown in [4] that the effective
theory on C should now be of the form as predicted in (3.6) with F (t, V ) given by
F (t, V ) =
X
X
(3.21)
Here Fg;n1 ,...,nh is the topological string amplitude on a genus-g surface with h boundaries.
The factors tr V ni are picked up by the boundary of the worldsheet, which wraps |ni |-times
the equator of the S 2 either clockwise ni > 0 or counterclockwise ni < 0 depending on
whether the worldsheet is mapped to the upper or the lower hemisphere.
To see that the ChernSimons computation (3.14) agrees with this expectation, we note
that, in the topological string computation, amplitudes are assumed to be analytic in t. By
performing the analytic continuation 5 , we can rewrite hZ(U, V )i as
"
#
X tr V n + tr V n
nt /2
e
.
(3.22)
hZ(U, V )iS 3 = exp i
2n sin(n/2)
n=1
This agrees with the general form (3.21) expected for the topological string amplitude. We
could make a more quantitative comparison by counting holomorphic maps. There are only
two basic holomorphic maps (with the image being a disc) with boundaries on C, which are
the upper and the lower hemispheres of the S 2 , together with their multicoverings, and with
the higher genus coverings of them (see Fig. 3). In particular, the comparison of (3.21) and
(3.22) suggests that all the relevant instantons have one boundary ending on C, wrapping
the equator of S 2 either clockwise or counterclockwise. It would be interesting to verify
the prediction of the ChernSimons computation (3.22) explicitly using the worldsheet
instanton calculus extending the results from the closed string case to open strings. In this
paper, we will take an alternative route, by giving the target space interpretation of F (t, V )
and evaluate it explicitly by the Schwinger-type computation, similar to what was done in
the closed string case in [10]. We will find that the prediction (3.22) is precisely reproduced
in this way.
any integer a.
430
where the integral is on the 4d N = 2 superspace, ti denote a vector multiplet with the
Khler expectation values as the top element. W denote the graviphoton multiplet (with
self-dual graviphoton field strength as the top component), , denote symmetric spinor
indices and
0
W 2 = W W 0 0 .
In fact it was through this connection where Fg (ti ) were reinterpreted in [10] in terms of
spectrum of wrapped M2/D2 branes in the CalabiYau threefold. In particular it was shown
that
X
2g2
Fg (ti )
NQ,s 2 sin(n/2)
2s2 entQ
,
n
where tQ = Q k is the area of the cycle and NQ,s denotes the (net) number of M2 brane
bound states of charge Q and SU (2)L content [2(0) + ( 12 )]s (for more detail see [10]).
This was obtained by computing the effective one-loop Schwinger-type correction to the
2 F 2g2 , with D2 brane bound states going around the loop [14]. The
terms of the form R+
+
sum over n above arises because every D2 brane can bind exactly once to an arbitrary
number of D0 branes, i.e., every M2 bound state can have arbitrary momentum around
the circle. In other words the sum over arbitrary number of D0 branes gives rise to a
delta function, which effectively replaces the Schwinger time integral by a discrete sum
represented by n above. The factor of (2 sin(n/2))2s2 arises from a (2 sin(n/2))2s
having to do with the extra contribution of a states of spin content [2(0) + (1/2)]s running
around the loop in the Schwinger computation, as compared to a spin 0 which would give
(2 sin(n/2))2 .
We would like to repeat an analogous scenario for reinterpretation of topological Amodel with D-branes which include a supersymmetric 3-cycle in the internal CalabiYau
threefold as its worldvolume. There are various cases one can consider. We will consider
in particular the type IIA compactification on a CalabiYau, with one additional D4 brane
wrapped around a supersymmetric 3-cycle S and filling an R 2 R 4 in the uncompactified
spacetime. Suppose the first Betti number of S is r = b1 (S). Then on R 2 subspace of R 4
live r (2,2) supersymmetric chiral superfields i corresponding to the scalar moduli of S
in the CalabiYau threefold [15]. The top component of this chiral field is a complex field
whose phase is related to the expectation value of the Wilson line of the gauge field A on the
D4-brane around the corresponding 1-cycle of S. Moreover, i can be viewed as a (2, 2)
vector multiplet on R 2 . The U (1) gauge field on R 2 corresponds to the magnetic 2-form B
field on the D4 brane dB = dA and taking the component of B along the corresponding
cycle in S, to yield a gauge field on R 2 . One could generalize this by considering M copies
of the D4 brane. We will get in this case M copies of the U (1) gauge field and so the
fields i will be naturally in the adjoint of U (1)M . The permutation groups SM which
is the symmetry of D4 branes acts on this set of fields to permute the i . Giving vev to
hi i = i allows us to think of each i-direction a diagonal U (M) matrix of holonomy
given by diagonal elements exp(ii ). Let us denote this U (M) matrix by Vi .
431
Now we are ready to state what physical amplitude the topological string computes in
the presence of D-branes. The topological strings in this case computes
Z
g
(4.2)
d4 x d4 2 (x) 2 ( )Fg,h (Vi , t) W 2 (W v)h1 ,
X
nij
Fg,ni (t)
h
Y
=1
tr
bO
1 (S)
ni
Vi ,
(4.3)
i=1
and Fg,ni (t) denotes the topological string amplitude at genus g with h holes, labeled by
= 1, . . . , h and where on each hole the circle on the Riemann surface is mapped to
the boundary of S characterized by the H1 (S) class ni . The trace factors above are just
the usual ChanPaton factors. The derivation of (4.2) is similar to that for the closed string
case and can be done most conveniently in the Berkovits formalism [16], similar to what
was done in the closed string case for the CalabiYau topological amplitudes in [17].
As in the closed string case, we would like to connect (4.2) with contributions due to
wrapped D2 branes. The main additional ingredient in this case is that the D2 brane can
end on the D4 brane S. This will give a state magnetically charged under the U (1)M living
on the D4 brane. One term included in (4.2) after doing the superspace integral is a term
of the form RF 2g2+h where F denotes the expectation value of the 4d graviphoton field
strength restricted to the uncompactified worldvolume of the D4 brane. If we give a vev to
R
the graviphoton, hF i = , this would compute correction to R as a function of
X
Fg,h (t, Vi )2g2+h .
F (t, Vi ) =
g,h
This is the summed up version of the topological string amplitudes over all genera and
holes, where the role of the string coupling constant is played by the vev of F .
We thus compute the contribution of magnetically charged D4 branes ending on S to
R
R in the presence of the background F . Each such particle will transform according
to some representation R for i Ui (1)M /SM , where i runs from i = 1, . . . , b1 (S). We
in principle do not know if they form representation of U (M) (for each element of
b1 (S)) 6 , but nevertheless we can assign them to representations of U (M) if we allow
negative multiplicity. This is because any SM symmetric spectrum for U (1)M can be
written as combination of weights appearing in various representations of U (M). From
this point on, we will therefore take R to be a representation of U (M) (for each b1 (S))
and allow negative multiplicities. In addition every such state is characterized by its bulk
D2 brane charge Q H2 (M, S), i.e., a 2-cycle in the CalabiYau threefold ending on S.
Every such field will be represented by some spin s field in 2 dimensions, where 2s is
6 There is a priori no reason why M coincident branes give rise to a magnetic U (M) gauge theory.
432
X Z d
1 is
NR,Q,s 2 sin( /2)
e
Tr e(mR,Q +i2n) .
n= R,Q,s +
0
Here NR,Q,s denotes the net number of magnetically charges states with charges given by
R, Q and spin s. The parameter is the Schwinger time, the sum over n is the sum over
the D0 brane bound states, and mR,Q + i2n denotes the BPS mass of the wrapped D2
brane, which is given by
Tr emR,Q +i2n = etQ +i2n TrR
bY
1 (S)
Vi ,
i=1
R
where tQ = Q k. To see how the above expression arises, note that for one D4 brane
R
this follows from the fact that Q k is just the bulk contribution to the BPS formula and
Qb1 (S)
Vi arises from the fact that giving vev to the U (1) fields for each one gives a
TrR i=1
BPS mass q, where q is the charge under U (1) and denotes the vev of a scalar in the
P
U (1) multiplet. Doing the sum over n in the above gives a delta function
n= ( n),
which converts the integral into a sum, and we obtain
F (t, Vi ) = i
X
X
n=1 R,Q,s
NR,Q,s 2n sin(n/2)
1
bY
1 (S)
Vin .
(4.4)
i=1
Note that to compare it with (4.3) one has to expand the trace from representation R in
terms of fundamental representation of U (M). Note that the above expression has strong
integrality predictions which would be interesting to verify.
Note that for the special case of g = 0, h = 1, i.e., the disc amplitude (4.2) computes
theta terms in gauge theory. Namely for each diagonal element of Vi , denoted by exp(ii )
the term
F0,1
(t, Vi ),
i
R
denotes the correction to the theta term Fi where Fi denotes the field strength for the
corresponding U (1) gauge field in 2d. From (4.4) we can read the prediction for this, which
is given by
433
1
NR,Q,s qi en(tQ +ivi i )
n
n=1 vi R,Q,s
X
= i
NR,Q,s qi log 1 etQ ivi i
vi R,Q,s
= i
m= vi R,Q,s
which is the expected correction to the theta angle in 2d N = 2 gauge theory from charged
matters with BPS masses tQ + iv + 2im and charge qi (see in particular a similar
correction which was studied in [18]). Note that from (4.4) we can write F0,1 in the form 7 ,
F0,1 (t, Vi ) = i
X
X
NR,Q,s
n=1 vR,Q,s
1 n(tQ +iv)
e
.
n2
(4.5)
X
X
n=1 vR,Q,s
NR,Q,s
1 n(tQ +iv)
e
.
n2
(4.7)
A special simple case of this is when we have a single brane where Vi can be viewed
as a complex superfield eii . Given that our derivation of this term seems to require
2-dimensional concepts, it is natural to ask if we could also reproduce this from a 4dimensional viewpoint. As we will see this is also possible. In the case of D6 branes with
worldvolume R 4 S, the magnetically charge branes are D4 branes ending on the D6
7 As before we are dropping terms polynomial in t and s which would have corresponded to n = 0 in the
above sum.
434
brane. This will correspond to a domain wall in R 4 . The expression (4.7) then suggests
that we should be able to relate the superpotential term, to the structure of domain walls
by integrating them out. However unlike the 2-dimensional case, we cannot send the
domain walls around the loop, so the question is how would we obtain such an expression
by integrating fields out in the 4d case.
A hint comes from the recent work [21] and a similar case studied in [22], where it
was shown how extra fields are relevant for reproducing the domain wall structure. For
each domain wall, we introduce a field Y as a chiral superfield, which characterizes it by
shifting by 2i as we go across the domain wall. Since we can have a priori an arbitrary
number of domain walls, we must thus have infinitely many vacua, given by shifting the
expectation value of Y by 2in. Moreover the tension for the domain wall should be given
by the BPS formula,
W (Y + 2i) W (Y ) = 2iZ.
(4.8)
The superpotential satisfying these constraints which was found in [21] in a similar context
is given by
W = Z Y + exp(Y ).
Note that the critical points obeying dY W = 0 are given by
exp(Y ) = Z ,
namely Y = log Z + 2in, and that the equation (4.8) is satisfied. If we integrate out
the hidden variable Y we obtain by replacing Y by its critical value a superpotential term
W = Z (1 logZ ).
(4.9)
In the case at hand for each element v in a representation v R of the magnetic charges,
and charge Q in the bulk we have NR,Q,0 net BPS domain walls for each integer m, with
BPS tension
Z = tQ + iv + 2im.
Plugging this into (4.9) and summing over all such states, we obtain the formula (4.7).
4.2. Application to D-brane on O(1) + O(1) over S2
In this section we will show how the results of the previous section are in agreement
with the above analysis, and in particular gives an independent derivation for topological
A-model in O(1) + O(1) over S 2 with the 3 cycle C we have discussed. In that case
the b1 (C) = 1 and so we have only one chiral field, which gives rise to one holonomy
matrix V . It is clear what the magnetically charges states are; they correspond to the D2
brane wrapping the northern hemisphere and ending on S or the one wrapping the southern
hemisphere and ending on S. The first one has (tQ , R) given by (t/2, fundamental) and
435
the second has (t/2, anti-fundamental). They both have spin 0, as there is no moduli for
them. We thus obtain from (4.4):
F (t, V ) = i
X
tr V n + trV n
n=1
2n sin(n/2)
ent /2 ,
which agrees with the knot invariant predicted for the unknot, as discussed in the previous
section, with t = 2iN/(k + N) and = 2/(k + N).
This is already rather difficult to do, even though in principle it should be possible. The
reason for this is the appearance of all the powers of tr U n . In particular we need to know all
correlations htr U n1 tr U nk i. For a general knot, the correlators do not decouple, unlike
the unknot (3.12). Even though it is in principle possible to compute them, they have not
been computed in the full generality we need. Nevertheless the structure of the answer
for the htr U trU i dictated by the Skein relations [11] are compatible with the general
answer expected for the knot invariants, which follows from the discussion in the previous
section, in particular (4.4). Note that we are mapping all the knot invariants for arbitrary
representations, into new integer invariants NQ,R,s , where two different Qs differ by an
integer (so they can be parametrized by an integer), s denotes a positive (or zero) spin
representation, and R is a representation of U (M) for any M. We expect that for each
knot R will stabilize for large enough M. What we mean by this is that it will given by
representations with finite number of boxes in the Young tableau (or whose conjugate has
finite number of boxes). Thus for M large enough we have probed the full content of
R representation (for example for the unknot we found that already M = 1 is sufficient
and the structure for higher Ms can be induced from that). This may be a very useful
reformulation of knot invariants, somewhat analogous to the reformulation of Gromov
Witten invariants, in terms of the new invariants defined in [10]. In particular the knot
436
1 n(t +is)
n
Q
NR,Q,s 2n sin(n/2) e
Tr V .
Z(U, V ) S 3 = exp i
n=1 R,Q,s
For understanding this new formulation of knot invariants, we also have to construct
a Lagrangian submanifold for an arbitrary knot, on the resolution of the conifold,
generalizing our explicit construction for the unknot. That there should be such a canonical
Lagrangian submanifold for each knot is natural. This is because we already have
identified, for an arbitrary knot, the Lagrangian submanifold on the T S 3 side, and
small resolution does not change the geometry of the Lagrangian submanifold at infinity.
So with some deformation near the origin we should be able to obtain the Lagrangian
submanifold after the conifold singularity is blown up. Then we are predicting that the
topological string amplitudes, whose answer must have the structure (4.4), compute the
knot invariants. This would be a very important subject to develop, not only for a deeper
understanding of knot invariants, but also for a better understanding of topological strings
with boundaries.
Also we have seen in this paper how we can compute superpotential terms on the
type IIA compactifications on CalabiYau threefold in the presence of D6 brane, at least in
some simple cases. In a more general case, doing the computation on the mirror should be
simpler [23]. Some examples of this have been recently studied in [20]. This may well lead
to a method for geometric engineering of N = 1 and its solution in terms of the type IIB
mirror. Namely, we start with the usual geometric engineering of N = 2, introduce additional D6 branes to break the N = 2 to N = 1 (effectively giving mass terms to the adjoint
fields) and then using the type IIB mirror to compute the superpotential terms generated,
very much the way prepotential for N = 2 theories were computed using mirror symmetry.
This would be very exciting to develop further.
437
Acknowledgements
We are grateful to N. Berkovits, R. Gopakumar, K. Hori, S. Sinha, C. Taubes and
T. Taylor for valuable discussions. H.O. would like to thank the theory group of Harvard
University, where this work was initiated and completed.
The research of H.O. was supported in part by NSF grant PHY-95-14797, DOE grant
DE-AC03-76SF00098, and the Caltech Discovery Fund. The research of C.V. is supported
by NSF Grant No. PHY-9218167.
References
[1] V. Periwal, Topological closed-string interpretation of ChernSimons theory, Phys. Rev. Lett. 71
(1993) 1295.
[2] M.R. Douglas, ChernSimonsWitten theory as a topological Fermi liquid, hep-th/9403119.
[3] R. Gopakumar, C. Vafa, On the gauge theory/geometry correspondence, hep-th/9811131.
[4] E. Witten, ChernSimons gauge theory as a string theory, hep-th/9207094.
[5] C. Faber, R. Pandharipande, Hodge integrals and GromovWitten theory, math.AG/9810173.
[6] P. Candelas, X.C. De La Ossa, P.S. Green, L. Parkes, A pair of CalabiYau manifolds as an
exactly soluble superconformal theory, Nucl. Phys. B 359 (1991) 21.
[7] S. Hosono, A. Klemm, S. Theisen, S.-T. Yau, Mirror symmetry, mirror map and applications to
complete intersection CalabiYau spaces, Nucl. Phys. B 433 (1995) 501; hep-th/9406055.
[8] M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa, Holomorphic anomalies in topological field
theories, Nucl. Phys. B 405 (1993) 279; hep-th/9302103.
[9] M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa, KodairaSpencer theory of gravity and exact
results for quantum string theory, Commun. Math. Phys. 165 (1994) 311; hep-th/9309140.
[10] R. Gopakumar, C. Vafa, M-theory and topological strings I, II, hep-th/9809197; hepth/9812127.
[11] E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989)
351.
[12] D.J. Gross, W. Taylor, Twists and Wilson loops in the string theory of two-dimensional QCD,
Nucl. Phys. B 403 (1993) 395; hep-th/9303026.
[13] I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor, Topological amplitudes in superstring theory,
Nucl. Phys. B 413 (1994) 162; hep-th/9307158.
[14] I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor, N = 2 type II heterotic duality and higher
derivative F -terms, Nucl. Phys. B 455 (1995) 109; hep-th/9507115.
[15] A. Strominger, S.-T. Yau, E. Zaslow, Mirror symmetry is T-duality, Nucl. Phys. B 479 (1996)
243; hep-th/9606040.
[16] N. Berkovits, private communication.
[17] N. Berkovits, C. Vafa, N = 4 topological strings, Nucl. Phys. B 433 (1995) 123; hepth/9407190.
[18] A. Hanany, K. Hori, Branes and N = 2 theories in two dimensions, Nucl. Phys. B 513 (1998)
119; hep-th/9707192.
[19] I. Brunner, M.R. Douglas, A. Lawrence, C. Romelsberger, D-branes on the quintic, hepth/9906200.
[20] S. Kachru, S. Katz, A. Lawrence, J. McGreevy, Open string instantons and superpotentials, to
appear.
[21] K. Hori, C. Vafa, Mirror symmetry, to appear.
[22] S. Gukov, C. Vafa, E. Witten, CFTs from CalabiYau four-folds, hep-th/990670.
438
Abstract
When the gauge groups of the two heterotic string theories are broken, over tori, to their SO(16)
SO(16) subgroups, the winding modes correspond to representations which are spinorial with
respect to those subgroups. Globally, the two subgroups are isomorphic neither to SO(16) SO(16)
nor to each other. Any attempt to formulate the T-duality of the two theories on topologically nontrivial compactification manifolds must therefore take into account various generalizations of the
spin structure concept. We give here a global formulation of T-duality, and show, with the aid of
simple examples, that two configurations which appear to be T-dual at the local level can fail to be
globally dual. 2000 Elsevier Science B.V. All rights reserved.
PACS: 11.15.Kc; 11.25.-w
Keywords: Duality; Spin structures
1. Introduction
A Riemannian structure on a smooth manifold M is a reduction [1] of its bundle of
linear frames to an O(n) bundle O(M), where O(n) is the group of n n orthogonal
matrices. This group is neither connected nor simply connected; however, it can in effect be
replaced by simpler groups if M satisfies certain topological conditions. If O(M) reduces
to a bundle with a connected structural group, then M acquires an orientation, and if
this connected group lifts to its universal cover, then M is said to have a spin structure.
These simplifications are of course not always possible: the O(4) associated with the
real projective space RP 4 cannot be reduced to SO(4), and the SO(4) associated with
the complex projective space CP 2 cannot be lifted to its non-trivial double cover Spin(4).
Gauge theory can be regarded as a generalisation of Riemannian geometry, in which
one considers arbitrary principal G-bundles [1] on M, instead of O(M). As in the
Riemannian case, G need be neither connected nor simply connected, and so one might
expect to confront problems analogous to the existence of orientations and spin structures.
matmcinn@nus.edu.sg
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 0 2 5 - 0
440
In fact, this analogy has not attracted a great deal of attention, partly because gauge
theory itself does not accord preference to any one of the Lie groups with a given Lie
algebra. If we had reason to believe, for example, that the gauge group of SO(10) grand
unification is really O(10), then indeed it would be of interest to investigate O(10)
gauge configurations (over necessarily topologically non-trivial base manifolds) which
cannot break to SO(10), and SO(10) configurations which cannot lift to Spin(10). (Such
configurations do exist.) The point, however, is that we have, a priori, no reason to prefer
O(10) to SO(10) or Spin(10) or Pin(10) or any of the other [2] Lie groups with this Lie
algebra. This is where the analogy with Riemannian geometry appears to break down.
The advent of string theories [3], however, has changed this situation. These theories
are so severely constrained that they do in fact dictate the precise global structures of
their gauge groups. In the case of the E8 E8 heterotic theory, the gauge group is
(E8 E8 ) G Z2 , where G denotes a semi-direct product, and Z2 acts by exchanging the
two E8 factors. (See, for example, Ref. [4] for a discussion of the significance of this
Z2 .) Thus, the gauge group is disconnected, and so a property analogous to orientability
must be considered. However, E8 E8 is simply connected, so there is no analogue of
spin structures here. By contrast, the full gauge group of the SO(32) heterotic theory
is the group usually denoted by Spin(32)/Z2 . Unlike E8 E8 (and SO(32)) this group
has no outer automorphism, and so [5] it has no non-trivial disconnected version; thus,
questions of orientability do not arise here. However, Spin(32)/Z2 is obviously not
simply connected, so the possible non-existence of spin structures is an issue [6].
We see, then, that the two heterotic string theories do require us to consider the
consequences of having topologically non-trivial gauge groups. The two theories appear,
however, to behave in opposite ways, with one gauge group being disconnected but with
a simply connected identity component, while the other is connected but not simply
connected. That such appearances are deceptive is, of course, the lesson of the duality
revolution [7]. The objective of this work is to understand the role of these two specific
kinds of topological non-triviality in maintaining (or obstructing) the T-duality between
the two heterotic string theories. (Note that, throughout this work, we interpret T-duality
in a very broad sense. Any mapping of the gauge and matter fields of one heterotic theory
to those of the other will be called T-duality here.)
The two heterotic theories have, of course, different gauge groups, and these groups
lead, as above, to different topological complications. The two Lie algebras, however, have
much in common: in particular, they have a common maximal, maximal-rank subalgebra
isomorphic to the algebra of SO(16) SO(16). The T-duality between the two heterotic
theories has to be established through (E8 E8 ) G Z2 and Spin(32)/Z2 configurations such
that each group is broken to the common SO(16) SO(16) subgroup. The problem here,
as was pointed out in Ref. [5], is that this common subgroup does not exist the two
SO(16) SO(16) algebras exponentiate to completely different subgroups. These are
[(Spin(16)/Z2 ) (Spin(16)/Z2 )] G Z2 (E8 E8 ) G Z2
and
[(Spin(16) (Spin(16))/(Z2 Z2 )] G Z2 Spin(32)/Z2 ,
where, in both cases, the semidirect product with Z2 is defined through the exchange of
the two local factors. Notice that the subgroups are certainly more closely related than the
441
original pair: they have the same Lie algebra, they both have two connected components,
and they both have Z2 Z2 as the fundamental group of the identity component; but they
are not isomorphic. However, it was also shown in Ref. [5] that they have a common double cover, which crucially is not the universal cover. T-duality, therefore, can only be
established at that level. This assumes, however, that the relevant gauge configurations obtained from breaking (E8 E8 ) G Z2 and Spin(32)/Z2 can indeed be lifted to this common
double cover. This, again, involves a gauge-theoretic analogue of the question of the existence of spin structures. Clearly, a full understanding of T-duality will require an analysis
of the theory of gauge spinors for these groups. It will also be useful to have concrete examples of (E8 E8 ) G Z2 and Spin(32)/Z2 configurations such that the above subgroups
do not lift to the double cover: cases in which T-duality is topologically obstructed.
We begin with a general analysis of the problem of comparing two distinct gauge
theories over a given manifold. Then we turn to the details of the specific groups involved
in string theory, before giving a general formulation of the topological aspects of T-duality.
We conclude with several very simple concrete examples which show that it is necessary
to take certain obstructions into account when discussing T-duality over topologically nontrivial spacetimes.
442
3. Gauge spinors
e be some
Let H be a connected, compact, non-simply connected Lie group, and let H
e
non-trivial finite cover (not necessarily the universal cover) of H , so that H has a finite
443
e/N = H . Let Q
e be a principal H
e-bundle over a manifold M;
central subgroup N with H
e
e then it may
then Q = Q/N
is a principal H -bundle over M. If f is a matter field on Q,
or may not be invariant with respect to N ; this will be determined by the representation in
which it takes its values. We shall say that f is vectorial with respect to H if it is invariant
with respect to N , and spinorial with respect to H otherwise. Clearly f descends to a
matter field on Q if and only if it is vectorial.
In grand unification [10], a given gauge group (usually the standard group) H is
regarded as a subgroup of some larger group G. The objective is to gain some control
over the kinds of matter fields in the theory: only those representations of H which can
be regarded as restrictions of G-representations are accepted. Of course, by the Peter
Weyl theorem, all H -representations are restrictions of some G-representation, but in
practice one requires the G-representation to be small. In addition, however, grand
unified theories have the effect of ruling out H gauge spinors. For if is a representation
e which does not descend to a representation of H , then there is no sense in which
of H
can be said to arise from any representation of G. For example, the unbroken gauge
group of the standard model [11] is the unitary group U (3), which is triply covered by
U (1) SU(3). The usual representations [11] of U (3) appearing in the standard model
can all be obtained from (small) SU(5) representations, but the opposite is true of those
(spinorial) representations of U (1) SU(3) which do not descend to representations of
U (3). The inclusion of matter fields defined by representations of that kind would defeat
the whole point of grand unification.
Now it is essential to understand that the gauge groups of (heterotic) string theories
are not grand unification groups in the usual sense; unification in string theory
is much more subtle. Consider, for example, either E8 in (E8 E8 ) G Z2 . This
group has a maximal, maximal-rank subgroup globally isomorphic to Spin(16)/Z2 .
In the fermionic formulation of the E8 theory, one has sixteen left-handed fermions
transforming as the 16 of Spin(16)/Z2 . However, no such representation exists
the 16 is a representation of Spin(16), but not of Spin(16)/Z2 . In other words, these
fields are spinorial with respect to Spin(16)/Z2 . This kind of behaviour, which cannot
occur in a true E8 grand unified theory, is of vital importance in string theory. Most
importantly for our purposes, when the heterotic theories are compactified on tori
or toral orbifolds, the winding modes [3] which play a crucial role in T-duality are
in fact gauge spinors with respect to certain groups. We shall return to this point
below.
In view of these remarks, we need to modify the formalism of the preceding section to
allow for the possibility that G1 and G2 have subgroups which are locally but not globally
isomorphic. That is, the respective subgroups, H1 and H2 , are not isomorphic, but they
have the same Lie algebra. Then there exists a group H with finite central subgroups N1
and N2 , such that Hi = H /Ni . (Note that H is not a subgroup of either G1 or G2 , and
that it need not be simply connected.) Suppose as before that P1 is a principal G1 bundle
admitting an H1 subbundle Q1 . Assume now that Q1 has a non-trivial N1 cover Q which
is an H bundle over the same base. Then Q2 = Q /N2 is a well-defined H2 bundle, and
Q2 can be extended as usual to a G2 bundle P2 . We have a pair of bundles reducing to
subbundles with a common gauge spinor bundle.
444
1 = Ad((h)) 1 = Ad(h) 1 ,
Rh 1 = ( Rh ) 1 = R(h)
445
446
447
makes sense to speak of mapping one to another under T-duality. Furthermore, since the
status of a given matter field on Q (as a gauge vector or a gauge spinor) is dependent on
whether one takes the Semispin(32) or the (E8 E8 ) G Z2 point of view, the characteristic
exchange of winding with non-winding modes can be easily accommodated. Finally, the
assumption that Q exists, to which we must of course return, in no way requires P1 to lift
to a Spin(32) bundle, and so we can handle DMW-like configurations.
Some concrete examples may be helpful. The adjoint of Semispin(32) coincides
with that of SO(32), which decomposes under SO(16) SO(16) as 496 = (16, 16)
(120, 1) (1, 120), where 120 is the adjoint of SO(16). The tensor product (16, 16) is
a legitimate representation of Spin(16) : Spin(16), though (16, 1) (1, 16) would not
be; for (I16 , I16 ) SO(16) SO(16) is in the kernel of (16, 16) but not that of
(16, 1) (1, 16), while K 16 :K 16 , the corresponding element of Spin(16) : Spin(16), equals
the identity. A matter field f1 on Q taking its values in (16, 16) is therefore a gauge
vector from the Semispin(32) point of view. On the other hand, a matter field h1 on Q
taking its values in the half-spin representation (128, 1) (1, 128) (which is a legitimate
representation of Spin(16) Spin(16)) will satisfy
h1 R(1)(1) = h1 ,
where Rg denotes the action of g Spin(16) Spin(16) on Q , so h1 is a gauge
spinor for Semispin(32); such fields will arise as winding modes when the corresponding
string theory is compactified on a torus. From the E8 E8 side, we can also have
a gauge vector h2 taking its values in (128, 1) (1, 128) (this being a legitimate
representation of Semispin(16) Semispin(16)) and a winding mode a gauge spinor
f2 , taking its values in (16, 16) (which is a representation of Spin(16) Spin(16)
but not of Semispin(16) Semispin(16)). In addition, we can have matter fields g1 , g2
from both sides, taking values in (120, 1) (1, 120); as this is a representation of both
Spin(16) : Spin(16) and Semispin(16) Semispin(16), g1 and g2 are gauge vectors from
both points of view. (The same would be true of matter fields, if any, taking values in
(128, 128).) Then T-duality acts by exchanging f1 f2 , g1 g2 , h1 h2 , and so on.
The details need not concern us here: the point is that we now have a meaningful global
formulation of T-duality. That is, the fields being exchanged live on the same manifold,
Q , and they take values in the same representations despite the fact that some of
these representations do not make sense for the original groups obtained by Wilson-loop
breaking of (E8 E8 ) G Z2 and Semispin(32). The fact that the respective SO(16)
SO(16) subgroups are not isomorphic leads to an ambiguity in the vector/spinor status of
matter fields but this ambiguity proves to be nothing but a facet of T-duality.
One question remains, however: does Q actually exist?
Sn
448
H 1 (M, Z2 ).
x1 (P ) = fP J16
If (and only if) x1 (P ) = 0, then by the exactness of the above sequence, H 0 (P , Z2 ) 6= 0,
that is, P is disconnected. Then P can be reduced to a Spin(16) : Spin(16) subbundle. This
is of course the analogue of orientability for this group.
Suppose that x1 (P ) = 0, so that, in effect, P is a Spin(16) : Spin(16) bundle. The
fibration Spin(16) : Spin(16) P M yields [12] another exact sequence,
0 H 1 (M, Z2 ) H 1 (P , Z2 ) H 1 (Spin(16) : Spin(16), Z2 ) H 2 (M, Z2 ).
Now Spin(16) : Spin(16) is defined by factoring Spin(16) Spin(16) by its subgroup
{(1, 1), (1, 1), (K 16, K 16 ), (K 16 , K 16 )}. Hence we can regard this as the fundamental group of Spin(16) : Spin(16), and, since it is isomorphic to Z2 Z2 , this is also
H 1 (Spin(16) : Spin(16), Z2 ). Thus, again denoting the last homomorphism in the exact sequence by fP , we define
w 2 (P ) = fP K 16 , K 16 H 2 (M, Z2 ),
x2 (P ) = fP ((1, 1)) H 2 (M, Z2 ).
(Since Spin(16) Spin(16) has an outer automorphism mapping (K 16 , K 16 ) to
(K 16 , K 16 ), fP ((K 16 , K 16 )) gives nothing new.) Now we have the following isomorphisms, in which the notation is self-explanatory:
Spin(16) : Spin(16) = (Spin(16) Spin(16))/ 1 1, K 16 K 16
= (Spin(16) Spin(16))/{1 1, (1) (1)}.
H 1 (P , Z2 )
449
In the same way, x2 (P ) = 0 means that P has another non-trivial double cover P ,
which is a Spin(16) Spin(16) bundle over M. We shall say in this case that P admits
an exceptional structure, because P /{1 1, K 16 1} is a Semispin(16) Semispin(16)
bundle which can be extended to an E8 E8 bundle.
We can now complete our global formulation of heterotic T-duality. We begin with a
Semispin(32) bundle P1 admitting a (Spin(16) : Spin(16)) G Z2 subbundle Q1 . We require
x2 (Q1 ) = 0.
Let Q be a non-trivial double cover of Q1 with structural group (Spin(16) Spin(16)) G
Z2 . Then Q is automatically a non-trivial double cover of a certain (Semispin(16)
Semispin(16)) G Z2 bundle Q2 = Q /{1 1, K 16 1}, which may be regarded as a
subbundle of an (E8 E8 ) G Z2 bundle P2 . As explained in the preceding section, Q
is the arena for the exchanges defining T-duality.
Several remarks should be made at this point. The first and most important is that the
condition x2 (Q1 ) = 0 will not, of course, always be satisfied for all base manifolds M. It
is satisfied for all bundles over R9 S 1 , since H 2 (R9 S 1 , Z2 ) = 0, but on more complex
manifolds we can certainly find Semispin(32) configurations such that Q1 does not lift to
any Q . In such a case, the configuration has no globally well-defined E8 E8 partner:
we can say that there is a topological obstruction to T-duality. This is a failure of T-duality
(see Ref. [16]) which would pass undetected at the local level. Notice that x2 and w 2
take values in the same cohomology group, so one must beware of this possibility in any
situation in which the existence of a vector structure is questionable. On the other hand, the
two obstructions are independent: the existence of a vector structure in no way guarantees
the existence of an exceptional structure or vice versa. (The existence of two obstruction
classes is of course due to the greater topological complexity of Spin(16) : Spin(16)
compared with SO(32) or Semispin(32).)
Secondly, we have throughout assumed that one begins with a Semispin(32) bundle and
works towards (E8 E8 ) G Z2 . However, one is entitled to begin with an (E8 E8 ) G Z2
bundle P2 admitting a (Semispin(16) Semispin(16)) G Z2 subbundle Q2 . There is an
element y1 (Q2 ) H 1 (M, Z2 ) which obstructs the reduction of Q2 to a Semispin(16)
Semispin(16) bundle. Assuming that y1 (Q2 ) = 0 and again calling the reduced bundle Q2 ,
we can ask whether Q2 lifts to a Spin(16) Spin(16) bundle Q ; the obstruction is a certain
element y2 (Q2 ) H 2 (M, Z2 ). If indeed y2 (Q2 ) = 0, then Q exists, and we can define
Q1 = Q /{1 1, (1) (1)},
a Spin(16) : Spin(16) bundle, and so T-duality goes through. We stress that w 2 , x2 , and
y2 are all distinct from each other and from the standard StiefelWhitney class w2 . The
latter would be relevant to the study of SO(16) SO(16) bundles lifting to Spin(16)
Spin(16) bundles; this may not seem to be directly relevant, since SO(16) SO(16) is not
a subgroup of either of the heterotic gauge groups. However, w2 also governs the lifting of
[SO(16) SO(16)]/Z2 bundles to Spin(16) : Spin(16) bundles (see the diagram in Ref. [5])
and so, since [SO(16) SO(16)]/Z2 is contained in the projective special orthogonal group
PSO(32), and the latter is the structural group of the ChanPaton bundles of open string
theory, w2 is important in Type I theory (and possibly in establishing the S-duality of Type I
with the heterotic Semispin(32) theory [17]).
450
Notice that the vanishing of the obstructions x2 and y2 does not in itself suffice to
establish T-duality. If Q1 and Q2 are bundles with x2 (Q1 ) = y2 (Q2 ) = 0, then both Q1
and Q2 have non-trivial (Spin(16) Spin(16)) G Z2 double covers, but these covers need
not be the same. If they are different, then T-duality cannot be implemented. This is why
we insist that if Q1 is a (Spin(16) : Spin(16)) G Z2 bundle with x2 (Q2 ) = 0, and Q is
the corresponding double cover, then Q /{1 1, K 16 1} is the T-dual (Semispin(16)
Semispin(16)) G Z2 bundle; for this bundle obviously does have Q as a double cover.
(Obviously y2 (Q /{1 1, K 16 1}) = 0, so x2 and y2 are related in this limited sense.)
Finally, there can be additional complications if the first cohomology group H 1 (M, Z2 )
does not vanish. It follows from the second exact sequence above that if Q1 is a
(Spin(16) : Spin(16)) G Z2 bundle with x2 (Q1 ) = 0, then the double cover Q need not be
unique: the number of possibilities is counted by |H 1 (M, Z2 )|, the order of H 1 (M, Z2 ).
Strictly speaking, then, we must label the various (Spin(16) Spin(16)) G Z2 bundles Qi ,
where i = 1 . . . |H 1 (M, Z2 )|, corresponding to a given Q1 . The winding modes might then
fall into topological sectors labelled by i. (For toral compactifications, one might interpret
these gauge spin structures in terms of boundary conditions.) The problem now is that, in
general, the (Semispin(16) Semispin(16)) G Z2 bundles
Q / 1 1, K 16 1 , i = 1 . . . H 1 (M, Z2 ),
i
will not be mutually isomorphic. This would mean that a certain given Semispin(32)
configuration might be T-dual not to one but rather to a whole collection of (E8 E8 ) G
Z2 configurations. (Note that requiring H 1 (M, Z2 ) = 0 would be very drastic, as it would
rule out most manifolds with finite fundamental groups of even order.) This situation can
be avoided only by confining all winding modes to the same gauge spin bundle.
We may summarise as follows. The fact that the SO(16) SO(16) subgroups of the
heterotic string gauge groups are not mutually isomorphic is not a problem in itself; indeed,
it plays a key role in T-duality. On the other hand, it also imposes topological conditions.
If we begin on the Semispin(32) side, this condition is that the reduced gauge bundle
Q1 must admit an exceptional structure, which means that a certain cohomology class
x2 (Q1 ), analogous to (but different from) the second StiefelWhitney class, must vanish.
If x2 (Q1 ) 6= 0, then T-duality is topologically obstructed.
Let us consider some simple examples of these phenomena.
6. Examples
There are of course many ways of constructing principal fibre bundles exhibiting various
kinds of topological non-triviality, but here we shall concentrate on three approaches which
are particularly relevant to string theory. All are straightforward and explicit.
6.1. Embedding the spin connection in the gauge group
The traditional method of constructing gauge vacua in string theory [18] is motivated
by the need to cancel anomalies. The curvature tensor of the compactification manifold
is equated to the gauge field strength, a procedure known as embedding the spin
451
connection in the gauge group. (One should not be misled by the terminology: in the
physics literature, the spin connection usually means the components of the Levi-Civit
connection with respect to an orthonormal basis. Globally, it could mean either the LeviCivit connection form or its pull-back to a spin bundle over the bundle of orthonormal
frames.) Let us examine the use of this technique in constructing various non-trivial
Semispin(32) and E8 E8 bundles.
An important string compactification manifold is the four-dimensional K3 space [19].
This manifold admits a Ricci-flat Riemannian metric. Let SO(K3) be a bundle of oriented
orthonormal frames over K3 (which is orientable). Then SO(K3) is an SO(4) bundle.
As K3 is a Ricci-flat Khler manifold, SO(K3) can be reduced to an SU(2) subbundle
SU(K3); however, SU(K3) need not be stable under the action of the (finite, but possibly
non-trivial) group of isometries of K3, so it is preferable to use SO(K3). This means that
we should embed SU(2) in the gauge group through some natural embedding of either
SO(4) or (if indeed we wish to use a spin connection in the true sense) of Spin(4).
In fact, both SO(4) and Spin(4) have natural embeddings in Spin(16) : Spin(16), as
follows. First, Spin(4) is a subgroup of Spin(16) in the obvious way, so the subgroup
{A : 1}, A Spin(4)
is isomorphic to Spin(4). On the other hand, since clearly (1) : (1) = 1 : 1, we see that
{A : A}, A Spin(4)
is isomorphic to SO(4). Let us concentrate first on this latter case, and determine the
structure of the covers of SO(4) in Spin(16) Spin(16) and Spin(16) Spin(16). We have
(1) (1) = 1 1,
(1) (1) 6= 1 1.
Thus {A A}, the cover of SO(4) in Spin(16) Spin(16), is still (perhaps surprisingly)
isomorphic to SO(4), whereas {A A}, the cover in Spin(16) Spin(16), is isomorphic to
Spin(4). Now take SO(K3), and extend it in the usual way to a Spin(16) : Spin(16) bundle.
(We shall ignore the disconnected version, since H 1 (K3, Z2 ) = 0 and so x1 (Q) = 0 for all
(Spin(16) : Spin(16)) G Z2 bundles over K3.) That is, define
Q(SO(K3)) = [SO(K3) Spin(16) : Spin(16)]/SO(4),
with SO(4) acting to the left on Spin(16) : Spin(16) as usual. We can define a Semispin(32)
bundle P (SO(K3)) in the same way. The Levi-Civit connection corresponding to a Ricciflat metric on K3 can be regarded as a connection one-form on SO(K3), and it pushes
forward to a gauge field on Q(SO(K3)) and P (SO(K3)); the gauge field strength will
coincide with the curvature form of K3; in short, we have embedded the spin (actually,
the Levi-Civit) connection in the gauge group, Semispin(32).
Now set
e
Q(SO(K3))
= [SO(K3) Spin(16) Spin(16)]/SO(4).
e
Clearly Q(SO(K3))
is a Spin(16) Spin(16) bundle over K3, and it can be extended to a
Spin(32) bundle. Now SO(4), as a subgroup of Spin(32), does not contain K 16 K 16 = K 32 ,
and so in fact
e
Q(SO(K3))/
1, K 32 = Q(SO(K3)).
e
That is, Q(SO(K3))
is a non-trivial double cover of Q(SO(K3)); it is a vector structure.
We have
w 2 (Q(SO(K3))) = 0.
452
Thus, embedding the Levi-Civit connection is Semispin(32) always yields a Semispin(32) configuration with a vector structure.
The K3 manifold is a spin manifold, that is, SO(K3) has a non-trivial double cover
Spin(K3) which is a Spin(4) bundle over K3. (This spin structure is unique.) As the SO(4)
subgroup of Semispin(32) is covered by Spin(4) in Spin(16) Spin(16), we can define
Q (SO(K3)) = [Spin(K3) Spin(16) Spin(16)]/Spin(4),
and this is a Spin(16) Spin(16) bundle over K3. As Spin(4) does contain (1) (1),
we have
Q (SO(K3))/{1 1, (1) (1)} = Q(SO(K3)),
so that Q (SO(K3)) is another non-trivial double cover of Q(SO(K3)). Evidently
x2 (Q(SO(K3))) = 0,
that is, embedding the Levi-Civit connection of K3 in Semispin(32) yields a
configuration with an exceptional structure. Thus, this configuration is globally T-dual to
an E8 E8 configuration
Q (SO(K3))/ 1 1, K 16 1
= [Spin(K3) Semispin(16) Semispin(16)]/Spin(4).
An alternative procedure is to use, instead of the Levi-Civit connection, the spin
connection in the true sense that is, the pull-back of the Levi-Civit connection to
Spin(K3). Embedding Spin(4) in Spin(16) Spin(16) as {A : 1}, we see that the covers
of Spin(4) in Spin(16) Spin(16) and Spin(16) Spin(16) are both isomorphic to Spin(4).
If therefore we define
Q(Spin(K3)) = [Spin(K3) Spin(16) : Spin(16)]/Spin(4),
then Q(Spin(K3)) is a principal Spin(16) : Spin(16) bundle over K3 with non-trivial
double covers
e
Q(Spin(K3))
= [Spin(K3) Spin(16) Spin(16)]/Spin(4),
Q (Spin(K3)) = [Spin(K3) Spin(16) Spin(16)]/Spin(4),
and so Q(Spin(K3)) is another Semispin(32) configuration with both vector and exceptional structures. Notice that, although embedding the Levi-Civit connection and embedding the spin connection both lead to Semispin(32) configurations possessing both
vector and exceptional structures, the two procedures are indeed quite different. A useful
way to show this is to note that the Levi-Civit connection breaks Spin(16) : Spin(16) to
[SU(2) Spin(12)] : [SU(2) Spin(12)] (since this is the centraliser of SU(2) with this
embedding), while the spin connection breaks it to [SU(2) Spin(12)] : Spin(16).
Apart from K3, the only known examples of compact Ricci-flat Riemannian fourdimensional manifolds are the flat manifolds and the Enriques [20] and Hitchin [21]
manifolds. The Enriques manifold is a Khler manifold of the form K3/Z2 , while the
Hitchin manifold is a non-Khler manifold, K3/[Z2 Z2 ]. Neither has attracted as much
interest as K3, partly because neither is a spin manifold; see Ref. [22] for a discussion
of this fact. (As a general rule, it is difficult for compact, locally irreducible, Ricci-flat
manifolds of dimension n = 4r to be spin if they are not simply connected. No example is
known for n = 4 or 12, and the only known examples for n = 8 have fundamental groups
453
isomorphic to Z2 .) However, Pope et al. [23] have argued that non-spin manifolds can
be of interest in string theory when winding modes are taken into account; for example,
the AdS5 S 1 CP 2 solution of IIA supergravity generates an acceptable BPS solution
of the full string theory, despite the fact that CP 2 is not a spin manifold. Furthermore, the
Enriques and Hitchin manifolds fail to be spin in a relatively innocuous way. To explain this
remark, we note first that, like all orientable four-dimensional manifolds [24], the Enriques
manifold K3/Z2 and the Hitchin manifold K3/(Z2 Z2 ) are both Spinc manifolds. This
essentially means that difficulties in constructing globally well-defined fermion fields can
be overcome by coupling to a specific U (1) gauge field. For the Enriques and Hitchin
manifolds, this U (1) field is particularly inconspicuous because its field strength vanishes;
hence it does not affect the Lagrangian or the formula for the square of the Dirac operator.
In view of all this, we suggest that these manifolds may repay further investigation.
If we let SO(K3/Z2 ) be a bundle of orthonormal frames over the Enriques manifold,
then with the above embedding of SO(4) in Spin(16) : Spin(16) we can define
Q(SO(K3/Z2 )) = [SO(K3/Z2 ) Spin(16) : Spin(16)]/SO(4).
One easily prove, in the same way as for K3, that
w 2 (Q(SO(K3/Z2 ))) = 0,
so we still have a vector structure in this case. But if Q(SO(K3/Z2 )) had a non-trivial
Spin(16) Spin(16) cover, then SO(K3/Z2 ) would have a non-trivial Spin(4) double cover,
and we know that this is not the case. Hence
x2 (Q(SO(K3/Z2 ))) 6= 0,
and so Q(SO(K3/Z2 )) is our first example of a Semispin(32) configuration with a vector
structure but without an exceptional structure. Precisely similar statements hold true of the
Hitchin manifold:
w 2 (Q(SO(K3/(Z2 Z2 )))) = 0,
x2 (Q(SO(K3/(Z2 Z2 )))) 6= 0.
454
over M M.) Clearly, Spinc structures are rarely unique when they exist, since there
may be many choices for L(M). For Khler manifolds, however, there is a natural choice:
take a bundle of unitary frames U (M), and set L(M) = U (M)/SU(n/2), where n is the
real dimension of M. For the Enriques manifold, which is Khlerian, this U (1) bundle
inherits from U (M) a connection with holonomy Z2 . The curvature is therefore zero, and
so the Spinc structure can be employed without changing any local structures. The Hitchin
manifold is not Khlerian, but for it, too, there is a canonical choice of L(M), and again
L(M) has a flat connection. (Even with L(M) fixed, however, there is still the usual source
of ambiguity in defining Spinc (M), namely H 1 (M, Z2 ).)
Let us consider, then, the bundles SO(K3/Z2 ) + L(K3/Z2 ) and SO(K3/(Z2
Z2 )) + L(K3/(Z2 Z2 )). Both are SO(4) SO(2) bundles over the respective base
manifolds. Now SO(4) SO(2) is a subgroup of SO(6), and the latter embeds naturally in
Spin(16) : Spin(16), as {A : A, A Spin(6)}. Hence we can define
Q(SO(K3/Z2 ) + L(K3/Z2 ))
= [(SO(K3/Z2 ) + L(K3/Z2 )) Spin(16) : Spin(16)]/(SO(4) SO(2)),
and similarly for Q(SO(K3/(Z2 Z2 )) + L(K3/(Z2 Z2 ))). The Levi-Civit connection
form on SO(K3/Z2 ), and the canonical flat connection form on L(K3/Z2 ), both
pull back to SO(K3/Z2 ) + L(K3/Z2 ), and their direct sum defines [1] a connection
there. This connection pushes forward to a connection on the Spin(16) : Spin(16) bundle
Q(SO(K3/Z2 ) + L(K3/Z2 )) and to the latters Semispin(32) extension.
It is easy to see that, since SO(4) SO(2) is covered in Spin(16) Spin(16) by a
subgroup which is again isomorphic to SO(4) SO(2), Q(SO(K3/Z2 ) + L(K3/Z2 )) and
Q(SO(K3/(Z2 Z2 )) + L(K3/(Z2 Z2 ))) both have vector structures:
w 2 (Q(SO(K3/Z2 ) + L(K3/Z2 ))) = 0,
w 2 (Q(SO(K3/(Z2 Z2 )) + L(K3/(Z2 Z2 )))) = 0.
Now, however, let Spinc (K3/Z2 ) and Spinc (K3/(Z2 Z2 )) denote Spinc structures over
the respective manifolds. Then
Q (SO(K3/Z2 ) + L(K3/Z2 ))
= [Spinc (K3/Z2 ) Spin(16) Spin(16)]/Spin(4) Spin(2)
is well defined, Spin(4) Spin(2) being the cover of SO(4) SO(2) in Spin(16) Spin(16),
and similarly for K3/(Z2 Z2 ). (Through this discussion we have used the elementary
isomorphisms [27] U (1) = SO(2) = Spin(2).) That is,
x2 (Q(SO(K3/Z2 ) + L(K3/Z2 ))) = 0,
x2 (Q(SO(K3/(Z2 Z2 )) + L(K3/(Z2 Z2 )))) = 0;
both of these Spin(16) : Spin(16) bundles have exceptional structures. (The reader who
wishes to investigate these configurations should note that while, for example, the
holonomy groups of the connections on SO(K3/Z2 ) and L(K3/Z2 ) are isomorphic
respectively to [Z4 SU(2)]/Z2 and Z2 , the holonomy group of the direct sum connection
on SO(K3/Z2 ) + L(K3/Z2 ) is not Z2 [Z4 SU(2)]/Z2 . Rather, it is again isomorphic to
[Z4 SU(2)]/Z2 . See Ref. [1], page 82. The corresponding connection on Spinc (K3/Z2 )
also has holonomy [Z4 SU(2)]/Z2 , not Z4 SU(2).)
455
We conclude, then, that if the Levi-Civit connections of the (Ricci-flat) Enriques and
Hitchin manifolds are embedded in Semispin(32), we obtain configurations with no
globally well-defined T-dual partners, even though both do have vector structures. On
the other hand, by exploiting the Spinc structures of these spaces, one can construct a
different pair of Semispin(32) bundles which do admit T-dual partners. It is interesting to
note here that the argument of Pope et al. [23], to the effect that non-spin manifolds are
acceptable in string compactifications, depends on the existence of Spinc structures and on
the application of T-duality.
Six-dimensional Ricci-flat Khler manifolds present fewer complications, because (in
the compact case) these are always spin whether they are simply connected or not.
Embedding SO(6) in Spin(16) : Spin(16) as {A : A, A Spin(6)}, we can extend SO(CY),
for a CalabiYau manifold CY, to a Spin(16) : Spin(16) bundle Q(SO(CY)); such a
bundle always has a vector structure, and, if Spin(CY) is a spin structure over CY, we
can use it to construct an exceptional structure. Alternatively, by embedding Spin(6) in
Spin(16) : Spin(16) in the obvious way, we can use any spin structure over CY to construct
Q(Spin(CY)), which again has both vector and exceptional structures. (As in the case
of K3, SO(CY) and Spin(CY) are reducible bundles, but we avoid using the reduced
bundles because they need not be mapped into themselves by isometries. Indeed, we
should actually use a sub-bundle of the full bundle of orthonormal frames, O(CY). See
Ref. [28] for a discussion of this point.) We remind the reader that Spin(CY) will not be
unique if, as is frequently the case, H 1 (CY, Z2 ) does not vanish. If more than one spin
structure is physically significant, then embedding the Levi-Civit or spin connections
in Semispin(32) will produce configurations which are T-dual to a family of apparently
distinct (E8 E8 ) G Z2 configurations. (Recall that we are using the term T-duality
in a very broad way in this work, to include any process of exchanging the gauge and
matter fields of the two heterotic theories. Such an exchange or comparison could be of
interest even if T-duality in the more restricted sense (involving inversions of radii or other
changes of moduli) cannot be implemented, which is apparently the case for CalabiYau
compactifications [16].)
6.2. Examples from abelian instantons
Gauge configurations with non-vanishing invariants such as w 2 or x2 are of course
topologically non-trivial. One of the simplest but physically most relevant ways to
construct such fields is to use the non-trivial U (1) bundles over the two-sphere (Abelian
instantons). These arise naturally when the singularities of orbifolds are blown up [6].
Throughout this section, the base manifold is either a two-cycle in some manifold, or a
two-sphere around an orbifold singularity. The construction is guided by the discussions in
Sections 4 and 5 of Ref. [6].
Recall that SO(16) contains the unitary group U (8), which is globally isomorphic to
[U (1) SU(8)]/Z8 . The corresponding subgroup of Spin(16) is isomorphic [5] to [U (1)
SU(8)]/Z4 . We denote the elements of the latter by [u, s]4 , so that [iu, s]4 = [u, is]4 and
so on. Now let J8 be the element of Spin(16) defined by
J8 = [i, I8 ]4 ;
456
457
Finally, notice that if (hJ , hL ) is any element of HJ + HL , then the maps (hJ , hL )
hJ , (hJ , hL ) hL are bundle homomorphisms onto HJ and HL respectively, and so
coverings of Q(HJ + HL ) yield coverings of Q(HJ ) and Q(HL ). Thus if Q(HJ + HL )
had a vector structure, so would Q(HJ ), and if Q(HJ + HL ) had an exceptional structure,
so would Q(HL ). Hence in fact Q(HJ + HL ) yields an example of a Semispin(32) gauge
configuration with neither a vector structure nor an exceptional structure. In summary, we
have
x2 (Q(HJ )) = 0,
w 2 (Q(HJ )) 6= 0,
x2 (Q(HL )) 6= 0,
w 2 (Q(HL )) = 0,
x2 (Q(HJ + HL )) 6= 0.
w 2 (Q(HJ + HL )) 6= 0,
In a similar way, let U (1)E be a U (1) subgroup of Semispin(16) containing J8 , the
projection of J8 , and let HE be a Hopf bundle with U (1)E as structural group. Extend
HE to a Semispin(16) Semispin(16) bundle Q(HE ). Then the pre-image of U (1)E
in Spin(16) Spin(16) must be isomorphic to U (1), since (J8 , 1) is covered by J8 1,
which squares to K 16 1, the non-trivial element of the kernel of Spin(16) Spin(16)
Semispin(16) Semispin(16). We conclude that Q(HE ) is an E8 E8 configuration with
no T-dual partner; that is, y2 (Q(HE )) 6= 0.
6.3. Examples involving orbifolds
T-duality was, of course, originally defined with respect to tori; however, one can attempt
to extend it to other manifolds (such as K3) by regarding them as desingularisations of
orbifolds. It is shown in Ref. [6] that Abelian instantons can hide in the singularities of
T 4 /Z2 , where T 4 is the four-torus. By blowing up the singularities, one can if necessary
bring the hidden instantons into the open, and the techniques of the preceding section can
be applied. The most striking examples of this kind have a hidden instanton which is a
Semispin(32) configuration similar to Q(HJ ) above: it has no vector structure, but it does
have an exceptional structure, and it is shown in Ref. [6] that it is T-dual to the E8 E8
DMW vacuum [15]. (In fact, the DMW vacuum is completely symmetric with respect to
the two E8 factors, and the gauge group is actually (E8 E8 ) G Z2 , so this is a case where
(see Section 5 above) y1 6= 0.)
Other vacua considered in Ref. [6] do have vector structures, but some of these do
not have exceptional structures. Before discussing these, however, we must clarify the
following point, first mentioned in Ref. [6]. When discussing T-duality on R9 S 1 , one
often begins with unbroken Semispin(32), and continuously turns on a Wilson line
that breaks Semispin(32) to (Spin(16) : Spin(16)) G Z2 . Then T-duality converts this to a
(Semispin(16) Semispin(16)) G Z2 theory, which becomes an (E8 E8 ) G Z2 theory
when the Wilson line is continuously turned off. However, it is intuitively clear that if we
insist that it should always be possible to turn Wilson lines on and off continuously, then
we shall exclude topologically non-trivial configurations such as Semispin(32) bundles
lacking vector structures. To see this, let P be a Semispin(32) bundle with a family of
gauge connections, {t , t [0, 1]}. Suppose that, for each t, the holonomy group of t is a
cyclic group (typically of infinite order) generated by an element Wt Semispin(32), such
that
458
W0 = 1,
W1 = K 16 : 1.
for all t.
Since W1 = diag(I16 , I16 ), we can set Wt = exp(tb), where, following Ref. [6], we put
0 I8
.
b = I8 0
016
Thus we need X(m, n)bX(m, n)1 = b, and so in this case m = 8, and so n = 4. Thus
X(8, 4) is the twisting matrix when the Wilson line is continuously turned on.
Now the elements of Spin(32) and Semispin(32) corresponding to X(8, 4) are,
respectively, K 8 K 4 and K 8 : K 4 , and the relevant element of Spin(16) Spin(16) is
K 8 K 4 . As (K 8 )2 = (K 4 )2 = 1, we see that K 8 K 4 , K 8 K 4 , and K 8 : K 4 are all of
order two. Any U (1) subgroup of Spin(16) : Spin(16) which contains K 8 : K 4 is therefore
459
7. Conclusion
The principal findings of this work can be stated very simply. On R9 S 1 , the two
heterotic string theories are related by T-duality: one theory compactified on a circle of
radius R is equivalent to the other compactified on a circle of radius proportional to 1/R.
As soon, however, as one goes to manifolds of greater complexity, the analogous statement
is questionable: T-duality can be obstructed topologically. This is true even on so simple
a manifold as T 2 R8 , even though one may wish to invert the radius of only one circle.
That is, whether T-duality works for a given circle depends on the context of that circle.
(See Ref. [16] for a much more subtle instance of this.) More mundanely, the essential
point here is that the two gauge groups, E8 E8 and Semispin(32), are not quite as similar
as their Lie algebras might lead one to expect.
From a practical point of view, our results mean that any discussion of the relationship
between the two heterotic theories must involve a computation of the exceptional Stiefel
460
Whitney class x2 (or of y2 if one begins on the E8 E8 side). If x2 fails to vanish, then a
comparison is not meaningful, whatever the local situation may suggest.
References
[1] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. I, Interscience, New
York, 1963.
[2] B. McInnes, J. Math. Phys. 38 (1997) 4354.
[3] J. Polchinski, String Theory, Cambridge Univ. Press, New York, 1998.
[4] W. Lerche, C. Schweigert, R. Minasian, S. Theisen, Phys. Lett. B 424 (1998) 53.
[5] B. McInnes, The semispin groups in string theory, hep-th/9906059, J. Math. Phys., in press.
[6] M. Berkooz, R.G. Leigh, J. Polchinski, J.H. Schwarz, N. Seiberg, E. Witten, Nucl. Phys. B 475
(1996) 115.
[7] J.H. Schwarz, Nucl. Phys. B (Proc. Suppl.) 55 (1997) 1.
[8] D. Husemoller, Fibre Bundles, Springer, Berlin, 1975.
[9] T. Brcker, T. Dieck, Representations of Compact Lie Groups, Springer, Berlin, 1985.
[10] G.G. Ross, Grand Unified Theories, AddisonWesley, Reading, 1984.
[11] L. ORaifeartaigh, Group Structure of Gauge Theories, Cambridge Univ. Press, Cambridge,
1987.
[12] H.B. Lawson, M.L. Michelsohn, Spin Geometry, Princeton Univ. Press, Princeton, 1989.
[13] A. Borel, Tohoku Math. J. 13 (1962) 216.
[14] S. Chaudhuri, G. Hockney, J. Lykken, Phys. Rev. Lett. 75 (1995) 2264.
[15] M.J. Duff, R. Minasian, E. Witten, Nucl. Phys. B 465 (1996) 413.
[16] P.S. Aspinwall, M.R. Plesser, T-duality can fail, hep-th/9905036.
[17] E. Witten, J. High Energy Phys. 9812 (1998) 019.
[18] M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, Cambridge Univ. Press, Cambridge,
1987.
[19] P.S. Aspinwall, K3 surfaces and string duality, hep-th/9611137.
[20] W. Barth, C. Peters, A. van de Ven, Compact Complex Surfaces, Springer, Berlin, 1984.
[21] N. Hitchin, J. Diff. Geom. 9 (1974) 435.
[22] B. McInnes, Commun. Math. Phys. 203 (1999) 349.
[23] C.N. Pope, A. Sadrzadeh, S.R. Scuro, Timelike Hopf duality and Type IIA string solutions,
hep-th/9905161.
[24] J.W. Morgan, The SeibergWitten Equations and Applications to the Topology of Smooth FourManifolds, Princeton Univ. Press, Princeton, 1996.
[25] A.L. Besse, Einstein Manifolds, Springer, Berlin, 1987.
[26] B. McInnes, J. Math. Phys. 34 (1993) 4287.
[27] M.L. Curtis, Matrix Groups, Springer, Berlin, 1984.
[28] B. McInnes, J. Math. Phys. 40 (1999) 1255.
[29] R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, Springer, Berlin, 1982.
[30] L. Dixon, J. Harvey, C. Vafa, E. Witten, Nucl. Phys. B 274 (1986) 285.
Abstract
Three inequivalent real forms of the complex N = 4 supersymmetric Toda chain hierarchy (Nucl.
Phys. B 558 (1999) 545) in the real N = 2 superspace with one even and two odd real coordinates are
presented. It is demonstrated that the first of them possesses a global N = 4 supersymmetry, while the
other two admit a twisted N = 4 supersymmetry. A new superfield basis in which supersymmetry
transformations are local is discussed and a manifest N = 4 supersymmetric representation of the
N = 4 Toda chain in terms of a chiral and an anti-chiral N = 4 superfield is constructed. Its relation
to the complex N = 4 supersymmetric KdV hierarchy is established. DarbouxBacklund symmetries
and a new real form of this last hierarchy possessing a twisted N = 4 supersymmetry are derived.
2000 Elsevier Science B.V. All rights reserved.
PACS: 02.20.Sv; 02.30.Jr; 11.30.Pb
Keywords: Completely integrable systems; Toda field theory; Supersymmetry; Discrete symmetries
1. Introduction
Recently the Lax pair representation of the even and odd flows of the complex N = 4
supersymmetric Toda chain hierarchy in N = 2 superspace were constructed in [1]. The
corresponding local and nonlocal Hamiltonians, the finite and infinite discrete symmetries,
the first two Hamiltonian structures and the recursion operator connecting all evolution
equations and the Hamiltonian structures were also studied. The goal of the present paper
is first to analyse the possible real forms of the N = 4 Toda chain hierarchy in N = 2
1 francois.delduc@ens-lyon.fr
2 sorin@thsun1.jinr.ru
3 UMR 5672 du CNRS, associe lEcole Normale Suprieure de Lyon.
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 2 1 - 8
462
/tk = Uk = [Uk ] = U k = k,
1
Dk = Q
k = k + .
2
(2)
v = v 000 3(D+ v)0 (D uv) + 3(D v)0 (D+ uv) 3v 0 (D+ u)(D v)
t3
+ 3v 0 (D u)(D+ v) 6vv 0 (D+ D u) + 6(uv)2 v 0 ,
u = u000 3(D u)0 (D+ uv) + 3(D+ u)0 (D uv) 3u0 (D v)(D+ u)
t3
+ 3u0 (D+ v)(D u) 6uu0 (D D+ v) + 6(uv)2 u0 ,
1
(uv),
D1 v = D v 2vD
v
v
Q
1 u = Q u ,
(1)
1
D1 u = D u 2uD
(uv),
1
1
(uv)),
U0 v = v (D v 2vD
2
1
1
(uv) ,
U0 u = u D u 2uD
2
1
1
(uv) D+ v 2vD
(uv) ,
U0 v = + D v + 2vD+
1
1
(uv) D+ u + 2uD
(uv) ,
U0 u = + D u 2uD+
v
v
1
+
U0
= (D+ + Q+ ) (D + Q )
.
2
u
u
(3)
(4)
(5)
(6)
(7)
(8)
(9)
+ ,
z u.
463
(10)
Q , Q = 2.
(11)
Using the explicit expressions of the flows (2)(9), one can calculate their algebra which
has the following nonzero brackets:
Q1 , Q
,
(12)
D1 , D1 = 2 ,
1 = +2
t1
t1
U 0 , U0 = U0 ,
U0 , U 0 = 2(U0+ U0 ),
(13)
U0 , U0 = U 0 ,
U0 , D1 = Q
1,
U0 , D1 = +Q
1,
U 0 , D1 = D1 ,
U0 , Q
1 = D1 ,
U0 , Q
1 = +D1 ,
U 0 , Q
1 = Q1 .
(14)
This algebra reproduces the algebra of the global complex N = 4 supersymmetry, together
with its gl(2, C) automorphisms. It is the algebra of symmetries of the nonlinear even flows
1
(15)
k = k = k ,
2
which are in one-to-one correspondence with the length dimensions (1) of the corresponding evolution derivatives.
2. Real forms of the N = 4 Toda chain hierarchy
It is well known that different real forms derived from the same complex integrable
hierarchy are inequivalent in general. Keeping this in mind it seems important to find as
many different real forms of the N = 4 Toda chain hierarchy as possible.
With this aim let us discuss various inequivalent complex conjugations of the superfields
u(z, + , ) and v(z, + , ), of the superspace coordinates {z, }, and of the evolution
derivatives {/tk , Uk , Uk , U k , Dk , Q
k } which should be consistent with the flows (2)
(9). We restrict our considerations to the case when iz and are coordinates of the real
N = 2 superspace which satisfy the following standard complex conjugation properties:
(16)
iz, = iz, ,
where i is the imaginary unit. We will also use the standard convention regarding complex
conjugation of products involving odd operators and functions (see, e.g., the books [6,7]).
4 Hereafter, we explicitly present only non-zero brackets.
464
(D+ D ) = D+ D
D = D ,
(17)
which we use in what follows. Here, are constant odd real parameters.
Direct verification shows that the flows (2)(9) admit the three inequivalent complex
conjugations:
z, = z, ,
(v, u) = (v, u),
= (1)p tp , Up , Up , U p , p Dp , p Q
tp , Up , Up , U p , p Dp , p Q
p
p , (18)
z, = z, ,
(v, u) = (u, v),
= tp , Up , Up , U p , p Dp , p Q
(19)
tp , Up , Up , U p , p Dp , p Q
p
p ,
?
z, = z, ,
(v, u)? = u(D D+ ln u + uv), 1/u ,
?
= tp , Up , Up , U p , p Dp , p Q
(20)
tp , Up , Up , U p , p Dp , p Q
p
p ,
where p and p are constant odd real parameters. We would like to underline that the
complex conjugations of the evolution derivatives (the second lines of Eqs. (18)(20))
are defined and fixed completely by the explicit expressions (2)(9) for the flows. These
complex conjugations extract different real forms of the algebra (12)(14). The real forms
of the algebra (12)(14) with the involutions (18), (19) correspond to a twisted real N = 4
supersymmetry, while the real form corresponding to the involution (20) reproduces the
algebra of the real N = 4 supersymmetry. This last fact becomes more obvious if one uses
the N = 2 basis of the algebra with the generators
1 U0 ,
3 U0 U0+ ,
2 iU 0 ,
+
D1 Q+
1 + D1 ,
D2 Q
1 + D1 ,
+
D 1 Q+
1 D1 ,
D2 Q
1 D1 .
U0 + U0+ ,
(21)
Then, the nonzero algebra brackets (12)(14) and the complex conjugation rule (20)
are the standard ones for the real N = 4 supersymmetry algebra together with its u(2)
automorphisms
,
a , b = 2iabc c ,
D , D = 4
t1
a , D = D (a ) ,
[a , D ] = (a ) D ,
, D = D ,
(22)
[, D ] = D ,
/t1 , a , , D , D
( , )? = ( , ).
?
= /t1 , a , , D , D ,
(23)
0 1
0
,
2
1 0
i
a b = ab + iabc c ,
1
i
,
0
3
465
1 0
,
0 1
(24)
and abc is a totally antisymmetric tensor (123 = 1). Therefore, we conclude that the
complex N = 4 supersymmetric Toda chain hierarchy with the complex conjugation (20)
possesses a real N = 4 supersymmetry, and due to this remarkable fact it can be called the
N = 4 supersymmetric Toda chain hierarchy (for some examples of N = 4 supersymmetric
integrable systems, see [814] and references therein).
Let us remark that a combination of the two involutions (20) and (19) generates the
infinite-dimensional group of discrete Darboux transformations [14]
?
z, = z, ,
(v, u)? = v(D D+ ln v uv), 1/v ,
?
= tp , Up , Up , U p , Dp , Q
(25)
tp , Up , Up , U p , Dp , Q
p
p .
This way of deriving discrete symmetries was proposed in [15,16] and applied to the
construction of discrete symmetry transformations of the N = 2 supersymmetric GNLS
hierarchies.
J uv,
(26)
1
1
J = J 00 2 J D+
D J + D+ D D+
D J ,
t2
0
2
1
1
D J + D+ D D+
D J ,
J = +J00 2 JD+
t2
1
2
1
000
J = J + 3 J 0 D+
D J + J + J D+
D J + J
t3
0
1
+ (D J )D+ J JJ 2 J 3
3
2 1
3D+ D J D+ D J + J ,
(30)
466
1
2
1
J = J000 3 J0 D+
D J + J J D+
D J + J
t3
0
1
+ D J D+ J + JJ2 + J3
3
2 1
3D+ D J D+ D (J + J ) ,
J
+J
J
J
= D
,
Q1 = Q ,
J
J
J
J
+J
J
J
+J
+
,
= D
,
U0
= ( D + D+ )
U0
J
J
J
J
J
J
1
U 0 = (D+ + Q+ ) + (D + Q ) .
2
J
J
D1
(31)
(32)
(33)
(34)
Notice that the supersymmetry and u(2) transformations (32)(34) of the superfields J ,
J are local functions of the superfields, while the evolution equations (30)(31) become
nonlocal.
J = J 00 D+ D 2 J 1 J D+ D 1 J ,
t2
h
0
2 i
J = +J 00 D+ D 2 J 1 J D+ D 1 J
,
(35)
t2
n h
2 i0
1
J = J 000 + D+ D 3 J 0 1 J + J 1 J D+ D 1 J D + D 1 J
t3
2
3
2
D+ D 1 J 3 D + D 1 J D+ D 1 J
o
+ 6J J D+ D 1 J ,
n h
2 i0
1
J = J 000 + D+ D 3 J 0 1 J + J 1 J D+ D 1 J + D+ D 1 J
t3
2
3
2 + 1
1
1
D+ D J 3 D+ D J D D J
o
(36)
+ 6J J D+ D 1 J ,
in terms of one chiral J (z, + , , + , ) and one antichiral J (z, + , , + , ) even
N = 4 superfield,
D J = 0,
D J = 0.
(37)
1
+
i
+
(
+
i
)
,
2
i
+
(
i
)
,
2
Dk , Dm = Dk , Dm = 0,
Dk , Dm = k m ,
467
k, m = ,
(38)
are two additional real odd coordinates. The relations between the independent
and
components of the N = 2 superfields {J (z, + , ), J(z, + , )} and those of the N = 4
superfields {J (z, + , , + , ), J (z, + , , + , )} are the following:
J | =0 = J,
J | =0 = J,
D J | =0 = D J,
D J | =0 = D J,
D + D J | =0 = D+ D J,
D+ D J | =0 = D+ D J.
(39)
J ?? = J D D+ ln J ?? ,
(40)
which can easily be derived using Eqs. (28)(29) and (39). They are discrete symmetries
of the even and odd flows of the N = 4 supersymmetric Toda chain hierarchy. In other
words, if the set {J , J } is a solution of the N = 4 Toda chain hierarchy, then the set
{J ?? , J ??}, related to the former by Eqs. (40), is a solution of the hierarchy as well.
Eqs. (40) reproduce the one-dimensional reduction of the two-dimensional N = (2|2)
superconformal Toda lattice [17,18].
Finally, we would like to remark that Eqs. (35)(36) can be rewritten in a local form, if
one introduces a new superfield basis defined by the following invertible transformations:
J D+ ,
J D ,
D+ 1 J ,
D 1 J ,
(41)
D = D + = 0
(42)
with the reality conditions which can be derived from Eqs. (27)(29) and (41). Then, these
equations become
2
= + 00 + 2D D D D+ D+ ,
t2
2
= 00 2D+ D+ D+ + D D ,
(43)
t2
h
0
2 1
2 i
= 000 + 3D D D + D D + D+ D D+
t3
2
i
h
3
2
+
D+ 3 D+ D 6 D+ D D+ ,
+D
h
0
2 1
2 i
= 000 + 3D+ D+ D+ + D+ D+ D+ D D
t3
2
h
3
2
i
(44)
+ D D 3 D D+ 6 D+ D D .
468
+ D, D J + ,
i J = + 2 J
t2
2
3
1
i = 2DD J0 J2 2 + ,
t2
4
2
3
1
2
(45)
i = 2DD J0 + J2 + ,
t2
4
2
where J, , ([J] = [] = [] = 1) are new unconstrained, chiral (D = 0) and
antichiral (D = 0) even N = 2 superfields, respectively, related to the superfields J, J
(26) by the following invertible transformations:
1
i
1
J + + iJ,
J J J ,
J + iJ,
2
2
2
DD 1 J + J ,
(46)
DD 1 J + J ,
and D, D are N = 2 odd covariant derivatives,
1
1
D (D+ iD ),
D (D+ + iD ),
2
2
D, D = D, D = 0.
D, D = ,
(47)
Now, one can easily recognize that Eqs. (45) is the second flow of the N = 4 KdV hierarchy
in a particular SU (2) frame (compare Eqs. (45) with Eqs. (4.5) and (4.3c) from Ref. [9]).
Moreover, in this basis the second Hamiltonian structure of the N = 4 Toda chain hierarchy
[1] reproduces the N = 4 SU (2) superconformal algebra and the flow (45) is generated by
the Hamiltonian H2t [1]
Z
Z
H2t = dz d + d uv 0 i dz d + d J D, D 1 J + J
Z
(48)
dz d + d J .
The same relationship is certainly valid for any other flow of the N = 4 Toda and N = 4
KdV hierarchies both in the N = 2 and N = 4 superspaces.
469
The relationship just established allows to apply the formalism of Ref. [1], developed
for the case of the N = 4 Toda chain hierarchy, for a more complete description of the
N = 4 KdV hierarchy. It can be used to construct new bosonic and fermionic flows and
Hamiltonians, new finite and infinite discrete symmetries, the tau function, etc. Let us
present as an example the three involutions
(49)
, , J = , , J ,
= + iDD ln + + 2iJ ,
= iDD ln + + 2iJ ,
1
J = J + D, D ln + + 2iJ ,
2
?
, , J = , , J
(50)
(51)
6. Conclusion
In this paper, we have described three distinct real forms of the N = 4 Toda chain
hierarchy introduced in [1]. It has been shown that the symmetry algebra of one of these
real forms contains the usual (untwisted) real N = 4 supersymmetry algebra. A set of
N = 2 superfields with simple conjugation properties in the untwisted case have been
introduced. It has then been shown how to extend these superfields to superfields in N =
4 superspace, and write all flows and conjugation rules directly in N = 4 superspace.
Finally, a change of basis in N = 4 superspace has allowed us to eliminate all nonlocalities
in the flows. As a byproduct, a relationship between the complex N = 4 Toda chain
and N = 4 KdV hierarchies has been established, which allows to derive Darboux
Backlund symmetries and a new real form of the last hierarchy possessing a twisted N = 4
supersymmetry.
It is obvious that there remain a lot of work to do in order to improve our understanding
of the hierarchy in N = 4 superspace. A first step in this direction would be to derive a Lax
formulation of the flows in terms of N = 4 operators.
470
Acknowledgements
A.S. would like to thank L. Freidel for useful discussions and the Laboratoire de
Physique Thorique de lENS Lyon for the hospitality during the course of this work.
This work was partially supported by the PICS Project No. 593, RFBR-CNRS Grant
No. 98-02-22034, RFBR Grant No. 99-02-18417, Nato Grant No. PST.CLG 974874 and
INTAS Grant INTAS-96-0538.
References
[1] F. Delduc, L. Gallot, A. Sorin, N = 2 local and N = 4 nonlocal reductions of supersymmetric
KP hierarchy in N = 2 superspace, Nucl. Phys. B 558 (1999) 545; solv-int/9907004.
[2] A.N. Leznov, A.S. Sorin, Two-dimensional superintegrable mappings and integrable hierarchies
in the (2|2) superspace, Phys. Lett. B 389 (1996) 494; hep-th/9608166.
[3] A.N. Leznov, A.S. Sorin, Integrable mappings and hierarchies in the (2|2) superspace, Nucl.
Phys. B 56 (1997) 258.
[4] O. Lechtenfeld, A. Sorin, Fermionic flows and tau function of the N = (1|1) superconformal
Toda lattice hierarchy, Nucl. Phys. B 557 (1999) 535; solv-int/9810009.
[5] O. Lechtenfeld, A. Sorin, Supersymmetric KP hierarchy in N = 1 superspace and its N = 2
reductions, ITP-UH-14/99; JINR E2-99-216; solv-int/9907021; Nucl. Phys. B, to appear.
[6] S.J. Gates Jr., M.T. Grisaru, M. Rocek, W. Siegel, Superspace or one Thousand and one Lessons
in Supersymmetry, Benjamin/Cummings Publishing, 1983.
[7] P. West, Introduction to Supersymmetry and Supergravity, 2nd edn., World Scientific,
Singapore, 1990.
[8] F. Delduc, E. Ivanov, N = 4 super KdV equation, Phys. Lett. B 309 (1993) 312; hep-th/9301024.
[9] F. Delduc, E. Ivanov, S. Krivonos, N = 4 super KdV hierarchy in N = 4 and N = 2 superspaces,
J. Math. Phys. 37 (1996) 1356; J. Math. Phys. 38 (1997) 1224, Erratum; hep-th/9510033.
[10] E. Ivanov, S. Krivonos, New integrable extensions of N = 2 KdV and Boussinesq hierarchies,
Phys. Lett. A 231 (1997) 75; hep-th/9609191.
[11] F. Delduc, L. Gallot, E. Ivanov, New super KdV system with the N = 4 SCA as the Hamiltonian
structure, Phys. Lett. B 396 (1997) 122; hep-th/9611033.
[12] E. Ivanov, S. Krivonos, F. Toppan, N = 4 super-NLS-mKdV hierarchies, Phys. Lett. B 405
(1997) 85; hep-th/9703224.
[13] L. Bonora, A. Sorin, The Hamiltonian structure of the N = 2 supersymmetric GNLS hierarchy,
Phys. Lett. B 407 (1997) 131; hep-th/9704130.
[14] Z. Popowicz, Odd bihamiltonian structure of new supersymmetric N = 2, 4 Kortewegde Vries
equation and odd SYSY Virasoro-like algebra, Phys. Lett. B 459 (1999) 150; hep-th/9903198.
[15] A. Sorin, The discrete symmetry of the N = 2 supersymmetric modified NLS hierarchy, Phys.
Lett. B 395 (1997) 218; hep-th/9611148.
[16] A. Sorin, Discrete symmetries of the N = 2 supersymmetric generalized nonlinear schroedinger
hierarchies, Phys. Atom. Nucl. 61 (1998) 1768; solv-int/9701020.
[17] J. Evans, T. Hollowood, Supersymmetric Toda field theories, Nucl. Phys. B 352 (1991) 723.
[18] V.B. Derjagin, A.N. Leznov, A. Sorin, The solution of the N = (0|2) superconformal f -Toda
lattice, Nucl. Phys. B 527 (1998) 643; solv-int/9803010.
[19] S. Krivonos, A. Sorin, The minimal N = 2 superextension of the NLS equation, Phys. Lett.
B 357 (1995) 94; hep-th/9504084.
[20] S. Krivonos, A. Sorin, F. Toppan, On the super-NLS equation and its relation with N = 2 superKdV within coset approach, Phys. Lett. A 206 (1995) 146; hep-th/9504138.
Abstract
In a symplectic framework, the infinitesimal action of symplectomorphisms together with suitable
reparametrizations of the two-dimensional complex base space generate some type of w-algebras.
It turns out that complex structures parametrized by Beltrami differentials play an important role in
this context. The construction parallels very closely two-dimensional Lagrangian conformal models
where Beltrami differentials are fundamental. 2000 Elsevier Science B.V. All rights reserved.
PACS: 11.10
Keywords: w-algebras
1. Introduction
In the last decade, w-algebras have provided a unifying landscape for various topics
like integrable systems, conformal field theory, as well as uniformization of 2-dimensional
gravity. They were originally discovered as a natural extension of the Virasoro algebra by
Zamolodchikov [1] and later implicitly by Drinfield and Sokolov [2]. The latter obtained
the classical w-algebras by equipping the coefficients of first order matrix differential
operators with the second GelfandDickey Poisson bracket [3].
In the study of two-dimensional conformal models in the so-called conformal gauge wgravity can be defined as a generalization of the reparametrization invariance such that in
the conformal gauge two copies of the corresponding w-algebra are obtained. Moreover,
it has be found that the matrix differential operator has to be supplemented by another
equation which is usually referred as the Beltrami equation [2].
Several attempts have been made in order to give a geometric picture of this very
rich structure provided by w-algebras. Various aspects of this issue have been tackled
beppe@genova.infn.it
1 sel@cpt.univ-mrs.fr
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 2 9 - 2
472
473
2. Geometrical approach
The starting point of a symplectic structure is the definition on a manifold of the
canonical 1-form on T (z, y) defined in a local chart frame U (z,y)
|U(z,y) = [yz dz + yz d z ].
(2.1)
(2.2)
(2.3)
Let us consider now a frame U(Z,Y ) : will take the form on T (Z, Y )
(2.4)
(2.5)
If we require the invariance of the theory under diffeomorphisms, we have to impose that
the local change of frame will generate a canonical transformation.
The change of charts will be canonical if in U(z,y) U(Z,Y )
|U(z,y) = |U(Z,Y )
(2.6)
(2.7)
F is a function on U(z,y) U(Z,Y ) . In the (z, Y ) plane we can define the function (z, Y )
as
= yz dz + yz d z + dYZ Z + d Y Z .
(2.8)
d(z, Y ) d F + (YZ Z + YZ Z)
Z
474
The function (z, Y ) is the generating function for the change of charts and it is defined
2
k.
(up to a total differential) on the space U(z,Y ) and has non-vanishing Hessian, k zY
In the region U(z,Y ) the differential operator takes the form
+ d YZ
(dz + dYZ )
d = dzz + d z z + dYZ
YZ
YZ
and d 2 = 0 will imply
= 0,
z ,
YZ
z ,
=0
YZ
(2.9)
(2.10)
yz = Z,
YZ
Z=
Z,
YZ
YZ
yz = Z
Y
(2.11)
(z, Y )
Z(z, Y ) =
YZ
(2.12)
(2.13)
are canonical. In particular for an arbitrary surface YZ = const we can construct a local
change of coordinates:
(z, Y )
.
(2.14)
(z, z ) (Z, Z)
for Z(z, z ) =
YZ
YZ = const
In the following, for writing convenience, we shall choose this constant equal to zero.
Going on we can verify that
|U(z,y) = dz (z, Y ).
(2.15)
YZ
= dY dz (z, Y ).
We shall now introduce some quantities which will be useful for our treatment.
Let us call
(2.16)
(z, Y ),
YZ
Y ) = z
(z, Y ),
(z,
Y
(z, Y ) = z
(z, Y )(z, Y ) = z
(z, Y ),
YZ
Y )(z,
(z,
Y ) = z
(z, Y ).
Y
475
(2.17)
(2.18)
(2.19)
(2.20)
(and their c.c.) which reveal a complex structure parametrized by an ordinary Beltrami
multiplier (z, Y ) in the (z, z ) plane by /
in the (YZ , YZ ) one.
So the Z(z, Y ) and y(z, Y ) coordinate systems can be defined in terms of a given
(z, Y ), by means of the equations
(z, Y ) Z(z, Y ) = 0,
(2.21)
(z, Y )(z,
Y)
(2.22)
yz (z, Y ) = 0.
YZ
(z, Y )
YZ
From the previous equations the Liouville theorem will follow
Z(z, Y )
= (z, Y )(z,
Y ) 1 (z, Y )(z,
Y)
det
z
yz (z, Y )
.
= det
Y
(2.23)
Using the previous parametrization, as is well known, we can write the derivative operators
Z , YZ as
z (z, Y )z
,
(z, Y )(1 (z, Y )(z,
Y ))
Dz (z, Y )D z
=
yz (z, Y ) 1 (z, Y )(z,
Y)
Z =
(2.24)
(2.25)
1
.
(z, Y ) YZ
(2.26)
Finally, if we work in the U(z,Y ) space, taking z and YZ as passive coordinates, the condition
d|U(z,Y ) = 0 will give
476
d|U(z,Y ) = (z,
Y ) d z dz dYZ + (z, Y )(z, Y ) dz d z dYZ
Y ) dz d z d Y + (z,
Y )(z,
+ (z,
Y ) d z dz d YZ
Z
+ d YZ
(z, Y ) dz + (z, Y ) d z dYZ
YZ
+ dYZ
(z, Y ) d z + (z,
Y ) dz d YZ = 0
YZ
which gives rise to the Beltrami identities
(z,
Y ) = (z, Y )(z, Y ) ,
Y )(z,
(z, Y ) =
(z,
Y)
Y
YZ
Z
(2.27)
(2.28)
(2.29)
= (z, Y )Dz (z, Y ) z , Dz ,
(2.30)
Y)
Dz , D z = Dz (z,
D z (z, Y )
yz (z, Y )
yz (z, Y )
,
= 0.
yz (z, Y ) yz (z, Y )
(2.31)
(2.32)
One remark is in order: in Eq. (2.16) the terms dYZ Z dYZ + dYZ Z d YZ and dz yz
dz + dz yz d z will identically vanish in . Accordingly, we can state the important
theorem [17].
Theorem 2.1. On the smooth trivial bundle 6 R2 , the vertical holomorphic change of
local coordinates,
Z (z, z ), F (YZ ), YZ ,
(2.33)
Z (z, z ), YZ , YZ
where F is a holomorphic function in YZ , while the horizontal holomorphic change of
local coordinates,
yz f (z), z , YZ , YZ ,
(2.34)
yz z, z , YZ , YZ
where f is a holomorphic function in z, are both canonical transformations.
477
In the first case an infinitesimal variation of Z(z, Y ) in YZ does not modify, for fixed
(z, z ) the form.
So the diffeomorphisms (z) Z((z, z ), YZ ); z Z((z, z ), YZ + dYZ ), will be related
to the same two form .
If we make the expansion around, say, YZ = 0, YZ = 0 the generating function will
be written as the series
X
n
1
1 (z, z ), YZ
YZn
(z, Y ) =
n!
YZ
YZ =0,YZ =0
n=1,...,nmax
X
n
1 n
Y
1 (z, z ), YZ
+
n! Z YZ
YZ =0,YZ =0
n=1,...,nmax
X
X
(2.35)
YZn Z (n) (z, z ) +
YZn Z (n) (z, z ) ,
n=1,...,nmax
where
Z (n) (z, z )
1
n
1 (z, Y )
.
n! YZ
YZ =0,Y =0
(2.36)
(2.37)
(2.38)
will be the origin of the w-algebras symmetry transformations. Obviously the choice of the
(YZ , YZ ) origin as starting point does not alter the treatment
The symplectic form will then be written
X
dY YZn dz Z (n) (z, z ) + dY YZn dz Z (n) (z, z ) . (2.39)
|U(z,Y ) = dY dz (z, Y ) =
n
Note that the conjugate momenta to the complex coordinates Z (n) , n 6= 1 are related to
the nth power of the conjugate momenta of the coordinate Z (1)(z, z ).
(3.1)
478
ai !
j =1,...,n
i=1,...,mj
X
Xi
ai = j ,
ai pi = n,
i
p 1 > p 2 > > p mj
M(j ) (z, z ),
(3.2)
where the
functions Mj (z, z )
M(j ) (z, z )
are given by
1
(D)j (z, Y )|YZ ,YZ =0 .
j!
(3.3)
Note that the Z (n) (z, z ) coordinate is no more independent from Z (r) , for r = 1, . . . ,
n 1; by the way it obeys differential consistency conditions induced by the M(j ) (z, z )
functions: we shall sketch the above system for further convenience
Z(z, z ) = ZM(1) (z, z ),
Z (2) (z, z ) = Z (2) (z, z )M(1) (z, z ) + (Z)2 M(2)(z, z ),
Z (3) (z, z ) = Z (3) (z, z )M(1) (z, z ) + 2Z(z, z )Z (2)(z, z )M(2)(z, z )
+ Z(z, z )3 M(3) (z, z ),
Z (N) (z, z ) = Z (N) (z, z )M(1)(z, z ) + + Z(z, z )
N
M(N) (z, z ).
(3.4)
dz0
1
+ ln Z(z, z ),
M1 (z0 , z )
(3.5)
dz0
1
M(1) (z0 , z )
(3.6)
which put into the second equation gives Z (2) as an integral relation of Z M(1)(z, z ) and
M(2)(z, z )
Zz
M(2) (z00 , z )Z(z00 , z )
(2)
dz00
,
(3.7)
Z (z, z ) = Z(z, z )
3
(M1 (z00 , z ))
in general
Z z
Z
(N)
(z, z ) = Z(z, z )
i 00
1
N
(j )
k 00
F Z M (z , z )k6j M (z , z ) , (3.8)
dz
Z(z00 , z )
00
479
where M(i) (z, z ), i = 1, 2, . . . , n, are fixed and Z (j ) (z00 , z ), j < n have been calculated at
the preceeding orders. So we can state:
Theorem 3.1. The set of functions M(i) (z, z ), i = 1, . . . , n, completely identify the set of
coordinates Z (j ) (z, z ), j = 1, . . . , n.
Now we want to solve another problem: given a local change of coordinates (z, z )
z )) is it possible to consider it as an element of arbitrary nth order of a w(Z(z, z ), Z(z,
hierarchy (z, z ) (Z (n) , Z (n) ), and to find a construction of the underlying (Z (i) , Z (i) ),
i = 1, . . . , n 1, in order to get a w-description of this local change?
The answer is positive, since using a standard construction of the (Z (i) , Z (i) ), i =
1, . . . , n 1, spaces (which do not interfere with (Z (n) , Z (n) )), we can use it in the last
equation of (3.4) and get in an algebraic way the suitable solution MN (z, z ). So we can
state the theorem:
z )) it is possible to
Theorem 3.2. For an arbitrary local change: (z, z ) (Z(z, z ), Z(z,
(r)
(r)
1
n
(z, Y )
Z (n) (z, z ),
n! YZ
YZ =0,YZ =0
n
1
(n)
z
(n) (z, z ).
Z
(z,
Y )
Z
z (z, z )z n, (z, z )
n! YZ
YZ =0,Y =0
(n)
Z
z (z, z )
(3.10)
So we can define n
(n,
(z, z ))
.
=
(n)
1 (n, (z, z ))(n,
(z, z ))
Z
In particular we get from Eq. (3.2)
ai
mj Z (pi )
n
X
Y
(z (z, z ))
j
!
ai !
Z (n)
j =1
i=1
(3.11)
!
X
Xi
ai = j,
ai pi = n,
i
p 1 > p 2 > > p mj
(j )
(z, z ) = 0.
(3.12)
This identity which now appears trivial, will acquire a particular meaning in the following.
It is so evident that the quantity zz (n, (z, z )) will label the complex structure of the
space Z (n) in the (z, z ) background and increasing the order n this complex structure will
explore the complex structure of all the (z, Y ) space.
480
(z, Y ) =
(n)
n1
nZ
.
z (z, z )YZ
(3.13)
i
(n)
(n)
n
d z .
z Z (z, z ) d YZn + Z
z (z, z ) n, (z, z ) dYZ
n=1,...,nmax
(3.14)
From the very definition, the Beltrami parameter will take the general form
(n) (z, z )
Z
zz n, (z, z )
Z (n) (z, z )
ai #
mj " Z (pi )
n
X
Y
z (z, z )
j!
=
(n)
Z
ai !z (z, z )
j =1 i=1
X
Xi
ai = j,
ai pi = n,
i
p 1 > p 2 > > p mj
(j ) (z, z ),
z
(3.15)
(3.16)
We remark that, due to Eq. (2.20) the presence of s in the YZ derivative compromises the
locality requirements but the parameters in Eq. (3.16) introduce a suitable parametrization
for a local Lagrangian Quantum Field Theory use. Furthermore these ones have to be
considered as the least common factors for all the Beltrami factors of the spaces Z (r) (z, z ),
r = 1, . . . , n.
Note that the Beltrami multiplier zz (n, (z, z )) will depend on the s parameters of the
spaces parametrized by the Z (i) coordinates with i 6 n.
Under a change of background coordinates the Beltrami multiplier transforms as
0
(3.17)
zz n, (z, z ) = zz 0 n, (z0 , z 0 ) ( 0 z)( z 0 ).
So we can derive
0
) 0 0
0 n 0
) = z(n
(n)
z (z, z
0 (z , z )( z) ( z ),
(3.18)
nz
481
(n)
(n)
z
Z
Z
=
z (z, z ) = z (z, z )z n, (z, z ) .
It is not only obvious that Z
z
(n)
(n)
(3.19)
(j )
zz (n, (z, z )) will depend, through Z , r < n, in a non-local way on the z s with
j 6 n.
(c) As a consequence of the previous statements and of Eq. (3.9) the coordinate
(j )
Z (n) (z, z ) is a non-local function of z (z, z ) with j 6 n.
These are important geometrical statements of the work and are the basis for the physical
discussion of the problem. So the mappings
(z, z ) (Z (n) , Z (n) ),
n,
(3.20)
0, r > 1. In this case the complex structure of all the (Z , Z ) space will coincide with
the (Z (1) , Z (1)) one. So necessarily r > 2
Y )|YZ ,Y =0
[Dz ]r (z,
Z
1
r X n (n)
=
YZ Z (z, z ) Y ,Y =0 = 0.
Z Z
(z, Y ) YZ
n
(3.21)
r > 2,
background, gives
which, expressed in terms of the (Z, Z)
(z, z ) Z (r) = 0, r > 2,
(3.22)
(3.23)
Y )|YZ ,Y =0
[Dz ]r (z,
Z
r X n (n)
1
=
YZ Z (z, z ) Y ,Y =0 = 0,
Z Z
(z, Y ) YZ
n
(3.24)
482
one recovers
(r) (z, z )
Z
Z (r) (z, z )
(l) (z, z )
Z
l, (z, z ) ,
Z (l) (z, z )
r > l + 1,
(3.25)
which leads to
Z (l) Z (r) = 0,
r > l + 1.
(3.26)
Thus Z (r) is holomorphic in Z (l) . Obviously the inverse is always true. So we can state:
(n) is an
Theorem 3.3. The set of conditions: (r)
z = 0, r = l + 1, . . . , n, will imply that Z
(l)
holomorphic function of Z and vice-versa.
(n)
Z
Z (r)
(3.28)
So that the quantity ZZ (r) (n, (Z (r) , Z (r) )) is the Beltrami multiplier of the coordinates Z (n)
in the (Z (r) , Z (r) ) background n. So we can relate these quantities to the corresponding
objects relatively to the (z, z ) background, n and r
(n)
Z (r) , Z (r) (z,z)
Z
Z (r)
(r)
) 1 (r, (z, z ))(n,
(z, z ))
Z
z (z, z
(3.29)
= (n)
n, (z, z ) ,
Z
z (z, z ) 1 (n, (z, z ))(n,
(z, z ))
(r)
and
(r)
ZZ (r)
n, Z (r) , Z (r)
(r)
Z (z, z ) (r, (z, z )) (n, (z, z ))
= (r)
.
(z,z)
Z (z, z ) 1 (r, (z, z ))(n,
(z, z ))
Also we derive
Z n ) (r) (r)
(n)
Z (r) n, Z , Z
1 Z (r) n, Z , Z
=
(n)
Z (n) (z, z )
Z
z (z, z )z
(r)
(r)
Z (z, z ) Z (z, z )
(z, z ))
.
1 (r, (z, z ))(r,
(z, z ))
(3.30)
(z,z)
(3.31)
483
4. w BRS algebras
The previous construction introduces to a BRS derivation of w-symmetry as shown
in [17]. Our aim is to construct a BRS differential which considers, in an infinitesimal
approach all the mappings (z, z ) (Z (n) , Z (n) ), for all n, on the same footing.
Consider first the infinitesimal variations (z, Y ) of the generating function (z, Y )
under the diffeomorphism action of the cotangent bundle. Then by taking the expansion
(2.35) one can proceed as follows [17]. Let S be the BRS diffeomorphism operator acting
on the (z, z ) basis and defined for each n by
1
n
(n)
(n)
(z, Y )
.
(4.1)
SZ (z, z ) (z, z )
n! Y
YZ ,YZ =0
(4.2)
We shall decompose
(n)
z
)
(n) (z, z ) Z
z (z, z )Kn (z, z
(4.3)
Z (n)
1
(z, Y )C(z, Y )
n! YZ
Z (n)
= z
YZ ,YZ =0
(4.4)
and consequently
SKnz (z, z ) = Knz (z, z )Knz (z, z ).
(4.5)
a
!
i
z
j =1 i=1
ai = j,
Xi
ai pi = n,
i
p 1 > p 2 > > p mj
n = 1, 2, . . . ,
C (j ) (z, z
),
(4.6)
(4.7)
which, for all n, provide, in Eq. (4.6), a geometric expansion with the same non-local
coefficients as in Eq. (3.15).
It is quite easy, from the very definition, to derive that these ghosts transform as
SC (n) =
n
X
r=1
r C (r)z C (nr+1) ,
(4.8)
484
revealing the holomorphic w-character of these ghosts. We remark that the BRS variations
of C (n) (z, z ) depends on the fields C (r) (z, z ) with r 6 n. The upper limit of the indices
of these ghosts coincides with the the upper index of the expansion (2.35), and will
characterize this symmetry: we do not fix it, and our conclusions hold their validity for
any value (finite or infinite) of this index.
The connection between Eqs. (4.6) and (3.15) spreads new light on the connection
between diffeomorphisms and w-algebras by putting into the game the complex structures.
Now the coefficients of these expansions are essentially geometrical factors.
On the other hand the quantities Z (n) (z, z ) in Eq. (3.16) will have the BRS variations
(n)
S(n)
z (z, z ) = C (z, z )
rC
(r)
n
X
(nr+1)
(z, z )
r (r)
z (z, z )C
r=1
(nr+1)
(z, z )z
(z, z )
(4.9)
and
(n)
(p) .
)zz n, (z, z ) =
S Z
z (z, z
(4.10)
So that
Szz n, (z, z )
nz (z, z ).
= Knz (z, z )zz n, (z, z ) zz n, (z, z ) Knz (z, z ) + K
(4.11)
So for each n a diffeomorphic structure with a ghost Knz (non-local in the complex structure
parameter) can be put into evidence from Eqs. (3.15) it is easy to find the form of these
variations in terms of w-components. We shall introduce
nz (z, z )
(z, z ))
(4.12)
for which
= nz (z, z ) + nz (z, z ),
(4.13)
so
Knz (z, z ) = n (z, z ) + n, (z, z ) nz (z, z ).
(4.14)
(4.15)
S n, (z, z ) = nz + nz n, (z, z ) n, (z, z )
nz + n, (z, z ) z
nz + n, (z, z ) nz ,
+
(4.16)
(n)
z
z Z (n)
SZ
z (z, z ) = n + n z (z, z )
z
(n)
z
(z,
z
n,
(z,
z
)
.
+ Z
z
n
485
(4.17)
(4.18)
ki !
r,s=0,
i
r+
r+s>0
li ki = j
c(r,s)(z, z ),
(4.19)
1 1
(p,q)
(z, z ) =
(z, Y )
c
p! q! yz (z, Y )
yz (z, Y )
(4.20)
YZ ,YZ =0
r=p,s=q
X
rc(r,s)(z, z )z c(pr+1,qs)(z, z )
r,s=0,
r+s>1
+ sc(r,s)(z, z )z c(pr,qs+1)(z, z ) .
(4.21)
c(p,q)
n
X
z
(r,s)
n, (z, z ) c(r,s)(z, z ),
c(r,s)
with lower
(4.22)
r,s=0,
r+s>1
where
z
(r,s)
n
X
n, (z, z ) =
j!
j =max(r,s)
ai #
mj " Z (pi )
Y
(z (z, z ))
i=1
ai !Z
z
(n)
ki
(l )
z i (z, z )
r!s! i
ki !
in particular
z
n, (z, z ) = 1,
(1,0)
ki = s,
i
X
r+
li ki = j,
Xi
ai = j,
Xi
ai pi = n,
i
p 1 > p 2 > > p mj
z
(0,1)
n, (z, z ) = zz n, (z, z ) .
(4.23)
(4.24)
486
nz (z, z ) = c(1,0)(z, z )
+
n
X
r,s=0,
r+s>2
z
z
(n, (z, z )) (n, (z, z )) (r,s)
(z, z )
(r,s)
(z, z ))
c(r,s)(z, z ).
(4.25)
= Z
(z,
z
)
(z,
z
)
(,) Z (n) , Z (n)
z
z
d z + n, (z, z ) dz
dz + n, (z, z ) d z
(,) (z, z ) dz + n, (z, z ) d z
d z + n, (z, z ) dz .
(4.26)
(4.27)
(4.28)
(4.29)
Now each (Z (n) , Z (n) ) space has a different complex structure, so using the background
representation each field living in this space carries into its transformation the imprinting
of this space
(n)
S(,) Z (n) , Z (n) = (n) Z (n) + Z Z (n) (,) Z (n) , Z (n)
(4.30)
= nz + nz (,) (z, z ).
The same can be done with respect to the background system of coordinates
S(,) (z, z ) = nz + nz (,) (z, z )
+ nz + n, (z, z ) nz (,) (z, z )
z
n (,) (z z )(z, z ).
+ nz + n, (z, z )
(4.31)
487
In conclusion the above ghost decompositions (4.6), (4.22) clarify our strategy towards a
treatment of w-algebras in a Lagrangian Quantum Field Theory framework. Since from the
former it is quite straightforward to derive in combination with the canonical construction
of the diffeomorphism BRS operator, the one induced by the w ordinary algebras. This will
be very useful in the next section.
The diffeomorphism variations of the matter fields (,) (Z (n) , Z (n) ), fix, from our
point of view, their w transformations, since it will be provided by the decomposition of
ghosts nz (z, z ) in terms of the true c(r,s)(z, z ) symplectomorphism ghost fields.
In particular, for the scalar field, the BRS variation (4.31) is rewritten as
S(z, z ) =
n
X
z
z
(z, z ).
c(r,s)(z, z ) n,(r,s)
+ n,(r,s)
(4.32)
r,s=0,
r+s>1
We remark that this description is totally different from the the various approach to w
matter found in the literature, e.g., [1416]. Moreover, according to this viewpoint, it gives
completely trivial the problem.
scalar = dZ Z (Z, Z)
Z
Z
(5.1)
dZ (n) Z (n) Z (n) , Z (n) d Z (n) Z (n) Z (n) , Z (n) .
So we shall start from a model which is invariant under a reparametrization (z, z )
z )), which is well defined but has quantum anomalies.
(Z(z, z ), Z(z,
488
Our strategy for the realization of a w-symmetry in this model will be to consider the
z ) space as an nth element of a w-space hierarchy as in Eq. (3.4).
Z(z, z ), Z(z,
A positive answer for our purposes comes from Theorem 3.2 but more care has to be
exercised.
For this reason the model is to be well defined with respect all the possible
backgrounds. Indeed the Lagrangian in (5.1) written in terms of the (z, z ) background
takes the form
Z
scalar dz d z Lz,z (z, z )
Z
(n,
(z, z ))
Moreover the model is well defined in each (Z (r) , Z (r) ) frame (r) since
Z
scalar = dZ (n) Z (n) Z (n) , Z (n) d Z (n) Z (n) Z (n) , Z (n)
Z
(r)
1
(r)
= dZ (r) d Z (r) 1 ZZ (r) n, (Z (r) , Z (r) ) ZZ (r) n, (Z (r) , Z (r) )
h
i
(r)
Z (r) ZZ (r) n, (Z (r) , Z (r)) Z (r) (z, z )
i
h
(r)
(5.3)
Z (r) ZZ (r) n, (Z (r) , Z (r)) Z (r) (z, z ).
This means that in this framework we can assume as symmetry transformations the changes
of charts
(5.4)
(z, z ) Z (r) (z, z ), Z (r)(z, z ) , r = 1, . . . , n, n
just defined in Eq. (3.4); and the dynamics of the particle, which is free and scalar in the
space (Z (n) , Z (n) ), if described by means of the background of the underlying complex
spaces (Z (r) (z, z ), Z (r) (z, z ), r = 1, . . . , n 1, need the parametrization of the Beltrami
multiplier zz (n(z, z )) just found in the Eq. (3.15).
So at the light of previous arguments and of Theorem 3.2.
Statement 5.1. A two-dimensional conformal model admits, at the classical limit, a wsymmetry of arbitrary order.
Anyhow the quantum extension requires some care.
s
Indeed the s, are non-local functions of the zz (z, z ), so in a local Lagrangian Quantum
Field theory approach, they are not primitive, but they are essential for the geometrical
meaning of our w-construction.
We have just seen in the preceding Lagrangian construction that, if we want to maintain
the well definition of the Lagrangian with respect all the (Z (r), Z (r) ) frames, they are
(r)
contained in the Beltrami ZZ (r) (n, (Z (r) , Z (r) )) due to Eqs. (3.30), (3.15).
If we want to analyze how the underlying complex structure contributes to the dynamics
(r)
fields, r = 1, . . . , n, induced by
the price to pay is to put into the game all the Z
z
489
the decomposition of the (Z (n) , Z (n) ) spaces Eq. (3.2). These fields (even if local in the
(r)
(z, z ) background) are non-local in the z (z, z ), r = 1, . . . , n, fields due to the Beltrami
(r)
(r)
equations (3.19) in each (Z , Z ), r = 1, . . . , n, sectors.
So, from this point of view, the model becomes intrinsically non-local in the fields
(unless n = 1).
We have now to choose the set of fields which exhausts the dynamical configuration
r
(r)
space: so our coordinates will be the fields , c(r,s), z , Z
z , r = 1, . . . , n, and their
derivatives. This means that all the Classical BRS diffeomorphism transformations of these
fields have to be written in terms of the c(p,q) ghosts using the expansion of the various
ghosts C (j ) , Knz and nz as written previously.
So we define as naive BRS functional operator the following c
"
nX
max Z
(n)
z
c =
) nz (z, z )
dz d z Z
z (z, z ) n + n, (z, z
Z (n) (z, z )
n=1
n
X
rc(r,s)(z, z )z c(pr+1,qs)(z, z ) + sc(r,s)(z, z )z c(pr,qs+1)(z, z )
r,s=0,
r+s>1
c(p,q)(z, z )
rC
+
(r)
n
X
(r)
(n) (z, z )
+ C
rz (z, z )C (nr+1) (z, z )
(nr+1)
(z, z )z
(z, z )
(n)
)
nz + nz Z
z (z, z
r=1
(n)
)
z (z, z
z
(n)
z
(z,
z
n,
(z,
z
+ Z
z
n
+
n
hX
(r,s)
(z, z )
r,s=0,
r+s>1
z
n,(r,s)
z
+ n,(r,s)
(n)
Z
)
z (z, z
+ c.c. ,
(z, z )
(z, z )
i
(5.5)
where both the ghosts nz , C (j ) have been expressed in terms of the c(p,q) ghosts according
to Eqs. (4.25), (4.19), respectively.
This is the ordinary diffeomorphism BRS operator, and its nilpotency is verified if the
Beltrami conditions (3.19) hold for all the s [34].
So the invariance of the Lagrangian scalar in Eq. (5.1) is written in a local form
(5.6)
c Lz,z (z, z ) = nz (z, z )Lz,z (z, z ) + nz (z, z )Lz,z (z, z ) .
We now define a set of local operators of zero F-P charge by
Z
nX
max
c(p,q)(z, z )T(p,q) (z, z ) + Sc(p,q) (z, z )
c = dz d z
p,q=0,
p+q>1
c(p,q) (z, z )
. (5.7)
Then thanks to both {c , c } = 0, and Eq. (4.21), it turns out that the T(p,q) (z, z )s fulfill
commutation rules of w-algebra type, see, e.g., [31] and references therein:
490
X
(r)
r, (z, z ) z ,Z (r) Sz(Z ) (z, z ) + c.c. .
r
(5.9)
So the complete Classical Action becomes
Classical = scalar + antifields
(5.10)
(Classical)
(Classical)
+
dz d z
)
z
z
(5.11)
By the way if we want to reproduce here one of the outstanding feature of conformal
models, that is the holomorphic properties of the object coupled in an invariant way to
Beltrami fields (i.e., the EnergyMomentum tensor) the task is not so simple.
This fact is, in this context, particularly fruitful: the presence of n independent complex
structures (and then n independent Beltrami fields) means that we can derive at least n
EnergyMomentum tensors and their related holomorphic properties.
The problem is that the Beltrami multipliers are non-local objects, so the Energy
Momentum tensor cannot be defined in terms of functional derivatives except for the case
n = 1.
491
We can provide a solution by the following shortcut: introduce the following lower
triangular nmax nmax matrix A with entries (r 1, 0)-differentials valued bilocal kernels
(j )
but highly non-local in the z s,
A n, r; (z, z ), (z0 , z 0 )
(r1)
zz (n, (z, z ))
(r)
z (z0 , z 0 )
n = 1, . . . , nmax , 0 6 r 6 n,
(5.12)
(5.13)
r=1
We shall suppose that A has an inverse B with entries (2 r, 1)-differentials valued bilocal
kernels, such that everywhere,
Z
nX
max
00
00
B n, r; (z, z )(z00 , z 00 ) (2r,1)A r, n0 ; (z00 , z 00 ), (z0 , z 0 ) (r1,0)
dz d z
r=1
= n,n0 (2) (z z0 ).
If we define
Pzz n, (z, z )
dz0 d z 0
(5.14)
X
B n, r; (z, z )(z0 , z 0 )
(2r,1) r (z0 , z 0 )
(5.15)
Pzz n, (z, z ) n0 (z0 , z 0 ) = n,n0 (2) (z z0 ).
(5.16)
The Ps play the role of the functional derivative operators with respect the Beltrami
parameters. They will be (as well as these last) non local (in (r) (z, z )) functional
operators and have a fundamental role in our context. If (n, (z, z )) is coupled at the
(Classical)
(n, (z, z )) in an invariant way, for each
tree approximation with a local field (zz)
n = 1, . . . , nmax we have
(Classical)
(5.17)
n, (z, z ) = Pzz n, (z, z ) , (Classical),
(zz)
so that the latter will transform at the classical level as
(Classical)
n, (z, z )
S(zz)
(Classical)
(Classical)
(z, z ) + 2zz
(z, z )Kn (z, z ).
= Kn (z, z )zz
(5.18)
(5.19)
n, (Z
(n)
, Z (n) )
1
Z
z
(Classical)
(zz)
(n) 2
n, (z, z )
(5.20)
492
(n) (n)
JZ(Classical)
,
Z
)
= 0.
n,
(Z
(n)
(n)
Z
Z n
(5.21)
)
dz
+
n,
(z,
z
)
d z .
=
(z,
z
)
z
Z (n)
(5.22)
(5.23)
(5.24)
(j )
n = 1, . . . , nmax ,
(5.25)
and a fortiori
= 0,
T(p,q) Q(Classical)
n
p, q 6 n = 1, . . . , nmax ,
(5.26)
+ (j + 1)z
(s + j + 1)z
(s)
z (z, z )
j =0
Z Classical()
(j +s)
(z, z )
= 0,
1
from which we derive by multiplying by (zz
0 ) and integration,
(5.27)
Z Classical()
(s)
z (z0 , z 0 )
nmax
Xs Z
Z Classical()
zz
(j+s)
(z, z )
j =0
(j +1)
dz d z z
(z, z )
s +j +1
(z z0 )2
(z z0 )
= 0.
493
(5.28)
(j )
Setting all the z s to zero and by quantum action principle one thus gets
Z Classical()
0 0 )(r) (z, z ) =0
(s)
z (z , z
z
Classical
()
Z
= 0,
(r+s1)
z
(z, z ) =0
nmax
Xs
j =0
j +1
s+r
(z z0 )2
(z z0 )
(5.29)
which gives the OPE for the tensors coupled to these objects.
This is valid only at the classical level: the local theory display at the quantum level
anomalies, while the non-local approach admits, as we shall see in the next section,
a rather painless cancellation mechanism.
5.1. Quantum extension and anomalies
The difficulties avoided using a local w-algebra using C (n) (z, z ) or c(r,s)(z, z ) ghosts will
create other problems for the Quantum extension of the model.
Due to quantum perturbative corrections the Action functional may violate the Ward
indentities, and according to the usual Lagrangian framework, one introduces the
corresponding linearized BRS operator,
= ,
"
X (Classical)
(Classical)
+
z,z (z, z ) (z, z )
(r+1,s+1)(z, z ) c(r,s)(z, z )
r,s
X
X (Classical)
(Classical)
+
+
(r)
(s+1)(z, z ) (s)(z, z )
z,Z (r) (r, (z, z )) Z
z (z, z )
s
r
z
X (Classical)
(Classical)
+
+
(z, z ) z,z (z, z )
c(r,s)(z, z ) (r+1,s+1)(z, z )
r,s
X (Classical)
+
(s+1)(z, z )
(s)
s
z (z, z )
#
X (Classical)
+ c.c.
(5.31)
+
(r)
z,Z (r) (r, (z, z ))
Z
z (z, z )
r
(5.30)
dz d z
so only by counterterms inclusion the symmetry will be restored at each order of the
perturbative expansion. As is well-known, this requires a cohomological approach and if
the cohomology is empty, the symmetry is restored at the quantum level.
494
This calculation is performed in the appendix, where we show that the cohomology
sector in the space of the local functions is isomorphic to the tensor product of
the cohomologies of the ordinary disjoint smooth changes of coordinates (z, z )
(Z (r) (z, z ), Z (r) (z, z )), r = 1, . . . , n.
This result shifts the problem to the quantum extension of a theory whose symmetry is
provided by n disjoint ordinary changes of coordinates, for which many known results are
at our disposal [34,35].
In particular if we add to our field content all the ln Z(r) (z, z ), r = 1, . . . , nmax , the
cohomology becomes empty.
The implicit non-locality on the fields of our model softens the possible disappointment coming from the introduction of logarithms.
In the usual Quantum Field Theory the local anomaly is a cocycle which has a
coboundary term which is log dependent and derives from the usual transgression
formulas coming from the FeiginFuchs cocycle, which becomes coboundary if we put
ln s into the game. In our case we get
\ (z, z ) =
=
n
X
r=1
n
X
(5.32)
r=1
modulo coboundary terms and total derivatives; the anomaly in the space of local
functionals is recovered by using the techniques of Ref. [34,36]
(z, z ) =
nX
max
cr Krz (z, z ) 3 zz r, (z, z ) .
(5.33)
r=1
Anyhow the BRS diffeomorphism operator is deeply related to the one of w-symmetry
when we render explicit the Kn (z, z ) ghosts in terms of c(r,s)(z, z ) ones. This allows to
calculate the w local anomalies:
T(r,s)(z, z ) =
nX
max
cn (r,s) n, (z, z ) 3 n, (z, z ) .
(5.34)
n=max(r,s)
(5.35)
(5.36)
(5.37)
495
where, as usual
2
1
(n)
(n)
ln Z
Szz Z (n) (z, z ) 2 ln Z
z (z, z )
z (z, z ) .
2
(5.38)
So we can define
(zz) n, (z, z ) = Pzz n, (z, z ) (Polyakov) .
(5.39)
)
,
(5.40)
P
n,
(z,
z
+
2
c
Z
n
zz
z
(n) 2
Z
z
which is the Quantum extension of the classical (2, 0)-covariant tensor JZ(Classical)
(n) Z (n)
(n, (Z (n) , Z (n) )). Note that due to the presence of the Schwarzian derivative the former
is no longer a tensor.
Moreover we can also define, for each n 6 nmax
Z
Qn = JZ (n) Z (n) Z (n) dZ (n) ,
(5.41)
which will be invariant even in the Quantum level.
6. Conclusions
The many aspects of two-dimensional reparametrization invariance provide a further
geometrical description of w-algebras. We have addressed the question of introducing
local (n, 1)-conformal fields generalizing the usual Beltrami differential appearing
in w-gravity. It was shown that the way out is based on the infinitesimal action of
symplectomorphisms on coordinate transformations dictated by very special canonical
transformations.
Also, it is both interesting and intriguing to note how intermingled the symplectic and
conformal geometries are relevant for all the present treatment. The combination of Beltrami parametrization of complex structures, canonical transformations and symplectomorphisms yields to a BRS formulation of w-algebras.
However, although the locality requirements are fundamental for the physical contents
within a Lagrangian Field Theory, we have overcome them in order to take ever present the
geometrical aspect of the problem. But we aim to treat the former in order to understand
better the role of the Quantum w local anomalies [1416,29,30,32] in relation to the point
of view expressed in the present paper.
496
Acknowledgements
We are grateful to Prof. A. Blasi for comments and discussions.
Appendix
The purpose of this appendix is to show that the cohomology space of our BRS
operator in the space of local functions, is isomorphic to the one of the n independent
reparametrizations (z, z ) (Z (n) , Z (n) ).
We have shown in [34] that the cohomology space in the functional of the BRS
operator will coincide with the local function cohomology of the nilpotent BRS operator
c(1,0) c(0,1).
This cohomology space will be computed by using the spectral sequences method. Let
us filter with
X
.
(A.1)
(p + q) m n c(p,q)(z, z )
=
m n c (p,q) (z, z )
p,q,m,n
At the zero eigenvalue the following operator is obtained,
"
Z
X (Classical)
(Classical)
+
0 dz d z
(s)
(z, z ) z,z (z, z )
z (z, z ) (s+1)(z, z )
s
X (antifields)
+
c(r,s)(z, z ) c=0 (r+1,s+1)(z, z )
r,s
#
X (Classical)
,
+
(r)
Z
) z,Z (r) (r, (z, z ))
z (z, z
r
where
(A.2)
(antifields)
c(p,q)(z, z ) c=0
is the c independent part of the BRS variation of the antifields induced by the
linearization of .
This operator is clearly nilpotent due to the neutrality of the Classical terms. Its
cohomology space can be calculated using again the spectral sequences method. Its adjoint
can be defined upon using the Dixon procedure [37] and the Laplacian kernel is isomorphic
to the cohomology space. The upshot of this calculation does not modify the final result;
anyhow for the sake of completeness we can calculate this space by first filtrating this
operator with the field operator counter, and by calculating the kernel of the Laplacian. It
is easy to convince one self that the cohomology space will be independent on the antifields
z,Z (r) (r, (z, z )),(s+1)(z, z ), z,z (z, z ), r+1,s+1(z, z ) and complicated combinations in the
matter fields and s and s.
The fundamental step takes place in the analyzis of the action on of the filtering
operator (A.1) at the eigenvalue equal to one. In this case we have to calculate the kernel
497
of this operator (and its adjoint) on the space previously calculated with 0 . It is easy to
derive that this operator is nothing else but the sum of the operators c(1,0) c(0,1)
(where is the ordinary diffeomorphism operator of the neutral fields containing the
ghosts c(1,0) and c(0,1) ) plus the total variation of c(p,q) .
This operator is still nilpotent so we can filter it again. We shall choose as filtering
operator the one which counts the c(1,0) and c(0,1) ghost fields, namely,
X
n m c(1,0)(z, z ) n m (1,0)
0 =
c
(z, z )
p,q,m,n
+ n m c(0,1)(z, z )
n m c(0,1)(z, z )
(A.3)
l)!(m
j
)!
j,l,m,n,r,s,p,q ,
p+q>r+s>1
i
( n m c(p,q)(z, z ))
(A.4)
After defining its adjoint according to the Dixon procedure it is easy to find that the
cohomology does not depend on the ghost fields c(p,q) and their derivatives, with the
condition (p + q) > 1.
At the end we are left with the BRS operator induced by the following transformation
rules, for any n
(1,0)
(0,1)
(n)
(1,0)(z, z ) n,1
(z, z )(n)
(z, z ) + c
S(n)
z (z, z ) = c
z (z, z ) + z (z, z )c
(1,0)
n(n)
(z, z )
z (z, z )c
X
(r)
(nr+1)
r z (z, z )z
(z, z ) c(0,1)(z, z ),
(A.5)
(n)
(n)
(1,0)
(z, z ) + c(0,1)(z, z ) Z
)
SZ
z (z, z ) = c
z (z, z
(n)
) c(1,0)(z, z ) + n, (z, z ) c(0,1)(z, z ) ,
+ Z
z (z, z
(A.6)
where (n, (z, z )) must be written according to the expansion (3.15), we thus get the
transformations for the Beltrami differentials at any level n
S n, (z, z ) = c(1,0)(z, z ) + c(0,1)(z, z ) n, (z, z )
(0,1)
(1,0)(z, z ) + n, (z, z ) c
(z, z ) n, (z, z )
+ c
(A.7)
c(1,0)(z, z ) + n, (z, z ) c(0,1)(z, z )
while for the scalar matter field,
S(z, z ) = c(1,0)(z, z ) + c(0,1)(z, z ) (z, z )
c(1,0)
(A.8)
c(0,1)),
Sc(1,0)(z, z ) = c(1,0)(z, z )z + c(0,1)(z, z )z c(1,0)(z, z ).
(A.9)
498
[28]
[29]
[30]
499
[31] X. Shen, W infinity and string theory, Int. J. Mod. Phys. A 7 (1992) 69536993.
[32] A.A. Bud, J.-P. Ader, L. Cappiello, Consistent anomalies of the induced W gravities, Phys. Lett.
B 396 (1996) 108116.
[33] J.-P. Ader, F. Biet, Y. Noirot, A geometrical approach to super W -induced gravities in two
dimensions, Nucl. Phys. B 466 (1996) 285314.
[34] G. Bandelloni, S. Lazzarini, Diffeomorphism cohomology in Beltrami parametrization, J. Math.
Phys. 34 (1993) 54135440.
[35] G. Bandelloni, S. Lazzarini, Diffeomorphism cohomology in Beltrami parametrization: the 1
forms, J. Math. Phys. 36 (1995) 129.
[36] S. Lazzarini, Phys. Lett. B 436 (1998) 73.
[37] J. Dixon, Cohomology and renormalization of gauge theories, I, II, III, unpublished reports.
Abstract
The knowledge of non usual and sometimes hidden symmetries of (classical) integrable systems
provides a very powerful setting-out of solutions of these models. Primarily, the understanding and
possibly the quantisation of intriguing symmetries could give rise to deeper insight into the nature
of field spectrum and correlation functions in quantum integrable models. With this perspective in
mind we will propose a general framework for discovery and investigation of local, quasi-local and
non-local symmetries in classical integrable systems. We will pay particular attention to the structure
of symmetry algebra and to the rle of conserved quantities. We will also stress a nice unifying point
of view about KdV hierarchies and Toda field theories with the result of obtaining a Virasoro algebra
as exact symmetry of sine-Gordon model. 2000 Elsevier Science B.V. All rights reserved.
PACS: 11.30.-j; 02.40.-k; 03.50.-z
Keywords: Integrability; Conserved charges; Symmetry algebra
1. Introduction
What is usually defined as a (1 + 1)-dimensional integrable system is a classical or
quantum field theory with the property to have an infinite number of local integrals of
motion in involution (LIMI), among which the Hamiltonian (energy) operator. This kind
of symmetry does not allow the determination of the most intriguing and interesting
features of a system because of its Abelian character. Instead, the presence of an infinitedimensional non-Abelian algebra could complete the Abelian algebra giving rise to the
possibility of building its representations, i.e., the spectrum (of the energy) and the
spectrum of fields. We may name this non-commuting algebra a spectrum generating
algebra. In different models and in a mysterious way the presence of this spectrum
dfiora@sissa.it
1 stanishkov@bo.infn.it; on leave of absence from INRNE Sofia, Bulgaria.
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 5 1 - 6
501
generating symmetry is very often connected to the Abelian one. This is the case of the
simplest integrable quantum theories the two-dimensional conformal field theories (2DCFTs) their common crucial property being covariance under the infinite-dimensional
Virasoro symmetry, a true spectrum generating symmetry. Indeed, the Verma modules
(highest weight representations) of this algebra classify all the local fields in 2D-CFTs and
turn out to be reducible because of the occurrence of vectors of null Hermitian product with
all other vectors, the so-called null-vectors. The factorization by the modules generated
over the null-vectors leads to a number of very interesting algebraic-geometrical properties
such as fusion algebras, differential equations for correlation functions, etc.
Unfortunately this beautiful picture collapses when one pushes the system away from
criticality by perturbing the original CFT with some relevant local field :
Z
(1.1)
S = SCFT + d 2 z (z, z ),
and from the infinite-dimensional Virasoro symmetry only the Poincar subalgebra
survives the perturbation. Consequently, one of the most important open problems in twodimensional quantum field theories (2D-IQFTs) is the construction of the spectrum of the
local fields and consequently the computation of their correlation functions. Actually, the
CFT possesses a bigger W-like symmetry and in particular it is invariant under an infinitedimensional Abelian subalgebra of the latter [1,2]. With suitable deformations, this Abelian
subalgebra survives the perturbation (1.1), resulting in the so-called LIMI. As said before,
this symmetry does not carry sufficient information, and in particular one cannot build the
spectrum of an integrable theory of type (1.1) by means of LIMI alone.
In the literature, there are several attempts to find spectrum generating algebras at least at
conformal point. For instance in [3,4], some progress was made in arranging the spectrum
generated through the action of objects called spinons in representations of deformed
algebras. The kind of structure built by spinons eliminates automatically all null-vectors
and consequently is unclear how to get (equations for) correlation functions. Besides, the
extension of this method in the scaling limit out of criticality is up to now unknown. More
recently, it has been conjectured in [5] that one could add to LIMI I2m+1 additional noncommuting charges J2m in such a way that the resulting algebra (actually it is not clear
from [5] if these objects close an algebra) would be sufficient to create all the states of a
particular class of perturbed theory (restricted sine-Gordon theory) of type (1.1). Therein
it was also discovered that a sort of null-vector condition appears in the above procedure
leading to certain equations for the form factors. However, what remains unclear in [5]
are the field theory expressions of null-vectors, the general procedure for finding them and
above all the symmetry structures lying behind their arising. Besides, heavy use is made
of the very specific form of the form factors of the RSG model, and it is not clear how
to extend this procedure to other integrable theories of the form (1.1). What is promising
in this work is the link the authors created between (quantum) form factors and classical
solutions of the equation of motion.
With this connection in mind, we present here a general framework to investigate
symmetries and related charges in 2D integrable classical field theories. The plan of the
paper is as follows.
502
In Section 2 we present the central idea, i.e., basing the whole construction on a
generalization of the so-called dressing symmetry transformations [68] connecting these
to the usual way of finding integrable systems [9]. In fact, our basic objects will be
the transfer matrix T (x; ), which generates the dressing, and the resolvent Z(x; ), the
dressed generator of the underlying symmetry. Although it is clear from the construction
that our method is applicable to any generalized KdV hierarchy [9], we will be concerned
(1)
with the semiclassical limit [1014] of minimal CFTs [15], namely the A1 -KdV and
(2)
the A2 -KdV systems [9]. Besides, we will show how to obtain in a geometrical way
from these conformal hierarchies the non-conformal sine-Gordon model (SGM), i.e., the
semiclassical limit of perturbed conformal field theories (PCFTs) (1.1). This kind of
geometrical point of view on the SGM (Toda field theories, in general), is very useful
in derivation of new symmetries and more general Toda field theories are linked to
generalized KdV equations.
In Section 3 we will present an alternative approach to the description of the spectrum
of the local fields in the classical limit of the 2D integrable field theories. We will build a
systematic and geometric method for deriving constraints or classical null-vectors without
the use of Virasoro algebra.
In Section 4 we will propose a further generalization of the aformentioned dressing
transformations. The central idea is that we may dress not only the generators of the
underlying KacMoody algebra but also differential operators in the spectral parameter,
m n , forming a w algebra. The corresponding vector fields close a w algebra as
well with a Virasoro subalgebra (realized for n = 1) made up of quasi-local and non-local
transformations. The regular non-local ones are expressed in terms of vertex operators and
complete the quasi-local asymptotic ones to the full Virasoro algebra. All these vector fields
do not commute with the KdV hierarchy flows, but have a sort of spectrum generating
action on them. Besides, it is very intriguing that only the positive index ones are exact
symmetries of the SGM (apparently the negative index ones do not matter particularly in
this theory). The apparition of a Virasoro symmetry, with its rich and well known structure,
is particularly useful and interesting.
(2)
In Section 5 we will deal with the A2 -KdV showing, as an example of generalization,
the building of the spectrum of local fields through the same geometrical lines as before.
In Section 6 we will summarize our results giving some hint on the meaning and
quantisation of these symmetries in critical and off-critical theories.
(1)
503
algebra G. Various Miura transformations [16] relate it to the generalized KdV hierarchies,
each one classified by the choice of a node cm of the Dynkin diagram of G. However, nodes
symmetrical under automorphisms of the Dynkin diagram lead to the same hierarchy. The
e
e is the
classical Poisson structure of this hierarchy is a classical w(G)-algebra,
where G
finite-dimensional Lie algebra obtained by deleting the cm node. In the simplest case the
usual mKdV equation
t v = 32 v 2 v 0 14 v 000
(2.1)
(2.2)
(2.4)
As all the generalized mKdV, the simplest one (2.1) can be rewritten as a null curvature
condition [17]
[t At , x Ax ] = 0
(2.5)
(1)
1 v 00
1 2
(v v 0 )e0 + (v 2 + v 0 )e1
v 3 h,
2
2 2
(2.6)
(1)
where the generators e0 , e1 , h are chosen in the canonical gradation of the A1 loop algebra
e0 = E,
e1 = F,
h = H,
(2.7)
[H, F ] = 2F,
[E, F ] = H.
(2.8)
(2.9)
The KdV variable u(x, t) is related to the mKdV variable (x) by the Miura transformation
[16]
u(x) = 0 (x)2 00 (x),
(2.10)
504
performs the parallel transport along the x-axis, i.e., the solution of the boundary value
problem
x M(x; ) = Ax (x; )M(x; ),
M(0; ) = 1.
(2.11)
Rx
0
Ax (y,)dy
(2.12)
RL
0
Ax (y,)dy
(2.13)
provided those among the entries of the connection Ax are known [20]. The result of this
calculation is that the Poisson brackets of the entries of the monodromy matrix are fixed
by the so-called classical r-matrix [20]
(2.14)
{M() , M()} = r(1 ), M() M() .
In our particular case the r-matrix is the trigonometric one (calculated, possibly, in the
fundamental representation of sl(2)):
2
+ 1 H H
+
(E F + F E).
(2.15)
1
2
1
By carrying through the trace on both members of the Poisson brackets (2.14) we are
allowed to conclude immediately that
r() =
() = tr M().
(2.16)
(2.17)
In other words, () is the generating function of the conserved charges in involution, i.e.,
it guarantees the integrability of the model la Liouville. Now, it is possible to expand
the generating function () in two different independent ways in order to obtain two
different sets of conserved charges in involution. One is the regular expansion in nonnegative powers of , the other is the asymptotic expansion in negative powers. In the first
case the coefficients in the Taylor series are non-local charges of the theory, instead in the
second one the coefficients in the asymptotic series are the LIMI. Likewise the transfer
matrix T (x, ) can be expanded in the two ways just mentioned giving rise to different
algebraic and geometric structures, as we will see in the following. The regular expansion
is typically employed in the derivation of PoissonLie structures for dressing symmetries
[68]. Instead, the second type of expansion plays a crucial rle in obtaining the flows of
the integrable hierarchy and the local integrals of motion that generate symplectically these
505
flows [9,20]. In this section we will see how the aforementioned approaches are actually
reconducible to a single geometrical procedure which, moreover, produces two different
kinds of symmetries.
2.2. Regular expansion of the transfer matrix and the dressing transformations
The coefficients of the regular expansion of (2.16) are non-local integrals of motion in
involution (NLIMI). However, these latter may be included in a larger non-Abelian algebra
of conserved charges, i.e. commuting with local Hamiltonians of the mKdV (2.1), but not
all between themselves. Actually to get those in a suitable form, we use a slightly different
procedure than the usual one [68], considering a solution of the associated linear problem
(2.11) with a different initial condition. Explicitly, we select the following solution of the
first equation in (2.12) which contains the fundamental primary field e [21]:
!
Zx
H (x)
2(y)
2(y)
P exp dy e
E+e
F
(2.18)
Treg (x; ) = e
0
X
k
K(x1 )K(x2 ) K(xk )dx1 dx2 dxk .
Treg (x; ) = eH (x)
k=0
(2.19)
Now we apply the usual dressing techniques [68] using the previous expression Treg (x; )
and stressing the point of contact with the derivation of an integrable hierarchy given in [9].
If we define a resolvent Z X (x, ) for the Lax operator L = x Ax (2.6) as a solution of
the equation
(2.20)
L, Z X (x; ) = 0,
it turns out that we may build several solutions by mean of a dressing reformulation of the
first equation of (2.11)
T x T 1 = x Ax .
(2.21)
In the specific case of the regular expansion (2.19), if X = H, E, F is one of the generators
(2.8) we get a regular resolvent by dressing
X
1
k ZkX .
(x, ) =
Z X (x, ) = Treg XTreg
(2.22)
k=0
The definition (2.20) of the resolvent is the key property for the construction of a
symmetry algebra, since, once the gauge connection
nX (x; ) = n Z X (x; )
n1
X
k=0
kn ZkX
(2.23)
506
is constructed, the commutator [L, nX (x; )] is in of the same degree of [L, (n Z X (x;
))+ ] and hence of degree zero. Therefore, to get a self-consistent gauge transformation
X
(2.24)
X
n L = n (x; ), L ,
we have to require only that the r.h.s. in (2.24) is proportional to H . This depends, for X
fixed, on whether n is even or odd. Indeed, a recursive relation between the terms ZnX in
(2.22) follows straightforward from the definition (2.20)
x Z0X = 0 H, Z0X ,
X
(2.25)
x ZnX = 0 H, ZnX + E + F, Zn1
and allows us to find the modes ZnX once the different initial conditions are established by
inserting the first term of the expansion (2.19) into (2.22)
Z0H = H,
Z0E = e2 E,
Z0F = e2 F.
(2.26)
The previous two relations yield the various terms of the expansion of the resolvent in the
form
H
H
(x) = a2m
(x)H,
Z2m
H
H
H
Z2m+1
(x) = b2m+1
(x)E + c2m+1
(x)F,
E
E
E
(x) = b2r
(x)E + c2r
(x)F,
Z2r
F
F
F
(x) = b2p
(x)E + c2p
(x)F,
Z2p
E
E
Z2r+1
(x) = a2r+1
(x)H,
F
F
Z2p+1
(x) = a2p+1
(x)H,
(2.27)
where anX ,bnX ,cnX are non-local integral expressions, containing exponentials of the field .
In addition, the variation (2.24) may be explicitly calculated as
X
(2.28)
X
n Ax = Zn1 , E + F
X cannot contain any term proportional to H . The conclusions
and hence it is clear that Zn1
about the parity of n of (2.28) are that
in the Z H case, n in (2.24) must be even,
in the Z E and Z F case, n must conversely be odd.
From (2.25) it is possible to work out simple recursion relations for the coefficients
anX , bnX , cnX in (2.27)
X
X
bn1
,
an0X = cn1
0X
X
2 0 bn+1
+ 2anX = 0,
bn+1
0X
X
+ 2 0 cn+1
2anX = 0,
cn+1
(2.29)
where n is even for X = H and odd for X = E, F . In fact, another way to obtain this
expansion could be to substitute directly the regular expansion (2.19) in (2.22), but the
recursive relations (2.29) (with the initial conditions (2.26)) will provide a contact with
the use of the recursive operator [22] in the theory of integrable hierarchies. Now, the
action (2.28) of the symmetry generators on the bosonic field is given in terms of anX (x)
by making use of (2.29)
0
X
X
n = x an (x)
(2.30)
507
in which n is even for X = H and odd for X = E, F . Instead, using the previous equations
of motions and the relations (2.29) it is simple to show that the action on the classical
X (x)
stress-energy tensor u(x) (2.10) is given in terms of bn1
X
X
n u = 2x bn1 (x),
(2.31)
in which n is even for X = H and odd for X = E, F . Now, we are interested in finding
X and vice versa. If we indicate with 1 =
a recursive relation giving anX in terms of an+2
x
Rx
0 dy, the system of Eqs. (2.29) yields easily
X
,
anX = Ra an+2
X
X
= R1
an+2
a an ,
Ra = vx1 vx + 14 x2 ,
1 2 1 2
R1
x e + x1 e2 x1 e2 ,
a = 2 x e
(2.32)
X and b X
in which n is even for X = H and odd for X = E, F . Similarly, for bn1
n+1
X
X
bn1
= Rb bn+1
,
Rb = 12 u + x1 ux + 12 x2 ,
X
X
= R1
bn+1
b bn1 ,
1
1 1
R1
b = 2x 2 Ra x ,
(2.33)
in which n is even for X = H and odd for X = E, F . The linear differential operators
Ra , Rb are called recursive operators [22] and they generate the integrable flows of an
hierarchy (next section). We have proven here that the proper dressing transformation
(2.30), (2.31) can be thought of as generated by the inverse power of the recursive
operators, i.e., in a compact notation,
n
2
0
X
n = x Ra
aX ,
n1 0
2
X
n u = 2x Rb
bX0 ,
(2.34)
(2.36)
(2.37)
508
X
l=0
nX ZlY
n
X
X
X
, Zl+nm
Zm
1
X
[X,Y ]
l Zl+n
.
(2.38)
l=n
m=0
Now, we must distinguish two cases. In the first case X 6= Y and the first negative powers
are
e = r Z [X,Y ] .
Z
0
(2.39)
But, given the first term of a resolvent, it is completely determined by the recursive relations
(2.25). In the second case X = Y and the last term in the previous equation vanishes.
e is expressed by the series
Therefore Z
!
n
X
X
X X
l
X
Y
e=
n Zl
Zm , Zl+nm
(2.40)
Z
l=0
m=0
and the first non-zero term must be a linear combination of Z0H ,Z0E ,Z0F (see the first of
e = 0. 2
Eqs. (2.25)). It is clear that this is not possible and consequently Z
Lemma 2.2. The equations of motion of the connections (2.23) under the flows (2.24) have
the form:
X Y
[X,Y ]
Y
Y X
(2.41)
X
n s s n = n , s n+s .
Proof. By using the definition of n (2.23) and the previous Lemma 2.1, we obtain
Y
Y X
X
n s s n
= s nX , Z Y ns Z [X,Y ] n sY , Z X + ns Z [Y,X]
[X,Y ]
= s Z Y , n Z X
s Z Y , n Z X
2n+s
[X,Y ]
= s Z Y , n Z X + n Z X
s Z Y , n Z X
2n+s
[X,Y ]
=
s Z Y + + s Z Y , n Z X +
s Z Y , n Z X
n+s
[X,Y ]
=
s Z Y , n Z X +
s Z Y , n Z X
n+s
,
(2.42)
Y
X
Y
X
Y
X
n , s L = n s , L s n , L
Y X X Y
Y
Y X
= X
n s s n , L + s , n , L n , s , L
[X,Y ]
,L ,
= n+s
509
(2.44)
To get from the previous Theorem 2.1 the usual form of the algebra, it is enough
to undertake the replacement and untwist. This kind of transformations are
historically called dressing transformations [68]. In consideration of the fact that all our
symmetries will be obtained by dressing, we will call them proper dressing transformations
(or flows). In the case of mKdV, these flows are non-local except the first ones which have
the form of a Liouville model equation of motion:
0
2(x)
,
E
1 (x) = e
F1 0 (x) = e2(x).
(2.45)
In particular, the Theorem 2.1 means that these infinitesimal variations (2.45) generate by
successive commutations all the proper dressing flows. In addition, from them it is simple
to get the sine-Gordon equation in light-cone coordinates x for the boson
i
,
2
if we define
x = x,
(2.46)
1 E
1 + F1 .
=
x+ 2i
(2.47)
(2.48)
ZL
QX
n
X
Jt,n
= anX (L),
(2.50)
which are not necessarily conserved (depending on the boundary conditions), due to nonlocality.
It is possible to verify by explicit calculations or from the Poisson brackets (2.14) that
the charges themselves close a (twisted) Borel subalgebra A1 C (of the loop algebra
(1)
A1 ).
It is interesting to note that the action by which these charges generate the transformaF
tions (2.30) is not always symplectic, but only in the case of the variations E
1 , 1 . For
instance the following Poisson brackets
510
E
E
F1 v = QF1 , v ,
1 v = Q1 , v ,
H
F
F
E
E
H
2 v = Q2 , v + Q1 Q1 , v Q1 Q1 , v
(2.51)
denote how in the first case the action is symplectic, while in the second it is of PoissonLie
type [68].
As a matter of fact, we will compute in the next section how the transformations (2.24)
act on t At (and the other higher time Lax operators of the hierarchy), finding that they
do as a gauge transformations.
2.3. The integrable hierarchy and the asymptotic dressing
It is however well-known [9,20] that besides the regular expansion of the transfer matrix
an asymptotic expansion exists for the latter. Since this will play an essential role in
our construction, we shall review a few important points in the procedure to obtain the
asymptotic expansion. The main idea is to apply a gauge transformation S(x) on the Lax
operator L in such a way that its new connection D(x; ) will be diagonal:
(2.52)
x Ax (x) S(x) = S(x) x + D(x) .
Because of the previous equation T (x; ) takes the form
T (x; ) = KG(x; )e
Rx
0
dyD(y)
(2.53)
(2.54)
A x = K 1 Ax K.
(2.55)
It is clear now that the previous equation can be solved by finding the asymptotic expansion
for D(x; )
X
d+ 0
i di (x)H i ,
(2.56)
=
D(x; ) =
0 d
i=1
X
1 g+
j Gj (x),
(2.57)
=H +
G(x; ) =
g 1
j =1
(2.58)
In addition, the off-diagonal part, gj (x), can be separated obeying an equation of Riccati
type. The latter is solved by a recurrence formula for the gj (x)
!
j 1
X
v
1 0
gj +1 =
g +v
gij gj .
(2.59)
g1 = ,
2
2 j
k=1
511
In addition, it is simple to see that the diagonal part dj (x), j > 0, is related to gj (x) by
dj = (1)j +1 vgj ,
(2.60)
and is given by d1 = 1, d0 = 0. Note that the d2n (x) are exactly the charge densities (of
the mKdV equation) resulting from the asymptotic expansion of
M() = T (L, )G1 (0, )K 1 ,
() = tr M(),
(2.61)
k ZkH ,
Z0H = H
(2.62)
k=0
(2.63)
The previous equation may be translated into a recursive system of differential equations
for the entries of Z H (x; ) and the solution turns out to have the form
H
(x) = b2k (x)E + c2k (x)F,
Z2k
H
Z2k+1
(x) = a2k+1 (x)H,
(2.64)
where
0
,
a2k+1 = 0 b2k 12 b2k
0
,
a2k+1 = 0 c2k + 12 c2k
0
= c2k b2k .
a2k1
(2.65)
In a way similar to what has been done for proper dressing transformation, we build
through Z H the hierarchy of commuting mKdV flows defining the gauge connections
H
(x; ) = 2k+1 Z H (x; )
2k+1
=
+
2k+1
X
k N,
(2.66)
j =0
(2.67)
H
given by Eq. (2.64) imposes the self-consistency requirement
The form of Z2k+1
H
[n (x; ), L] H satisfied only for odd subscript n = 2k + 1.
The action (2.67) of the mkdV-flows on the bosonic field can be recast in terms of a2k+1
by using the recursive system (2.65)
2k+1 0 = x a2k+1 .
(2.68)
Instead, using the previous equations of motion and the relations (2.65) it is simple to show
that the action on the classical stress-energy tensor u(x) (2.10) is given in terms of b2k (x)
2k+1 u = 2x b2k+2 .
(2.69)
These relations have a form similar to that of proper dressing symmetries and consequently
we are again interested in separating the recursive system (2.65) into one single recursion
512
relation for a2k+1 . It is simple to show that the desired equation involves exactly the same
recursion operator Ra of Eq. (2.32):
a2k+1 (x) = Ra a2k1(x),
Ra = vx1 vx + 14 x2 .
(2.70)
This equation determines uniquely a2k+1 (x), once the initial value of a1 (x) has been given.
For a similar reason we obtain a recursive differential equation for b2k+2 (x)
0
0
000
(x) = 12 u0 b2k + ub2k
+ 14 b2k
.
b2k+2
(2.71)
This equation determines uniquely b2k+2(x), once the initial value of b0 has been given.
Indeed it implies
b2k+2 = Rb b2k ,
(2.72)
where Rb is the same as in Eq. (2.33). The arbitrariness in the initial condition for a2k+1
and b2k will be fixed in the following using the geometrical interpretation of the resolvent
(Eq. (2.74)). The recursive operators Ra , Rb [22] generate the integrable flows of the
hierarchy as implied by (2.67), (2.68):
2k+1 0 = x Rka 0 ,
2k+1 u = 2x Rkb 1.
(2.73)
b0 = 1,
(2.75)
throughout which all the other a2k+1 and b2k can be determined via (2.72) and (2.70). As
a consequence of this fact the b2k are the densities of the LIMI. Indeed, the differential
relation (2.71) (or equivalently (2.72)) coincides (up to a normalization factor of u) with
that satisfied by the expansion modes of the diagonal of the resolvent of the Sturm
Liuoville operator x u [23]. The initial condition b0 = 1 makes the b2k proportional
to the aforementioned modes [23].
The observation (2.74) makes evident the same geometrical origin of integrable
hierarchies and of their proper dressing symmetries. In addition it will allow us in the
sequel to build a more general kind of symmetries and find out their algebra. Indeed, the
first generalization of (2.74) consists in the construction of the flows deriving from the
resolvents
Z F (x, ) = T F T 1 (x, ).
(2.76)
Z E (x, ) = T ET 1 (x, ),
Unlike the previous case, these resolvents possess an expansion in all the powers of
Z E (x, ) =
+
X
i=
i ZiE ,
Z F (x, ) =
+
X
j =
j ZjF .
(2.77)
513
In terms of the data (2.56), (2.57) of the asymptotic transfer matrix, they take the form
2 1
1
(g + 1)2
g
E
2I (x)
e
Z (x; ) =
(2.78)
2
(g 1)2
1 g
2(1 + g+ g )
and
2
1
g+ 1
2I (x)
e
Z (x; ) =
(g+ + 1)2
2(1 + g+ g )
F
(g+ 1)2
,
2
1 g+
(2.79)
1
2
Zx
X
2k1 d2k+1 (y) dy,
d (y) d+ (y) dy =
k=1
(2.80)
which generates the LIMI once calculated in x = L. Now, it is easy one to convince himself
that the entries of the resolvents (2.78), (2.79) admit an expansion in all the (positive and
negative) powers of , i.e., that the modes ZiE and ZjF of the series (2.77) are made up
of linear combinations of all the three Lie algebra generators E, F, H . This implies the
impossibility to satisfy the self-consistency condition L H . Nevertheless, we can go
over this difficulty by defining two other resolvents, combinations of the previous ones
Z (x, ) = T (E F )T 1 (x, ). (2.81)
Z + (x, ) = T (E + F )T 1 (x, ),
By using the expressions (2.78), (2.79), we obtain the following formul for the entries of
Z
h
i
1
2
2
2
2
g
(Z + )11 =
g+
2 cosh 2I + g+
g
sinh 2I ,
2(1 + g+ g )
h
1
2
2
g
g+
+ 2g + 2g+ cosh 2I
(Z + )12 =
2(1 + g+ g )
i
2
2
+ g+
2g 2g+ + 2 sinh 2I ,
g
h
1
2
2
g+
g
+ 2g + 2g+ cosh 2I
(Z + )21 =
2(1 + g+ g )
i
2
2
+ g+
2g + 2g+ + 2 sinh 2I ,
+ g
(Z + )22 = (Z + )11
and
h
i
1
2
2
2
2
g
g+
+ g
2 sinh 2I ,
cosh 2I g+
2(1 + g+ g )
h
1
2
2
g
+ g+
+ 2g 2g+ + 2 cosh 2I
(Z )12 =
2(1 + g+ g )
i
2
2
g
2g 2g+ sinh 2I ,
g+
h
1
2
2
g+
+ g
+ 2g+ 2g+ + 2 cosh 2I
(Z )21 =
2(1 + g+ g )
i
2
2
g+
2g 2g+ sinh 2I ,
+ g
(Z )11 =
(2.82)
514
(Z )22 = (Z )11 .
(2.83)
If we assume to denote by (e) and (o) series with only even and odd powers of ,
respectively, we have
g+ g = (e),
2
2
g+
+ g
= (e),
g+ + g = (o),
g+ g = (e).
2
2
g+
g
= (o),
(2.84)
,
Z =
,
Z =
(o) (e)
(e) (o)
(2.85)
or equivalently by
Z+ =
Z =
+
X
i=
+
X
+
X
+
+
+
2i1 b2i+1
E + c2i+1
F +
2i a2i
H,
i=
+
X
2j b2i
E + c2i
F +
j =
2j 1 a2i+1
H.
(2.86)
j =
It follows that self-consistency requirement may now be satisfied and we are allowed to
define two new series of dressing transformations through the connections
+
(x; ) = 2i Z + (x; )
2i
=
+
2i
X
i Z,
l=
2j +1
Z (x; )
2j
+1 (x; ) =
=
+
2j
+1
X
2j +1l Zl (x),
j Z,
(2.87)
l=
which are no more finite sums. Finally, the following gauge transformations of the Lax
operator L
+
+
Ax = 2i
(x; ), L ,
2j
(2.88)
2i
+1 Ax = 2j +1 (x; ), L
yield this compact form for the additional mKdV flows
+
,
2i 0 = x a2i
2j +1 0 = x a2j
+1 .
(2.89)
These flows are complicated series in x with quasi-local coefficients, so that it would be
very difficult to find their commutation relations by direct computation.
Therefore, to find the algebra of these additional dressing transformations (2.88), we use
the previous procedure based on three steps.
Lemma 2.3. The equations of motion of the resolvents (2.81), (2.74) under the flows (2.88)
have the form:
(2.90)
nX Z Y = nX (x; ), Z Y n Z [X,Y ] ,
where now X, Y = H, E + F, E F .
515
Proof. We omit the specific proof because it can be carried out along the lines of the
analogous Lemma 2.1. 2
Lemma 2.4. The equations of motion of the connections (2.66), (2.87) under the flows
(2.88) have the form:
[X,Y ]
.
(2.91)
nX s sY n = nX , sY n+s
Proof. The proof is analogous to that of Lemma 2.2. 2
Theorem 2.2. The algebra of the asymptotic dressing vector fields (2.67), (2.88) is:
H
+
, k N, i Z,
= 22k+2i+1
2k+1 , 2i
H
+
2k+1 , 2j +1 = 22k+2j +2, j Z,
H
H
= 0, l N,
2k+1 , 2l+1
+
H
(2.92)
2i , 2j +1 = 22i+2j
+1 ,
H
= 0 if k < 0.
where in the last relation we have defined 2k+1
Proof. As in the proof of Theorem 2.1, the action on L of the commutators in the l.h.s.
can be calculated by using the equation of motion of L (2.88), (2.67), the previous Lemma
2.4 and the Jacobi identity. 2
The previous Theorem 2.2 proves that the KdV flows form an hierarchy (they commute
with each other). Besides, they are local and the Lemma 2.4 ensures that each Lax
connection transforms in a gauge way under a generic flow. This is why we may attach
H
and think to each flow as a true symmetry of all the
a time tk to each flow 2k+1
+
and 2j
others. Instead, the additional asymptotic flows 2i
+1 do not commute with the
hierarchy flows, but close an algebra in which they are (in some sense) spectrum generating
symmetries.
It is easy to prove that the proper dressing transformation are true symmetries of the
hierarchy as well.
Lemma 2.5. The transformation of the resolvent (2.74) under the regular flows (2.24) and
the evolution with the times tk of the regular resolvents (2.22) have the same form of the
hierarchy flow of L:
H
H
Z X = 2k+1
, ZX ,
2k+1
(2.93)
nX Z H = nX , Z H ,
where for the regular resolvents we have X = H, F, F .
Lemma 2.6. The mKdV flows (2.67) act as gauge transformations on the connections
(2.23) of the proper dressing flows:
H
H
X
n X
(2.94)
2k+1
n 2k+1 = 2k+1 , n .
516
Theorem 2.3. The proper dressing vector fields (2.24) commute with the mKdV flows
(2.67):
H
(2.95)
2k+1 , X
n = 0.
In particular, the previous Theorem 2.3 implies that the light cone evolution + commutes
with all the KdV flows, i.e., a different way to say that the KdV hierarchy is a symmetry of
H
maps, at infinitesimal level,
the light-cone SG. In particular, the symmetry generator 2k+1
solution of SG into solution.
In consideration of the fact that these theories are classical limits of CFTs and PCFTs,
let us concentrate our attention on the phase spaces of mKdV and KdV systems, i.e., those
objects which at the quantum level constitute the spectrum of fields.
(1)
(3.1)
n
X
0
0
b2i a2i1
b2m+2k2i+2 ,
a2n+2k2i+1
i=0
2k+1 b2n = 2
n1
X
a2n+2k+1 b2n+2k2i a2n+2k2i+1b2i .
(3.2)
i=0
Therefore, according to our conjecture the linear generators of the mKdV identity Verma
module V1mKdV are made up of the repeated actions of the 2k+1 on the polynomials
P{(b2, b4 , . . . , b2N , a1 , a3 , . . . , a2P +1 )} in the b2n and the a2k+1
V1mKdV = {linear combinations of 2k1 +1 2k2 +1 2kM +1 P},
(3.3)
with a natural gradation provided by the subscripts. Actually, the Verma module V1mKdV
exhibits several null-vectors, i.e., polynomials in the b2n and the a2k+1 which are zero.
This is due to the very simple constraint on Z H
2
(3.4)
ZH = 1
517
originating from the dressing relation with the transfer matrix T (2.74). The constraints
(3.4) may be rewritten through the modes of a2k+1 and b2k
C2n =
n
X
n1
X
0
b2n2i b2i + a2i1
a2n2i1 a2i+1 = 0,
+
i=0
(3.5)
i=0
and produce null-vectors under the application of mKdV flows 2k+1 . These latter generate
linearly the graded vector space (Verma module) of all null-vectors
N = {linear combinations of 2k1 +1 2k2 +1 2k3 +1 2kQ +1 C2n }.
(3.6)
In conclusion our conjecture is that the (conformal) family of the identity [1]mKdV of the
mKdV hierarchy is obtained as a factor space:
[1]mKdV = V1mKdV /N .
(3.7)
On the other hand, in order to deduce the form of the Verma module V1KdV of the identity
for the KdV hierarchy we have to make three observations:
1. the recursive formula (2.72) proves that b2n are polinomials of the KdV field u(x)
and its derivatives, whereas the a2k+1 do not enjoy this property;
2. the variation of b2n in (3.2) can be written accidentally in terms of the b2k alone,
using the relationships (2.65) between a2k+1 and b2k
H
2k+1
b2n =
n1
X
0
0
b2i b2i
b2m+2k2i ;
b2n+2k2i
(3.8)
i=0
n
X
i=1
0
b2n2i b2i 2b2 b2n2i b2i2 12 b2n2i b2j
2
0
0
+ 14 b2n2j
b2j
2 = 0,
(3.9)
(3.10)
It turns out to be a sort of reduction of the Verma module V1mKdV (3.3) of the mKdV
hierarchy. As for the mKdV case the (conformal) family of the identity [1]KdV of the KdV
hierarchy is obtained as a factor space of V1KdV over N given by (3.6) and (3.9):
[1]KdV = V1KdV /N .
(3.11)
Therefore we are led to the same scenario that arises also in the classical limit of the
construction [5]. Nevertheless, in our approach the generation of null-vectors is automatic
and geometrical (see Eqs. (3.8) and (3.9)). In addition, our approach is applicable to any
other integrable system, based on a Lax pair formulation. We will illustrate this fact below
518
m
X
mj
Treg 2k+1Treg .
(3.13)
j =1
By successive applications of (3.2), (3.9) and (3.13) we obtain the whole null-vector set
NmKdV . In conclusion the spectrum is again the factor space
mKdV
/NmKdV .
[m] = Vm
(3.14)
Similarly, the construction of the identity operator family suggests the following form for
KdV of the primary e m , m = 0, 1, 2, 3, . . . :
the Verma module Vm
n
KdV
= linear combinations of 2k1 +1 2k2 +1 2kM +1
Vm
o
P(b2 , b4 , . . . , b2N )em ,
(3.15)
with P(b2, b4 , . . . , b2N ) polinomials in b2n . Again, by successive applications of (3.8),
(3.9) and (3.13) we obtain the whole null-vector linear space NmKdV . In conclusion, the
spectrum is again a factor space
KdV
/NmKdV .
[m] = Vm
(3.16)
Of course, we have checked all our conjectures up to high gradation of the null-vectors.
Nevertheless, we did not manage to generate the spectrum of fields only through the
asymptotic symmetry of Theorem 2.2. For dimensional arguments it is plausible to make
the substitution
+
,
a2k+1 2k+1
k Z,
,
b2k 2k
but now the null-vector meaning and origin should be completely different.
(3.17)
519
(4.1)
m < 0,
1
= Tasy lm Tasy
,
m > 0,
V
Zm
(4.2)
where we have to use the different regular and asymptotic transfer matrices, Treg and Tasy .
Of course, they satisfy the usual definition of resolvent
V
(x; ) = 0
(4.3)
L, Zm
and, as in the previous cases, they have two different kinds of expansions
n=0
V
Z1
V
Z1
1
= Treg l1 Treg
=
reg
asy
1
= Tasy l1 Tasy
=
reg
n Zn+1 ,
n Zn1
asy
(4.4)
n=0
and consequently the mode expansion of the more general Virasoro resolvent (4.2). In the
same way, (4.3) authorizes us to define gauge connections
V
mV = Zm
m2
X
reg
m < 0,
n=0
V
mV = Zm
=
+
m+1
X
asy
m > 0,
(4.5)
n=0
(4.6)
=
Finally, we have to verify the consistency of this gauge transformation requiring
V
0
H m for positive and negative m. It is very easy to see that this requirement imposes m
to be even. Indeed, from (4.3) or (4.2) it is simple to derive the form of the generic term of
the expansions (4.4)
VA
m
x
reg
reg
V
V
E + c2n1
F,
Z2n1 = b2n1
V
Z2n = a2n
H,
asy
Z2n3
asy
Z2n2
V
= 2n3
E
V
+ 2n3
F,
n > 0,
V
= 2n2
H,
n > 0.
(4.7)
In addition, we can easily find recursive relations for the regular coefficients
V
V
(x) = R1
b2k+1
b b2k1 (x),
V
V
a2n+2
(x) = R1
a a2n (x),
(4.8)
520
V
V
2n
(x) = Ra 2n2
(x),
(4.9)
where the recursive operators Ra , Rb are given in (2.32), (2.33) and (4.2) fixes the initial
conditions:
Zx
Zx2
V
2 1 2
,
a2 = dx2 dx1 2 cosh (x1 ) (x2 ) ,
b1 = e x e
0
V
= x,
1
0V = x 0 .
(4.10)
m < 0,
m > 0,
(4.11)
V
0
2
=e
V
0
4
V
0
6
=e
2(x)
dy e
2(y)
2(x)
3B3 (x) A2 (x)B1 (x) e2(x) 3C3 (x) D2 (x)C1 (x) ,
= e2(x) 5B5 (x) 3A4 (x)B1 (x) + A2 (x)B3 (x)
e2(x) 5C5 (x) 3D4 (x)C1 (x) + D2 (x)C3 (x) ,
2(x)
0V 0 = 0 + x 00 ,
2V 0
Zx
= 2xa30 + 6g3
2g13
+ 2g10
Zx
d1 ,
(4.12)
where we have used a convenient notation for the entries of the regular expansion
A B
,
(4.13)
Treg (x, ) =
C D
P
P 2n+1
2n
0V e = (xx + )e ,
2V e
Zx
!
d1 e .
(4.14)
521
It is understood of course that these fields are primary with respect to the usual spacetime
Virasoro symmetry. The actions of the variations (4.11) on the generator of this symmetry
can be easily calculated as usual by means of Miura transformation (2.10) and the recursive
relations implied by (4.3)
V
V
u = 2x b2m1
,
2m
V
V
u = 2x 2m+1
,
2m
m < 0,
m > 0.
(4.15)
For example
2V u = x
Zx
3 0
1 0
000
00
2
u uu + u 2u u
u.
2
2
(4.16)
We now apply our usual procedure in three parts to find transformation equations and
symmetry algebra in this case. We will omit the proofs in consideration of the fact that they
are very similar to the previous ones.
Lemma 4.1. The equations of motion of the resolvents (4.2) under the flows (4.6) have the
form:
V V
V V
V
Z2m = 2n
, Z2m (2n 2m)Z2n+2m
, m, n Z.
(4.17)
2n
Lemma 4.2. The transformations of the connections (4.5) under the flows (4.6) have the
form:
V V
V V
V V
V
2m 2m
2n = 2n
, 2m (2n 2m)2n+2m
, m, n Z.
(4.18)
2n
Theorem 4.1. The algebra of the vector fields on L (4.6) forms a representation of the
centerless Virasoro algebra:
V V
V
, m, n Z,
(4.19)
2m , 2n = (2m 2n)2m+2n
after the redefinition V V .
The Theorem 4.1 gives us a very non-trivial information because of the different
character of the asymptotic and regular Virasoro vector fields. Indeed, the asymptotic
ones are quasi-local (they can be made local after differentiating a certain number of
times), the regular ones instead are essentially non-local being expressed in terms of vertex
operators. In addition, it is easy to compute the most simple relations [0 , 2n ] = 2n2n ,
n Z, which means that 0 counts the dimension or level. We want to stress once more
that this Virasoro symmetry is different from the spacetime one and is essentially nonlocal. The additional symmetries coming from the regular dressing are very important for
applications. They complete the asymptotic ones forming an entire Virasoro algebra and
provide a possibility of a central extension in the (generalized) KdV hierarchy, which is
the classical limit of CFTs . However, this central term may appear only in the algebra of
the Hamiltonians of the above transformations, as it is for the case of CFT s as well.
522
With the aim of understanding the classical and quantum structure of integrable systems,
we present here the complete algebra of symmetries. The Virasoro flows commute neither
with the mKdV hierarchy (2.67) nor with the (proper) regular dressing flows (2.24). In fact
one can show, following the lines of the three steps procedure, these statements.
Lemma 4.3. The equations of motion of the resolvents (4.2) under the mKdV flows (2.67)
and of the mKdV resolvent (2.74) under the Virasoro flows (4.6) have the same form of the
variation of L:
H
H
V
V
Z2m
= 2k+1
, Z2m
, k N, m Z,
2k+1
V
V
H
H
(4.20)
2m Z2k+1 = 2m , Z2k+1 .
Lemma 4.4. The mixed transformations of the connections (4.5) under the mKdV flows
(2.67) are not of gauge type:
H
H
V
V
V
H
2m
2m
2k+1 = 2k+1
, 2m
.
(4.21)
(2k + 1)2k+2m+1
2k+1
Theorem 4.2. The algebra of the hierarchy flows and of the Virasoro flows is not Abelian:
H
V
H
,
(4.22)
= (2k + 1)2m+2k+1
2k+1 , 2m
H
= 0 if k < 0 in the r.h.s.
where we have put 2k+1
The content of the previous theorem is that Virasoro symmetry shifts along the KdV
hierarchy.
Likewise, it may be proven that
X V
(4.23)
n , 2m = nX
n2m
after putting the proper dressing flows (2.24) with negative n 2m: X
n2m = 0 in the r.h.s.
As a very important consequence of this fact, the light-cone sine-Gordon flow + (2.47),
(2.48) commutes with all the positive Virasoro modes, i.e., we have obtained a half Virasoro
algebra as exact symmetry of SGM. We note again that this infinitesimal transformation is
quasi-local in the boson .
One remark is necessary at this stage. It is quite interesting to have a Virasoro algebra
not commuting (spectrum generating) with the KdV flows, but one may transform the
Virasoro flows into true symmetries commuting with the mKdV hierarchy by adding a
term containing all the times t2k+1
V
V
2m
2m
H
(2k + 1)t2k+1 2m+2k+1
.
(4.24)
k=1
From the view point of CFT it is very difficult to give a physical meaning to these times,
but from the restriction of the action of the positive part of the Virasoro algebra on u
(4.15) we can check that the previous formula yields the half Virasoro algebra described in
[2426] by using the pseudodifferential operator method. Actually, it plays an important
role in the study of the matrix models where it leads to the so-called Virasoro constraints:
523
Lm = 0, m > 0. Here is the -function of the hierarchy and is connected to the partition
function of the matrix model. Moreover, it seems that it should play an important role also
in the context of the Matrix string theory [27], which is now intensively studied. Note also
that these Virasoro constraints are the conditions for the highest weight state and, because
we also have L0 [25,26], the -function is a primary state for the Virasoro algebra.
But, we uncovered the negative modes of the Virasoro algebra, which build the highest
weight representation over the -function. We are analyzing this intriguing scenario even
in off-critical theories like sine-Gordon [28].
Let us also note that actually the symmetry of mKdV is much larger. Indeed, the
differential operators l2m,2n = 2m+1 2n+1 close a (twisted) w which is isomorphic to
its dressed version
(4.25)
2m,2n Ax = 2m,2n (x; ), L ,
where we have defined the connections and the resolvents
V
) ,
2m,2n = (Z2m,2n
1
Z2m,2n = Treg l2m,2n Treg
,
m < 0,
V
)+ ,
2m,2n = (Z2m,2n
1
Z2m,2n = Tasy l2m,2n Tasy
,
m > 0.
(4.26)
(4.27)
n, m > 0.
(4.28)
As far as we know, this observation is new and we suggest that the diagonal flows are
connected to the higher CalogeroSutherland Hamiltonian flows in their collective field
theory description. This could give a geometrical explicit explanation of the mysterious
connection between CalogeroSutherland systems and KdV hierarchy [29].
5. Generalization: the A(2)
2 -KdV
Let us show that our approach is easily applicable to other integrable systems. Here we
(2)
consider the case of the A2 -KdV equation. The reason is that it can be considered as
a different classical limit of the CFTs [14,30]. Consider the matrix representation of the
A(2)
2 -KdV equation:
t L = [L, At ]
(5.1)
where
L = x Ax ,
and
0 0
e0 = 0 0
0 0
Ax = 0 h + (e0 + e1 ),
0,
0
0 0
e1 = 0
0
0
0,
0
(5.2)
1 0
h = 0 0
0 0
0
0
1
(5.3)
524
!
Zx X
1
h+
h+
1
2
fi i ,
(5.5)
Tasy (x; ) = 0 1 + h03
h01 exp
i=0
0
h2
1 + h3
0
0
0
where h
i , hi are certain polinomials in , f6k , f6k+2 are the densities of the local
(2)
conserved charges of the A2 -KdV and 3 = e0 + e1 . Complying with our approach let
us introduce the asymptotic resolvents
1
,
Z1 = Tasy 3Tasy
1
Z2 = Tasy 32 Tasy
(5.6)
1 a1 + 4 a4 + 2 b2 + 5 b5 +
1 + 6 b6 +
(1)
(1)
(1)
(1)
3
4
2
1 + c3 +
2 a4 +
b2 5 b5 + ,
(1)
(1)
(1)
(1)
(1)
2 c2 +
1 3 c3 + 6 c6 + 1 a1 + 4 a4 +
(1)
Z1 =
and
(1)
(1)
(1)
(1)
2 a + 5 a5 + 1 + 3 b3 + 6 b6 +
4 b4 +
1 2(2)
(2)
(2)
(2)
(2)
4
2
3
Z2 = c1 + c4 +
2 a2 +
1 b3 6 b6 + .
(2)
(2)
(2)
(2)
(2)
1 + 6 c6 +
1 c1 + 4 c4 +
2 a2 5 a5 +
(2)
(2)
(2)
(2)
(2)
For example, some expressions for the fields in the entries of Zi , i = 1, 2, as a function of
v = 0 and its derivative, are:
(1)
a1 = v;
b2(1) = 13 v 2 + 13 v 0 ,
c2(1) = 13 v 2 23 v 0 ;
c3 = 13 v 3 + 13 vv 0 13 v 00 ,
(1)
b3 = 23 vv 0 + 13 v 00 ;
(2)
(5.7)
525
The Eq. (5.1) is invariant under a gauge transformation of the form (2.67). This latter will
be a true symmetry provided the variation is proportional to h: Ax = h 0 . We construct
the appropriate gauge parameters by means of the resolvents (5.6) in a way similar to what
(1)
we did in the A1 -mKdV case:
6k1(x; ) = 6k1 Z2 (x; ) + ,
(5.8)
6k+1 (x; ) = 6k+1 Z1 (x; ) + ,
which results in the following transformations for the A(2)
2 -mKdV field
6k+1 0 = x a6k+1 ,
6k1 0 = x a6k1 .
(1)
(2)
(5.9)
One can easily recognize in (5.9) the infinite tower of the commuting A(2)
2 -mKdV flows.
Now, in accordance with our geometrical conjecture, we would like to treat the entries of
the transfer matrix T and of the resolvents Zi , i = 1, 2, as independent fields and to build
(2)
(1)
the spectrum of the local fields of A2 -KdV by means of them alone. As in the A1 case,
it turns out that not all of them are independent. If the defining relations of the resolvents
are used, it is easy to see, that the entries of the lower triangle of both Zi can be expressed
in terms of the rest. Therefore, taking also into account the gauge symmetry of the system,
one is led to the following proposal about the construction of the Verma module of the
identity:
n
o
(1) (2) (1) (2)
, (5.10)
V0mKdV = l.c.o. 6k1 +1 6kM +1 6l11 6lN 1 P bi , bj , ak , al
where l.c.o. means linear combinations of. Again, null-vectors appear in the r.h.s. of the
(5.10) due to the constraints:
Z12 = Z2 ,
Z1 Z2 = 1
(5.11)
i = 1, 2.
(5.12)
A(1)
1
(1)
level 3: b3 x b2 = 0,
2
level 4: b4(2) b2(1) + 23 x b3(2) = 0,
(1)
(2)
(1) (2)
(2) 2
= 0,
b2(1)
3
+ b3(2)
2
= 0. (5.13)
(2)
Therefore, in order to obtain the true spectrum of the family of the identity (i.e., the A2 KdV spectrum), one has to factor out, from the linearly generated Verma module
n
o
(1) (2)
,
(5.14)
V0KdV = l.c.o. 6k1 +1 6kM +1 6l11 6lN 1 P bi , bj
the Verma module of null-vectors N0KdV , i.e.,
[0] = V0KdV /N0KdV .
(5.15)
526
m
X
T j 6k1 T mj
(5.17)
j =1
obtaining the whole set of null-vectors NmKdV . The first non-trivial examples of these
additional null-vectors are given by:
(1)
level 2: x2 + 3b2 e = 0,
(2)
(1)
level 3: x3 6b3 + 12x b2 e2 = 0,
(1) 2
(2)
(2) 3
+ 30x2b2(1) 27
(5.18)
level 4: x4 + 135
2 b2
2 b4 30x b3 e = 0,
where the operator x acts on all the fields to its right. As a result, the (conformal) family
of the primary field em , m = 0, 1, 2, 3, . . ., is conjectured to be in this case
KdV
/NmKdV .
[m] = Vm
(5.19)
T (x,
) = P exp
Z0
dy e
2(y)
E+e
2(y)
527
eH (x)
,
(6.1)
(6.2)
(6.3)
528
corresponding hyperelliptic Riemann surfaces, and on the basic objects of the Witham
hierarchy [25,26]. This suggests we may found the quantum action of our symmetries (in
particular the Virasoro one) in SGM using the form factor formalism developed in [5].
Acknowledgments
We are indebted to E. Corrigan, P. Dorey, G. Mussardo, I. Sachs and F. Smirnov for
discussions and interest in this work. D.F. thanks the INFNSISSA, the Mathematical
Sciences Departement in Durham and the EC Commission (TMR Contract ERBFMRXCT960012) for financial support. M.S. acknowledges SISSA for the warm hospitality
over part of this work.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
United Kingdom
b Center for Gravitational Physics and Geometry, Physics Department, The Pennsylvania State University,
Abstract
We present a general procedure for constructing new Hilbert spaces for loop quantum gravity on
non-compact spatial manifolds. Given any fixed background state representing a non-compact spatial
geometry, we use the GelfandNaimarkSegal construction to obtain a representation of the algebra
of observables. The resulting Hilbert space can be interpreted as describing fluctuations of compact
support around this background state. We also give an example of a state which approximates
classical flat space and can be used as a background state for our construction. 2000 Elsevier
Science B.V. All rights reserved.
PACS: 04.60.Ds; 04.60.-m
1. Introduction
Remarkable progress has been made in the field of non-perturbative (loop) quantum
gravity in the last decade or so and it is now a rigorously defined kinematical theory. One
of the most important results in this area is that geometric operators such as area and
volume have discrete spectra. However, before loop quantum gravity can be considered a
complete theory of quantum gravity, we must show that the discrete picture of geometry
that it provides us reduces to the familiar smooth classical geometry in some appropriate
limit. One aspect of this is the recovery of the weak-field limit of quantum gravity which
is described by gravitons and their interactions.
In standard perturbative quantum field theory, gravitons are fields which describe the
fluctuations of the metric field around some classical vacuum metric (usually Minkowski
space). The graviton is then a spin-two particle as defined by the representations of the
1 m.arnsdorf@ic.ac.uk
2 gupta@gravity.phys.psu.edu
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 9 6 - 6
530
Poincar group at infinity. Thus, in order to study graviton physics in the context of
loop quantum gravity, a basic requirement is the construction of a state corresponding
to the Minkowski metric, which in turn necessitates a proper quantum treatment of
asymptotically flat spaces. Given this framework one could then construct asymptotic
states (corresponding to gravitons) describing fluctuations of the background metric and
the action of the generators of the Poincar group at infinity.
These are non-trivial requirements in loop quantum gravity, which can be thought of
as describing excitations of the three-geometry itself. Hence, the zero excitation state
of the theory corresponds to the metric gab = 0 and not the Minkowski metric. In
this description, Minkowski spacetime is a highly excited state of the quantum geometry
containing an infinite number of elementary excitations. The situation, in fact, is analogous
to that in finite-temperature field theory where the thermal ground state is a highly excited
state of the zero-temperature theory which does not even lie in the standard Fock space.
Excitations are then constructed by building a representation of the standard algebra of
creation and annihilation operators on the thermal vacuum. In this paper, we shall give an
analogous construction in loop quantum gravity which enables us to describe fluctuations
of essentially compact support around a flat background metric. It is important to note that
these fluctuations can be arbitrarily large and hence we are not quantising linearised general
relativity. Indeed, our framework aims to identify quantum linearised general relativity as
a sector of the non-perturbative theory.
This work is comprised of two main parts. After a brief review of concepts from loop
quantum gravity, we describe how new Hilbert spaces for loop quantum gravity applicable
to non-compact spaces can be constructed by finding new representations of the standard
algebra of observables given some notion of vacuum or background state.
We proceed to present a detailed example of a background state Q, which approximates
the Euclidean metric on three-space. This state is related to the weave construction, but
differs from it as it is peaked not only in the spin-network basis but also in the connection
basis. We show that even though this state is not an element of the standard Hilbert space, as
is generically the case for states approximating geometries on non-compact spaces, it can
be used as a vacuum in the above construction to give genuine Hilbert spaces describing
fluctuations around this state.
531
ec . The
lie algebra su(2). The triads can be considered as duals to two-forms eabi abc E
i
dynamics of general relativity on spatially compact manifolds is then completely described
by the Gauss constraints which generate SU(2)-gauge transformations, the diffeomorphism
constraints which generate spatial diffeomorphisms on , and the Hamiltonian constraint,
which is the generator of coordinate time evolution. In the non-compact case, true
dynamics is generated by the boundary terms of the Hamiltonian.
For compact spatial manifolds , a well defined quantisation procedure for the above
setup has been developed, which we review before discussing our extension to the noncompact case. The strategy is to specify an algebra of classical variables Baux and then
to seek a representation of this algebra on some auxiliary Hilbert space Haux . The second
step is to obtain operator versions of the classical constraints and to then impose these on
the Hilbert space to obtain a reduced space of physical states along with a representation
of the subalgebra of observables that commute with the constraints.
2.1. The classical algebra of observables
To obtain the classical algebra of elementary functions which can be implemented in
the quantum theory, we need to integrate the canonically conjugate variables, Aia and
eiab , against suitable smearing fields. In usual quantum field theory, these fields are threedimensional. However, in canonical quantum general relativity, due to the absence of a
background metric, it is more convenient to smear n-forms against n-dimensional surfaces
instead of the usual three-dimensional ones [3,6,20].
Configuration observables can be constructed through holonomies of connections. Given
an embedded graph which is a collection of n paths {1 , . . . , n } , and a smooth
function f from SU(2)n to C, we can construct cylindrical functions of the connection:
f, (A) = f (H (A, 1 ), . . . , H (A, n )).
H (A, i ) SU(2) is the holonomy assigned to the edge i of by the connection A A.
We denote by C, the algebra generated by all the functions of this form. This is the space
of configuration variables. To obtain momentum variables, we smear the two-forms eabi
against distributional test fields t i which take values in the dual of su(2) and have two
dimensional support. This gives us
Z
Et,S eabi t i dS ab ,
S
532
AA
The key to representing C is to define a positive linear form on it. This can be done using
the Haar measure dg on SU(2) as follows [7]
Z
Z
dg1 dgn f (g1 , . . . , gn ),
(1)
(f, ) = d(A) f,
SU(2)n
where gi SU(2). Note that the right hand side does not depend on . Nevertheless, our
definition makes sense since, if 5 f, = f0 0 , 0 , then () = ( 0 ). This allows us to
define the (standard) inner product:
Z
h1 |2 is = (1 2 ) =
f1 (g1 , . . . , gn )f2 (g1 , . . . , gn ) dg1 dgn .
(2)
SU(2)n
5 This holds if consists of analytic paths but extensions to the non-analytic case are possible, cf. [11].
533
Here we make use of the fact that if the functions f1 and f2 have a different number of
arguments, say f1 : SU(2)m C with m < n, we can trivially extend f1 to a function on
SU(2)n , which does not depend on the last n m arguments. Since this product is already
positive definite, we can proceed directly with the completion of C to obtain our auxiliary
Hilbert space Haux carrying a multiplicative representation of the algebra C. Haux can
also be regarded as space of square integrable functions defined with respect to a genuine
measure on some completion A of A as is done in [5].
We are left with the task of representing the momentum variables on this Hilbert
bt,S on C which can
space. This done by constructing essentially self adjoint operators E
be extended to Haux . These operators are derivations on C, i.e., linear maps satisfying
the Leibnitz rule, which act on functions f, C only at points where intersects the
oriented surface S. The precise definition of these operators is not needed for our purposes,
but it can be found, e.g., in [3]. This choice of operators gives the correct representation of
the classical algebra Baux , which provides us with a kinematical framework for canonical
quantum gravity. In the following this representation will be referred to as the standard
representation s . To obtain physical states we need to introduce the quantum constraints
and study their action.
2.3. The constraints
The simple geometrical interpretation of the Gauss and diffeomorphism constraints
allows us to bypass the attempt to construct the corresponding constraint operators. Instead,
we can construct unitary actions of the gauge group G and the diffeomorphism group D
on Haux and demand that physical states be invariant under these actions. The imposition
of the Hamiltonian constraint is still an open issue and we will not discuss it. In this sense,
our entire discussion is at the kinematical level.
The group G has a natural action on the space of connections which induces a unitary
action of G on Haux . Gauge invariance is simply achieved by restricting to the subspace
HG Haux of gauge invariant functions. It can be shown that this space is spanned by the
so-called spin networks states [9,10].
If we try to follow the same procedure for the diffeomorphism constraint, we find that
there are no non-trivial diffeomorphism invariant states in HG . This problem is overcome
by looking for distributional solutions to the constraints. Again the natural pull-back action
of the diffeomorphism group on the connections induces a unitary representation of D on
Haux because of the diffeomorphism invariance of the inner product (2). One then considers
a Gelfand triple construction F HG F 0 , where F is a dense subspace 6 of HG and F 0
its topological dual and identifies the reduced Hilbert space Hkin with a subspace of F 0 that
is invariant under the dual action of the diffeomorphism constraint. Hkin carries a natural
dual action of the algebra Bkin , which is the subalgebra of Baux containing the elements
that commute with the constraints.
6 F is usually chosen to be C.
534
2.4. Problems
As we stated in the introduction, we want to study states which represent asymptotically
flat classical metrics, especially Minkowski space. We now argue that such states
generically do not lie in the Hilbert space Haux constructed above, which is the reason
that this representation is not adequate for non-compact .
States which describe non-compact geometries should either be based on curves of
infinite length or an infinite number of curves. This is because the area and volumes of
regions of which do not contain edges and vertices of graphs vanish. A particular
example of an attempt to construct a state which approximates a chosen flat Euclidean
3-metric gab on is the so-called weave given in [14]. This weave is based on an infinite
S
collection of graphs r, =
i=1 Di , where Di is the union of two randomly oriented
circles ia and ib of radius r, which intersect in one point. To ensure isotropy these graphs
are sprinkled randomly in , with sprinkling density , where is defined with respect to
gab . Given n double circles Di we can construct the cylindrical function Wn :
Wn (A) =
n
Y
Tr 1 H ia , A H ib , A ,
(3)
i=1
where 1 denotes the fundamental representation of SU(2). The weave state W should
arise in the limit n . This limit does not exist in the Hilbert space Haux , since the
above sequence Wn is not Cauchy in either the sup norm or the L2 norm based on the
inner product (2). This holds even if we impose physically reasonable (non-uniform) falloff conditions on the connections, such as those for asymptotically flat gravity. The basic
problem is that for any curve embedded in we can always find a connection that will
assign to this curve any holonomy we choose. 7 In particular, this means that if we have a
state f (H ( , A)) then sup |f (H ( , A))| will be independent of the location of . This is a
generic result, and we conclude that a large class of physically interesting states based on
infinite collections of graphs do not exist in Haux . Similar arguments can be used to show
that states based on curves of infinite length do not lie in Haux either.
A very natural way of dealing with states based on a finite number of curves of
infinite length was introduced in [1]. The key point is to consider a compactification of
and show that these states belong to the auxiliary Hilbert space constructed on the
compactified manifold. Problems arise when trying to extend this approach to discuss
cylindrical functions based on graphs with infinite number of edges and vertices as cluster
points of vertices necessarily arise in the compactified manifold. This just illustrates the
fact that the Hilbert space Haux , along with the representation of observables it carries,
was constructed for compact spatial slices and is not adequate to describe the case of noncompact . In the next section, we propose a different solution to the above problems by
giving a procedure to construct new Hilbert spaces for quantum general relativity, which
describe fluctuations around specified background states. In particular, these states can
have non-compact support on the spatial manifold.
7 We thank John Baez for this observation.
535
536
highly delocalized in their time derivative. Intuitively, this means that while a weave may
approximate the 3-metric at one instant of time, evolving the state for even an infinitesimal
time will completely destroy this approximation. In this section, we will construct a more
satisfactory set of states that can also be used to approximate 3-metrics. In particular, these
states can be used to define a positive linear form on Baux as is needed for the GNS
construction even in the case that we want to approximate a non-compact geometry.
3.1. Approximating 3-metrics
Let us now take a closer look at the weaves. Their construction is made possible by the
existence of operators on Haux which measure the area of a surface and the volume of a
region [4,8,13,21]. This allows us to approximate classical metrics by requiring that the
expectation values of areas and volumes of macroscopic surfaces and regions agree with
the classical values.
For concreteness, in the rest of this section, we shall restrict ourselves to the problem
of approximating the flat Euclidean metric on R3 . Let w be a state which approximates
the flat space at scales larger than a cut-off scale lc . The approximation problem can then
be stated as follows: Given any object (of characteristic size larger than lc ) with bulk R
and surface S in R3 , we wish to make repeated measurements of the volume of the region
V [R] and the area of the surface A[S] in the state w while placing the object at different
points in space. If we wish to recover values corresponding to the flat metric gab on R3 at
large scales, we require:
(1) The average values of the area and volume for S and R obtained during the
measurements should be given by the classical values:
Z q
hAS i = Ag [S]
det 2 gab ,
(4)
S
hVR i = Vg [R]
Z p
det gab ,
(5)
R
2g
ab
denotes the induced 2-metric on S. Here a bar over the value indicates
where
an average with respect to position in space whereas the angle brackets indicate the
expectation value in the quantum state.
(2) The standard deviation of the measurements should be small compared to the
length scale `c accessible by current measurements:
V `3c
and A `2c ,
(6)
537
n
Y
qi (A).
i=1
The state that we are interested in, is the limit Q = Q , which is again not an element
of the standard Hilbert space. Nevertheless, as we will see, this product can serve as a
background for the construction of new Hilbert spaces, describing excitations of Q.
Let us denote the connection that gives a trivial holonomy on all paths by A0 . By
construction the functions Qn (A) take on their maximum values at A0 . Conversely,
knowledge of the holonomies on all paths in allows us to determine a corresponding
connection uniquely. Hence as n the function Qn becomes increasingly peaked
around A0 , the sharpness of the peak being determined by . This is one of the reasons for
calling our weave a quasi-coherent state, the other being the exponential dependence on
group elements which is characteristic of coherent states. We will explore these properties
of the state further in future work. For the present, we are interested in showing that Q is
a good weave. In order to do so, we need to show that it satisfies the weave conditions (4),
(5) and (6). We shall do this by demonstrating that standard
deviations of area
p
p and volume
measurements are roughly of the order of `c `P and (`c `P )3 , where `P = h GNewton/c3
is the Planck length and hence much smaller than the bounds set by Eq. (6). If only
interested in how Q can be used to construct new Hilbert spaces the reader may skip to
the next section.
Let us start by expanding the state qi (A) in into an eigenbasis of the area operator
and calculate the area expectation values and deviations. We do this by noting that the
cylindrical function fp (A) = Tr[p (H (ia , A)H (ib , A))] based on the graph Di is an
eigenstate of the area operator, 9 where p is a representation of SU(2) in colour notation,
to some
i.e., p = dim() 1. The eigenvalues ap of the area operator corresponding
q
surface S, which intersects Di exactly once, are given by 16`2P p2 ( p2 + 1). Thus, to
evaluate the area expectation value hai and the deviation a of this operator we start by
P
expanding the function qi (A) in terms of the area eigenstates fp (A): qi = p sp fp . We
9 This follows since f can be expanded in terms of spin-network functions that all assign to each edge of
p
p
Di . Hence all spin-network functions in the expansion have the same area eigenvalues.
538
want to determine the coefficients of this expansion. We start by noting that qi is defined
by its series expansion (the i index labelling the graph will be suppressed in the following):
2 f12 3 f13
+
+ .
q = e2 1 + f1 +
2!
3!
Hence, we need to expand f1n in terms of fp to determine the sp s. This can be done by
using the decomposition rules for tensor products of representations of SU(2):
Trn [1 (g)] = Tr p cpn p (g) .
P
Hence, it follows that f1n = p cpn fp . To determine the coefficients cpn , we use the fact
that p 1 = p1 p+1 and that:
1 . . . 1 = cpn1 p 1 .
|
{z
}
n
n, p > 0, p 6 n.
Using the condition c00 = 1, we can solve this recursion relation to get:
cpn =
(p + 1)n!
for
n+p
( np
2 )!( 2 + 1)!
np
N,
2
(8)
and cpn = 0 otherwise. The expansion coefficients sp (as a function of ) are given by
sp () = N ()
n n
X
cp
n=0
n!
P
2
where N () is defined such that
p=0 sp = 1. Substituting Eq. (8) into the right hand side
of the above expression, we find:
(p + 1)
2k+p
= N ()
Ip+1 (2)
k!(k + p + 1)!
k=0
= N () Ip (2) Ip+2 (2) ,
sp () = N ()(p + 1)
where Ip (x) is the modified Bessel function of order p. Using the properties of Bessel
functions, we can then evaluate the normalisation constant to be
1/2
.
N () = I0 (4) I2 (4)
Fig. 1 shows the numerical values of the coefficients sp2 for a few values of .
The final form of the expansion of qi in terms of the area eigenstates fp is
q(A) =
X
Ip (2) Ip+2 (2)
fp (A).
I0 (4) I2 (4)
p=0
(9)
We needed to show that the states qi are peaked in area and volume. Let us consider, in
particular, = 30. In this case, we have:
q
and a ha 2 i hai2 = 1.87 16`2P ,
hai = 4.33 16`2P
539
Fig. 1. Normalised coefficients sp2 for = 10, = 30 and = 50. The sharpness of the peak
decreases with increasing .
and hence both values are of order `2P . The explicit calculation for the volume expectation
value and deviation, hvi and v , in the state qi is somewhat more complicated since the
states fp have to be expanded in volume eigenstates. After doing this, we find:
q
3/2
3/2
and v hv 2 i hvi2 = 2.21 16`2P
.
hvi = 3.61 16`2P
Again, both values are of the order of a few `3P .
In general, the area expectation value for the area as a function of can be written as
s
X
Ip (2) Ip+2 (2) 2 p p
2
+1 .
hai = 16`P
2 2
I0 (4) I2 (4)
p=0
We do not have a closed form for hvi since the evaluation of the volume expectation
value involves the diagonalization of a matrix. However, since the largest eigenvalue of
the volume operator in the state fp increases with p as p3/2 , for a value of for which
the expansion is dominated by a few small ps, we can see that hvi has to be of the order
of a few `3P . Given the fact that both hai and hvi have value of the order of a few Planck
units, it is reasonable to assume that the bound (6) is satisfied for area and volumes in the
state Q, which is a product of the qi s, as well. We now show that this is indeed the case.
bR depends upon the number
The expectation value of the volume operator for a region V
NR of double circles in the region R, namely:
hVR i = NR hvi.
Similarly, the expectation value for the area operator of a surface A S depends on the
number of intersections NS of the double circles with the surface:
hAS i = NS hai.
540
Because the double circles Di are sprinkled randomly in the number of the graphs in
any given region R of volume Vg [R] is given by a Poisson distribution. In particular, the
average value is given by N R = Vg [R]. Hence average values of the above measurements
are given by:
hVR i = N R hvi
(10)
hAS i = N S hai,
(11)
and
where N S = 23 V [S] and V [S] = 6rAg [S] denotes the volume of a shell surrounding S
with thickness 3r on either side. The factor 2/3 is the average number of crossings between
a double circle within this shell with S as determined in [14].
To determine the standard deviations V and A around hVR i and hAS i we use an
bR and A S are given by the product of
approximation: we assume that the eigenvalues of V
two independent quantities, i.e., NR (VR /NR ) and NS (AS /N S ) respectively, where VR and
vp and ap . The deviations of the quantities
AS are sums of any n elementary eigenvalues
q
p
2
NR and NS are given by N R and 3 N S , since we are dealing with a Poisson distribution
p
p
of graphs. Deviations of VR /N R and AS /N S on the other hand are v / N R and a / N S .
Our approximation implies that:
q q
q r
2 2
2
V N R 2v + hvi2 .
A NS a + hai ,
3
From Eqs. (4), (5), (10), and (11), we find: N S = Ag [S]/hai and N R = Vg [R]/hvi.
Because of the scale of hai, a and hvi, v we conclude that Eq. (6) is satisfied. Thus, we
have shown that Q is a weave state. For any particular value of , we can determine and
r using Eqs. (10), (11), (4) and (5):
1
= hvi1 and r = 4hai .
S
We will denote the infinite collection
i=1 Di of double circles Di with r and
determined by the above conditions by Q .
4. New Hilbert spaces
We now show how we can use Q to define a new representation of the algebra
Baux following the steps outlined in Section 2.5. We begin by noting that we have a
representation s of Baux on Haux , but as we have seen Q does not belong to this Hilbert
space. Nevertheless, we can define the action of an element of Baux on Q. The crucial
point is that the elementary quantum observables the elements of C and the derivations
on C have support on a compact spatial region, which is a direct consequence of the
smearing needed to make sense of the classical expressions. But if we restrict Q to any
compact region of we obtain an element of Haux by restricting the underlying graph Q
to that region. Hence given an arbitrary element a Baux we proceed as follows:
(1) Denote the closure of the support of a by R .
541
In other words, consider the union of all double circles which have a non-zero
intersection with the support of a. This graph is finite, since R is compact and the
double circles Di are sprinkled in with finite density . We obtain the state Q|R
Haux which is given by restricting Q to the graph Q |R :
Y
qi ,
Q|R I
Di Q |R
where I(A) = 1 for all A is the identity function. This state has unit norm in Haux
since all the qi s are normalised.
(3) Since Q|R Haux , the action of a on Q|R denoted by (a)s Q|R is well-defined. It
is understood that the region R will depend on a.
This allows us to define Q (a):
Z
(12)
Q (a) = hQ|R |s (a)Q|R is = Q|R s (a)Q|R d(A),
where the integral is defined as in Eq. (1). This is well-defined since the integrand is an
element of Haux . It follows from the fact that Eq. (2) defines a true inner product h|is on
Haux that is indeed a positive (not necessarily strictly positive) linear form on Baux .
Given this positive linear functional, we could proceed with steps 2 and 3 of the GNS
construction outlined in Section 2.2 to obtain a representation of the algebra Baux . Instead,
to get a more intuitive representation we make use of the theorem below to construct a
unitarily equivalent representation Q of Baux on a Hilbert space HQ obtained by defining
a new inner product on C.
Theorem 1. Any representation of a -algebra A with cyclic vector 10 such that:
h|(a)i = (a),
for all a A is unitarily equivalent to the GNS representation , with cyclic vector
(corresponding to the unit element in A).
Proof. The proof of this theorem is analogous to the one of Proposition 4.5.3 in [18]. To
proceed, we note that for each a A,
k(a)k2 = h(a)|(a)i = (a)|(a a)
= (a a) = kak2 ,
where kk2 is the norm in the Hilbert space H which carries the GNS representation. This
means that there exists a norm-preserving, linear operator U0 such that U0 (a) = .
10 The vector H is cyclic with respect to the representation of A on H if (A) is dense in H.
542
(13)
f1 ,1 |f2 ,2 Q = Q|R Q|R 1 2 d(A),
where R here is the union of the graphs 1 and 2 . Completion of C with respect to
this positive definite inner product gives us the Hilbert space HQ .
(2) We construct a representation Q of Baux on C, which is dense in HQ by:
Q (a) = Q|1
R s (a)(Q|R ),
(14)
Q|1
R
543
we have presented is very general and can be applied to a large class of background states
provided they satisfy the necessary invertability condition.
4.1. Constraints and asymptotic symmetries
We can proceed to reduce the Hilbert space obtained in the previous section by
imposing the Gauss and diffeomorphism constraints. In general, when considering GNS
representations of an algebra A we can implement actions of symmetry groups G using
the following theorem (Eq. (III.3.14) in [15]):
Theorem 2. Given an action of G on A: a ga such that (ga) = (a) for all a A
and g G then we can define a unitary representation U of G on H by:
U (g)(a) = (ga) ,
where is a cyclic vector in H .
In practice, when considering the representation Q we can proceed as in Section 2.3 to
reduce the Hilbert space HQ . As before, there is no problem in implementing the Gauss
constraint. Since the state Q|R is invariant under gauge transformations, the inner product
defined in Eq. (13) is also gauge invariant and we have a unitary action of the gauge group
on the state space. Again we implement the Gauss constraint by restricting to the subspace
of HQ consisting of gauge invariant states, which is spanned by spin-networks.
When considering diffeomorphisms, we notice that Q|R is invariant only under
diffeomorphisms that leave the graph Q , on which Q is based, invariant. Let us denote this
subgroup of diffeomorphisms by Diff . Invariance of the inner product under Diff gives
us a unitary representation of this group and we can use the Gelfand triple construction
kin that is invariant under Diff .
detailed earlier to obtain a space of kinematical states HQ
This space then naturally carries a dual representation of the subalgebra Bkin Baux of
operators that commute with constraints.
To discuss the significance of the breaking of diffeomorphism invariance to the group
Diff , we note that given two different background states Q and Q0 which are defined with
respect to diffeomorphic graphs: Q0 = Q , where is any diffeomorphism, we obtain
unitarily equivalent representations of Baux :
U [Q (a)I] = Q0 (a)I,
where unitarity of U follows from the diffeomorphism invariance of the inner product
given by (2). Hence, the choice of a particular member of the diffeomorphism class of Q
is simply a partial gauge fixing. A physically equivalent way of getting the same results
would be to average over the group Diff .
HQ should have the same property, i.e.: Q|1
s (Et,S )(Q|S ) = 0 for all derivations Et,S . But since has
S
essentially compact support in and since we can chose S so that s (Et,S )Q|S 6= 0 outside any compact region,
this cannot be satisfied for all regions S.
544
To conclude, we note that in the context of asymptotically flat general relativity, which
is the prime case of interest involving non-compact spatial manifolds, the invariance
group of the theory is restricted to the connected component of the asymptotically trivial
diffeomorphisms. In the neighbourhood of infinity we would like to have a unitary action
of the Poincar group on our state space. Since physically relevant operators are typically
evaluated at infinity, this invariance is what is of prime interest.
We shall discuss the construction of the action of the full Poincar group in future
work. Here, we show how a unitary action of the Euclidean group E acting on can
be incorporated in our scheme. From Theorem 2, it follows that to do this we need a linear
form on the algebra of observables that is invariant under the action of E. Such a form can
be obtained by using the fact that the Euclidean group is locally compact to group average
the form given in Eq. (12). Given an increasing sequence of compact subsets Sk E,
Sk Sk+1 , Sk = E we define:
Z
1
(ga) d(g),
k (a) = (Sk )
Sk
where d(g) is the invariant measure on E and g E. It can be shown [15] that the
sequence k converges to a positive linear form E , which is invariant under E. Using
this form we obtain a representation of Baux on a state space HE carrying a unitary
representation of E. Equivalently, we could have used this procedure to average the
background state Q, to obtain the desired representation.
545
(3) We are in the process of computing the spectra of the area and volume operators
in the new Hilbert space defined by Q to verify the intuitive picture of areas and
volumes fluctuating around flat space values.
(4) A quantum positivity of energy theorem was proved in [22]. However, as we have
shown, the Hilbert space on which that result was proved is not applicable to the
study of non-compact spatial geometries. We believe that our construction provides
the proper arena for questions of this nature and are currently investigating the
properties of suitably defined ADM energy and momentum operators on our Hilbert
space.
This work is a step in the direction of making contact between the non-perturbative
quantisation of gravity and the picture of graviton physics which arises from standard
perturbative quantum field theory. A lot more work needs to be done before the relation
between the two is completely clarified.
Acknowledgements
We would like to thank Chris Isham, Carlo Rovelli and Lee Smolin for discussions and
motivation. We would also like to thank Abhay Ashtekar for his comments on an earlier
version of this manuscript. S.G. was supported in part by NSF grant PHY-9514240 to the
Pennsylvania State University and a gift from the Jesse Phillips Foundation.
References
[1] M. Arnsdorf, R.S. Garcia, Existence of spinorial states in pure loop quantum gravity, Class.
Quant. Grav. 16 (1999) 34053418, gr-qc/9812006.
[2] A. Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett. 57 (1986) 2244.
[3] A. Ashtekar, A. Corichi, J.A. Zapata, Quantum theory of geometry III: Non-commutativity of
riemannian structures, gr-qc/9806041.
[4] A. Ashtekar, J. Lewandowski, Quantum theory of geometry I: Area operators, Class. Quant.
Grav. 14 (1997) A43, gr-qc/9602046.
[5] A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao, T. Thiemann, Quantization of diffeomorphism invariant theories of connections with local degrees of freedom, J. Math. Phys. 36 (1995)
6456, gr-qc/9504018.
[6] A. Ashtekar, C. Rovelli, L. Smolin, Weaving a classical geometry with quantum threads, Phys.
Rev. Lett. 69 (1992) 237.
[7] A. Ashtekar, J. Lewandowski, Representation theory of analytic holonomy c algebras, in:
J. Baez (Ed.), Knots and Quantum Gravity, Oxford University Press, Oxford, 1994, grqc/9311010.
[8] A. Ashtekar, J. Lewandowski, Quantum theory of geometry II: Volume operators, Adv. Theor.
Math. Phys. 1 (1998) 388, gr-qc/9711031.
[9] C. Rovelli, L. Smolin, Spin networks and quantum gravity, Phys. Rev. D 52 (1995) 5743.
[10] J.C. Baez, Spin networks in nonperturbative quantum gravity, in: L. Kauffman (Ed.), The
Interface of Knots and Physics, Am. Math. Soc., Providence, RI, 1996, p. 167, gr-qc/9504036.
[11] J.C. Baez, S. Sawin, Functional integration on spaces of connections, J. Func. Anal. 150 (1997)
1, q-alg/9507023.
546
[12] F. Barbero, Real ashtekar variables for lorentzian signature space-times, Phys. Rev. D 51 (1995)
5507, gr-qc/9410014.
[13] R. DePietri, C. Rovelli, Geometry eigenvalues and scalar product from recoupling theory in
loop quantum gravity, Phys. Rev. D 54 (1996) 2664, gr-qc/9602023.
[14] N. Grot, C. Rovelli, Weave states in loop quantum gravity, Gen. Rel. Grav. 29 (1997) 1039.
[15] R. Haag, Local Quantum Physics, Springer-Verlag, 1993.
[16] J. Iwasaki, C. Rovelli, Gravitons as embroidary on the weave, Int. J. Mod. Phys. D 1 (1993)
533.
[17] J. Iwasaki, C. Rovelli, Gravitons from loops: non-perturbative loop space quantum gravity
contains the graviton-physics approximation, Class. Quant. Grav. 11 (1994) 1653.
[18] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. 1,
Academic Press, 1983.
[19] N.P. Landsman, Ch.G. Weert, Real and imaginary-time field theory at finite temperature and
density, Phys. Rep. 145 (3/4) (1987) 141.
[20] C. Rovelli, L. Smolin, Loop representation of quantum general relativity, Nucl. Phys. B 331
(1990) 80.
[21] C. Rovelli, L. Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys. B 442
(1995) 593.
[22] T. Thiemann, QSD VI: Quantum Poincar algebra and a quantum positivity of energy theorem
for canonical quantum gravity, Class. Quant. Grav. 15 (1998) 1463, gr-qc/9705020.
Abstract
In this paper we discuss candidate superconformal N = 2 gauge theories that realize the AdS/CFT
correspondence with M-theory compactified on the homogeneous Sasakian 7-manifolds M 7 that
were classified long ago. In particular we focus on the two cases M 7 = Q1,1,1 and M 7 = M 1,1,1 , for
the latter the KaluzaKlein spectrum being completely known. We show how the toric description
of M 7 suggests the gauge group and the supersingleton fields. The conformal dimensions of the
latter can be independently calculated by comparison with the mass of baryonic operators that
correspond to 5-branes wrapped on supersymmetric 5-cycles and are charged with respect to the
Betti multiplets. The entire KaluzaKlein spectrum of short multiplets agrees with these dimensions.
Furthermore, the metric cone over the Sasakian manifold is a conifold algebraically embedded in
some Cp . The ring of chiral primary fields is defined as the coordinate ring of Cp modded by
the ideal generated by the embedding equations; this ideal has a nice characterization by means
of representation theory. The entire KaluzaKlein spectrum is explained in terms of these vanishing
relations. We give the superfield interpretation of all short multiplets and we point out the existence
of many long multiplets with rational protected dimensions, whose presence and pattern seem to be
universal in all compactifications. 2000 Elsevier Science B.V. All rights reserved.
PACS: 11.30.Pb; 04.65
Keywords: Supergravity; AdS/CFT; M-theory
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 0 9 8 - 5
548
1. Synopsis
In this paper we consider M-theory compactified on anti-de-Sitter four-dimensional
space AdS4 times a homogeneous Sasakian 7-manifold M 7 and we study the correspondence with the infrared conformal point of suitable D = 3, N = 2 gauge theories describing the appropriate M2-brane dynamics. For the readers convenience we have divided our
paper into three parts.
Part I contains a general discussion of the problem we have addressed and a summary
of all our results.
Part II presents the superconformal gauge theory interpretation of the KaluzaKlein
multiplet spectra previously obtained from harmonic analysis and illustrates the nontrivial predictions one obtains from such a comparison.
Part III provides a detailed analysis of the algebraic geometry, topology and
metric structures of homogeneous Sasakian 7-manifolds. This part contains all
the geometrical background and the explicit derivations on which our results and
conclusions are based.
Part I
General discussion
2. Introduction
The basic principle of the AdS/CFT correspondence [13] states that every consistent
M-theory or Type II background with metric AdSp+2 M dp2 in d dimensions, where
M dp2 is an Einstein manifold, is associated with a conformal quantum field theory
living on the boundary of AdSp+2 . The background is typically generated by the near
horizon geometry of a set of p-branes and the boundary conformal field theory is identified
with the IR limit of the gauge theory living on the world-volume of the p-branes.
One remarkable example with N = 1 supersymmetry on the boundary and with a nontrivial smooth manifold M 5 = T 1,1 was found in [4] and the associated superconformal
theory was identified. Some general properties and the complete spectrum of the T 1,1
compactification have been discussed in [57], finding complete agreement between gauge
theory expectations and supergravity predictions. In this paper we will focus on the case
p = 2 when M is a coset manifold G/H with N = 2 supersymmetry.
Backgrounds of the form AdS4 M 7 arise as the near horizon geometry of a collection
of M2-branes in M-theory. The N = 8 supersymmetric case corresponds to M 7 =
S 7 . Examples of superconformal theories with less supersymmetry can be obtained by
orbifolding the M2-brane solution [8,9]. Orbifold models have the advantage that the gauge
theory can be directly obtained as a quotient of the N = 8 theory using standard techniques
[10]. On the other hand, the internal manifold M 7 is S 7 divided by some discrete group
and it is generically singular. Smooth manifolds M 7 can be obtained by considering M2branes sitting at the singular point of the cone over M 7 , C(M 7 ) [4,1114]. Many examples
where M 7 is a coset manifold G/H were studied in the old days of KK theories [15]
549
(see [1626] for the cases S 7 and squashed S 7 , see [2732] for the case M p,q,r , see [33,
34] for the case Q1,1,1 , see [26,29,3537] for general methods of harmonic analysis in
compactification and the structure of Osp(N|4) supermultiplets, and finally see [38] for a
complete classification of G/H compactifications).
For obvious reason, AdS4 was much more investigated in those days than his simpler
cousin AdS5 . As a consequence, we have a plethora of AdS4 G/H compactifications for
which the dual superconformal theory is still to be found. If we require supersymmetric
solutions, which are guaranteed to be stable and are simpler to study, and furthermore we
require 2 6 N 6 8, we find four examples: N 0,1,0 with N = 3 and Q1,1,1, M 1,1,1 , V5,2
with N = 2 supersymmetry. These are the natural AdS4 counterparts of the T 1,1 conifold
theory studied in [4].
In this paper we shall consider in some detail the two cases Q1,1,1 and M 1,1,1 . They
have isometry SU (2)3 U (1) and SU (3) SU (2) U (1), respectively. The isometry of
these manifolds corresponds to the global symmetry of the dual superconformal theories,
including the U (1) R-symmetry of N = 2 supersymmetry. The complete spectrum of 11dimensional supergravity compactified on M 1,1,1 has been recently computed [39,40]. The
analogous spectrum for Q1,1,1 has not been computed yet 1 , but several partial results exist
in the literature [41], which will be enough for our purpose. The KK spectrum should
match the spectrum of the gauge theory operators of finite dimension in the large N limit.
As a difference with the maximally supersymmetric case, the KK spectrum contains both
short and long operators; this is a characteristic feature of N = 2 supersymmetry and was
already found in AdS5 T 1,1 [5,7].
We will show that the spectra on Q1,1,1 and M 1,1,1 share several common features with
their cousin T 1,1 . First of all, the KK spectrum is in perfect agreement with the spectrum
of operators of a superconformal theory with a set of fundamental supersingleton fields
inherited from the geometry of the manifold. In the abelian case this is by no means a
surprise because of the well known relations among harmonic analysis, representation
theory and holomorphic line bundles over algebraic homogeneous spaces. The non-abelian
case is more involved. There is no straightforward method to identify the gauge theory
living on M2-branes placed at the singularity of C(M 7 ) when the space is not an orbifold.
Hence we shall use intuition from toric geometry to write candidate gauge theories that
have the right global symmetries and a spectrum of short operators which matches the KK
spectrum. Some points that still need to be clarified are pointed out.
A second remarkable property of these spaces is the existence of non-trivial cycles and
non-perturbative states, obtained by wrapping branes, which are identified with baryons
in the gauge theory [43]. The corresponding baryonic U (1) symmetry is associated with
the so-called Betti multiplets [30,35]. The conformal dimension of a baryon can be
computed in supergravity, following [6], and unambiguously predicts the dimension of
the fundamental conformal fields of the theory in the IR. The result from the baryon
analysis is remarkably in agreement with the expectations from the KK spectrum. This can
be considered as a highly non-trivial check of the AdS/CFT correspondence. Moreover,
1 This spectrum is presently under construction [42].
550
we will also notice that, as it happens on T 1,1 [5,7], there exists a class of long multiplets
which, against expectations, have a protected dimension which is rational and agrees with
a naive computation. There seems to be a common pattern for the appearance of these
operators in all the various models.
551
The relevant degrees of freedom at the superconformal fixed points are in general
different from the elementary fields of the supersymmetric gauge theory. For example,
vector multiplets are not conformal in three dimensions and they should be replaced by
some other multiplets of the superconformal group by dualizing the vector field to a
scalar. Let us again consider the simple example of N = 8. The degrees of freedom at
the superconformal point (the singletons, in the language of representation theory of the
superconformal group) are contained in a supermultiplet with eight real scalars and eight
fermions, transforming in representations of the global R-symmetry SO(8). This is the
same content of the N = 8 vector multiplet, when the vector field is dualized into a scalar.
The change of variable from a vector to a scalar, which is well-defined in an abelian theory,
is obviously a non-trivial and not even well-defined operation in a non-abelian theory. The
scalars in the supersingleton parametrize the flat space transverse to the M2-branes. In
this case, the moduli space of vacua of the abelian N = 8 gauge theory, corresponding
to a single M2-brane, is isomorphic to the transverse space. The case with N M2-branes
is obtained by promoting the theory to a non-abelian one. We want to follow a similar
procedure for the conifold cases.
For branes at the conifold singularity of C(M 7 ) there is no obvious way of reducing the
system to a simple configuration of D2-branes in type IIA and read the field content by
using standard brane techniques. 3 We can nevertheless use the intuition from geometry
for identifying the relevant degrees of freedom at the superconformal point. We need an
abelian gauge theory whose moduli space of vacua is isomorphic to C(M 7 ). The moduli
space of vacua of N = 2 theories have two different branches touching at a point, the
Coulomb branch parametrized by the vev of the scalars in the vector multiplet and the Higgs
branch parametrized by the vev of the scalars in the chiral multiplets. The Higgs branch
is the one we are interested in. Each of the two branches excludes the other, so we can
consistently set the scalars in the vector multiplets to zero (see Appendix A for a discussion
of the scalar potential in general N = 2, D = 3 theories). We can find what we need in
toric geometry. Indeed, this latter describes certain complex manifolds as Khler quotients
associated to symplectic actions of a product of U (1)s on some Cp . This is completely
equivalent to imposing the D-term equations for an abelian N = 2, D = 3 gauge theory and
dividing by the gauge group or, in other words, to finding the moduli space of vacua of the
theory. Fortunately, both the cone over Q1,1,1 and that over M 1,1,1 have a toric geometry
description. This description was already used for studying these spaces in [46,47]. In
this paper, we will consider a different point of view. We can then easily find abelian gauge
theories whose moduli space of vacua (the Higgs branch component) is isomorphic to these
two particular conifolds. In the following subsections, we briefly discuss the geometry of
the two manifolds and the abelian gauge theory associated with the toric description. More
complete information about the geometry and the homology of the manifolds are contained
in Part III. Here we briefly recall the basic information needed to discuss the matching of
the KK spectrum with the expectations from the conformal theory.
3 This possibility exists for orbifold singularities and was exploited in [8,9,49] for N = 4 and in [50] for N = 2.
552
(3.1)
The cone over Q1,1,1 is a toric manifold obtained as the Khler quotient of C6 by the
symplectic action of two U (1)s. Explicitly, it is described as the solution of the following
two D-term equations (momentum map equations in mathematical language)
|A1 |2 + |A2 |2 = |B1 |2 + |B2 |2 ,
|B1 |2 + |B2 |2 = |C1 |2 + |C2 |2 ,
(3.2)
modded by the action of the corresponding two U (1)s, the first acting only on Ai with
charge +1 and on Bi with charge 1, the second acting only on Bi with charge +1 and on
Ci with charge 1.
The manifold Q1,1,1 can be obtained by setting each term in (3.2) equal to 1, i.e., as
3
S S 3 S 3 /U (1) U (1). This corresponds to taking a section of the cone at a fixed value
of the radial coordinate (an horizon in Morrison and Plessers language [14]). Indeed, in
full generality, this radial coordinate is identified with the fourth coordinate of AdS4 , while
the section is identified with the internal manifold 4 M 7 [4,51].
Given the toric description, the identification of an abelian N = 2 gauge theory whose
Higgs branch reproduces the conifold is straightforward. Equations (3.2) are the D-terms
for the abelian theory U (1)3 with doublets of chiral fields Ai with charges (1, 1, 0),
Bi with charge (0, 1, 1) and Ci with charges (1, 0, 1) and without superpotential.
The theory has an obvious global symmetry SU (2)3 matching the isometry of Q1,1,1 .
We introduced three U (1) factors (one more than those appearing in the toric data, as
the attentive reader certainly noticed) for symmetry reasons. One of the three U (1)s is
decoupled and has no role in our discussion. Since we do not expect a decoupled U (1) in
the world-volume theory of M2-branes living at the conifold singularity, we should better
consider the theory U (1)3 /U (1)DIAGONAL.
The fields appearing in the toric description should represent the fundamental degrees of
freedom of the superconformal theory, since they appear as chiral fields in the gauge theory.
They have definite transformation properties under the gauge groups. Out of them we
can also build some gauge invariant combinations, which should represent the composite
operators of the conformal theory and which should be matched with the KK spectrum.
Geometrically, this corresponds to describing the cone as an affine subvariety of some Cp .
This is a standard procedure, which converts the definition of a toric manifold in terms
4 In the solvable Lie algebra parametrization of AdS [52,53] the radial coordinate is algebraically characterized
4
as being associated with the Cartan semisimple generator, while the remaining three are associated with the three
nilpotent generators spanning the brane world-volume. So we have a natural splitting of AdS4 into 3 + 1 which
mirrors the natural splitting of the eight-dimensional conifold into 1 + 7. The radial coordinate is shared by the
two spaces. This phenomenon, that is the algebraic basis for the existence of smooth M2-brane solutions with
horizon geometry AdS4 M 7 , was named dimensional transmigration in [52].
553
i, j, k = 1, 2.
(3.3)
They satisfy a set of binomial equations which cut out the image of our conifold C(Q1,1,1 )
in C8 . These equations are actually the 9 quadrics explicitely written in Eqs. (7.72) of
Part III. Indeed, there is a general method to obtain the embedding equations of the cones
over algebraic homogeneous varieties based on representation theory. 5 If we want to
summarize this general method in few words, we can say the following. Through Eq. (3.3)
we see that the coordinates Xij k of C8 are assigned to a certain representation R of
the isometry group SU (2)3 . In our case such a representation is R = (J1 = 1/2, J2 =
1/2, J3 = 1/2). The products Xi1 j1 k1 Xi2 j2 k2 belong to the symmetric product Sym2 (R),
which in general branches into various representations, one of highest weight plus several
subleading ones. On the cone, however, only the highest weight representation survives
while all the subleading ones vanish. Imposing that such subleading representations are
zero corresponds to writing the embedding equations. This has far reaching consequences
in the conformal field theory, since provides the definition of the chiral ring. In principle
all the representations appearing in the kth symmetric tensor power of R could correspond
to primary conformal operators. Yet the attention should be restricted to those that
do not vanish modulo the equations of the cone, namely modulo the ideal generated
by the representations of subleading weights. In other words, only the highest weight
representation contained in the Symk (R) gives a true chiral operator. This is what matches
the KaluzaKlein spectra found through harmonic analysis. Two points should be stressed.
In general the number of embedding equations is larger than the codimension of the
algebraic locus. For instance, 8 4 < 9, i.e., the cone is not a complete intersection.
The 9 equations (7.72) define the ideal I of C[X] := C[X111 , . . . , X222 ] cutting the
cone C(Q1,1,1). The second point to stress is the double interpretation of the embedding
equations. The fact that Q1,1,1 leads to N = 2 supersymmetry means that it is Sasakian,
i.e., it is a circle bundle over a suitable complex three-fold. If considered in C8 the ideal I
cuts out the conifold C(Q1,1,1 ). Being homogeneous, it can also be regarded as cutting out
an algebraic variety in P7 . This is P1 P1 P1 , namely the base of the U (1) fibre-bundle
Q1,1,1 .
It follows from this discussion that the invariant operators Xij k of Eq. (3.3) can be
naturally associated with the building blocks of the gauge invariant composite operators
of our CFT. Holomorphic combinations of the Xij k should span the set of chiral operators
of the theory. As we stated above, the set of embedding equations (7.72) impose restrictions
on the allowed representations of SU (2)3 and hence on the existing operators. If we put
the definition of Xij k in terms of the fundamental fields A, B, C into Eqs. (7.72), we see
that they are automatically satisfied when the theory is abelian. Since we want eventually
5 The 9 equations were already mentioned in [46] although their representation theory interpretation was not
given there.
554
555
Fig. 1. Gauge group SU(N)1 SU(N)2 SU(N)3 and color representation assignments of the
supersingleton fields Ai , Bj , C` in the Q1,1,1 world-volume gauge theory.
(see Section 2), the entire spectrum is fully determined by the structure of the ideal
above. Indeed, as it should be clear from the previous group theoretical description of
the embedding equations, the result of the constraints is to select chiral operators which
are totally symmetrized in the SU(3) and SU(2) indices.
556
fundamental fields of the CFT, which are non-zero only for U (N) gauge groups. A second
evidence is the existence of states dual to baryonic operators in the non-perturbative
spectrum of these Type II or M-theory compactifications; baryons exist only for SU(N)
groups. We will find baryons in the spectrum of both Q1,1,1 and M 1,1,1 : this implies that,
for the compactifications discussed in this paper, the gauge group of the CFT is SU(N).
In the non-abelian case, we expect that the generic point of the moduli space corresponds
to N separated branes. Therefore, the space of vacua of the theory should reduce to the
symmetrization of N copies of Q1,1,1 . To get rid of unwanted light non-abelian degrees
of freedom, we would like to introduce, following [4], a superpotential for our theory.
Unfortunately, the obvious candidate for this job
ij mn pq Tr(Ai Bm Cp Aj Bn Cq )
(4.2)
is identically zero. Here the close analogy with T 1,1 and Ref. [4] ends.
We consider now the spectrum of KK excitations of Q1,1,1 . The full spectrum of Q1,1,1
is not known; however, the eigenvalues of the Laplacian were computed in [41]. As shown
in [39], the knowledge of the Laplacian eigenvalues allows to compute the entire spectrum
of hypermultiplets of the theory, corresponding to the chiral operators of the CFT. The
result is that there is a chiral multiplet in the (k/2, k/2, k/2) representation of SU(2)3 for
each integer value of k, with dimension E0 = k. We naturally associate these multiplets
with the series of composite operators
Tr(ABC)k ,
(4.3)
where the SU(2)s indices are totally symmetrized. A first important result, following from
the existence of these hypermultiplets in the KK spectrum, is that the dimension of the
combination ABC at the superconformal point must be 1.
We see that the prediction from the KK spectrum are in perfect agreement with the
geometric discussion in the previous section. Operators which are not totally symmetric in
the flavor indices do not appear in the spectrum. The agreement with the proposed CFT,
however, is only partial. The chiral operators predicted by supergravity certainly exist in
the gauge theory. However, we can construct many more chiral operators which are not
symmetric in flavor indices. They do not have any counterpart in the KK spectrum. The
superpotential in the case of T 1,1 [4] had the double purpose of getting rid of the unwanted
non-abelian degrees of freedom and of imposing, via the equations of motion, the total
symmetrization for chiral and short operators which is predicted both by geometry and
by supergravity. Here, we are not so lucky, since there is no superpotential. We cannot
consider superpotentials of dimension bigger than that considered before (for example,
cubic or quartic in ABC) because the superpotential (4.2) is the only one which has
dimension compatible with the supergravity predictions. 6 We need to suppose that all the
non-symmetric operators are not conformal primary. Since the relation between R-charge
and dimension is only valid for conformal chiral operators, such operators are not protected
and therefore may have enormous anomalous dimension, disappearing from the spectrum.
6 For a three-dimensional theory to be conformal the dimension of the superpotential must be 2.
557
Fig. 2. Gauge group U (N)1 U (N)2 and color representation assignments of the supersingleton
fields V A and U i in the M 1,1,1 world-volume gauge theory.
Simple examples of chiral but not conformal operators are those obtained by derivatives
of the superpotential. Since we do not have a superpotential here, we have to suppose that
both the elimination of the unwanted colored massless states as well as the disappearing of
the non-symmetric chiral operators emerges as an non-perturbative IR effect.
4.2. The case of M 1,1,1
Let us now consider M 1,1,1 . The non-abelian theory is now SU(N) SU(N) with chiral
matter in the following representations of the gauge group
U i Sym2 (CN ) Sym2 (CN ),
(4.4)
The representations of the fundamental fields have been chosen in such a way that
they reduce to the abelian theory discussed in the previous section, match with the KK
spectrum and imply the existence of baryons predicted by supergravity. Comparison with
supergravity, which will be made soon, justifies, in particular, the choice of color symmetric
representations.
The field content can be conveniently encoded in the quiver diagram in Fig. 2.
The global symmetry of the gauge theory is SU(3) SU(2), with the chiral fields U and
V transforming in the fundamental representation of SU(3) and SU(2), respectively.
We next compare the expectations from gauge theory with the KK spectrum [39]. Let us
start with the hypermultiplet spectrum (the full spectrum of KK modes will be discussed in
Part II). There is exactly one hypermultiplet in the symmetric representation of SU(3) with
3k indices and the symmetric representation of SU(2) with 2k indices, for each integer
k > 1. The dimension of the operator is E0 = 2k. We naturally identify these states with
the totally symmetrized chiral operators
Tr(U 3 V 2 )k .
(4.5)
has
dimension 2 at the superconformal fixed point.
Once again, we are not able to write any superpotential of dimension 2. The natural
candidate is the dimension two flavor singlet
ij k AB (U i U j U k V A V B )color singlet
(4.6)
558
which however vanishes identically. There is no superpotential that might help in the
elimination of unwanted light colored degrees of freedom and that might eliminate all
the non-symmetric chiral operators that we can construct out of the fundamental fields.
Once again, we have to suppose that, at the superconformal fixed point in the IR, all the
non-totally symmetric operators are not conformal primaries.
4.3. The baryonic symmetries and the Betti multiplets
There is one important property that M 1,1,1 , Q1,1,1 and T 1,1 share. These manifolds
have non-zero Betti numbers (b2 = b5 = 2 for Q1,1,1 , b2 = b5 = 1 for M 1,1,1 and b2 =
b3 = 1 for T 1,1 ). This implies the existence of non-perturbative states in the supergravity
spectrum associated with branes wrapped on non-trivial cycles. They can be interpreted as
baryons in the CFT [6,43].
The existence of non-zero Betti numbers implies the existence of new global U (1)
symmetries which do not come from the geometrical symmetries of the coset manifold,
as was pointed out long time ago. The massless vector multiplets associated with these
symmetries were discovered and named Betti multiplets in [30,35]. They have the
property that the entire KK spectrum is neutral and only non-perturbative states can be
charged. The massless vectors, dual to the conserved currents, arise from the reduction
of the 11-dimensional 3-form along the non-trivial 2-cycles. This definition implies that
non-perturbative objects made with M2- and M5-branes are charged under these U (1)
symmetries.
We can identify the Betti multiplets with baryonic symmetries. This was first pointed
out in [7,63] for the case of T 1,1 and discussed for orbifold models in [14]. The existence
of baryons in the proposed CFTs is due to the choice of SU(N) (as opposed to U (N))
as gauge group. In the SU(N) case, we can form the gauge invariant operators det(A),
det(B) and det(C) for Q1,1,1 and det(U ) and det(V ) for M 1,1,1 . The baryon symmetries
act on fields in the same way as the U (1) factors that we used for defining our abelian
theories in Sections 3.1.1 and 3.1.2. They disappeared in the non-abelian theory associated
to the conifolds, but the very same fact that they can be consistently incorporated in
the theory means that they must exist as global symmetries. It is easy to check that no
operator corresponding to KK states is charged under these U (1)s. The reason is that
the KK spectrum is made out with the combinations X = ABC or X = U 3 V 2 defined in
Sections 3.1.1 and 3.1.2 which, by definition, are U (1) invariant variables. The only objects
that are charged under the U (1) symmetries are the baryons.
Baryons have dimensions which diverge with N and cannot appear in the KK spectrum.
They are indeed non-perturbative objects associated with wrapped branes [6,43]. We see
that the baryonic symmetries have the right properties to be associated with the Betti
multiplets: the only charged objects are non-perturbative states. This identification can
be strengthened by noticing that the only non-perturbative branes in M-theory have an
electric or magnetic coupling to the 11-dimensional 3-form. Since for our manifolds, both
b2 and b5 are greater than 0, we have the choice of wrapping both M2- and M5-branes.
M2-branes wrapped around a non-trivial 2-cycle are certainly charged under the massless
559
vector in the Betti multiplet which is obtained by reducing the three-form on the same
cycle. Since a non-trivial 5-cycle is dual to a 2-cycle, a similar remark applies also for
M5-branes. We identify M5-branes as baryons because they have a mass (and therefore a
conformal dimension) which goes like N , as discussed in Section 5.2.
What follows from the previous discussion and is probably quite general, is that there
is a close relation between the U (1)s entering the brane construction of the gauge theory,
the baryonic symmetries and the Betti multiplets. The previous remarks apply as well to
CFT associated with orbifolds of AdS4 S 7 . In the case of T 1,1 ,Q1,1,1 and M 1,1,1 , the
baryonic symmetries are also directly related to the U (1)s entering the toric description of
the manifold.
4.4. Non-trivial results from supergravity: a discussion
In the previous sections, we proposed non-abelian theories as dual candidates for the Mtheory compactification on Q1,1,1 and M 1,1,1 . We also pointed out the difficulties related
to the existence of more candidate conformal chiral operators than those expected from
the KK spectrum analysis. We have no good arguments for claiming that these non-flavor
symmetric operators disappear in the IR limit. If they survive, this certainly signals the
need for modifying our guess for the dual CFTs. In the latter case, new fields may be
needed. The theories we wrote down are based on the minimal assumption that there is
no superpotential in the abelian case; 7 if we relax this assumption, more complicated
candidate dual gauge theories may exist. In the case of T 1,1 , the CFT was identified in two
different ways, by using the previous section arguments and also by describing the conifold
as a deformation of an orbifold singularity. Since orbifold CFT can be often identified using
standard techniques [10], this approach has the advantage of unambiguously identifying the
conifold CFT. It would be interesting to find an analogous procedure for the case of AdS4 .
It would provide a CFT which flows in the IR to the conifold theory after a deformation
[4,14] and it would help in checking whether new fields are necessary or not for a correct
description of the CFTs. Attempts to find associated orbifold models in the case of Q1,1,1
have been made in [46,47]; the precise relation with our approach is still to be clarified. 8
In any event, whatever is the microscopic description of the gauge theory flowing to
the superconformal points in the IR, it is reasonable to think all the relevant degrees of
freedom at the superconformal fixed point corresponding to the M-theory on Q1,1,1 and
M 1,1,1 has been identified in the previous geometrical analysis. We will make, from now
on, the assumption that the fundamental singletons of the CFT for Q1,1,1 are the fields
A, B, C and for M 1,1,1 the fields U, V with the previously discussed assignment of color
and flavor indices and that they always appear in totally symmetrized flavor combinations.
Given this simple assumption, inherited from the geometry of the conifolds, we can make
several non-trivial comparisons between the expectation of a CFT (in which the singletons
7 If there is a superpotential the toric description may contain extra U (1)s related to the F -terms of the theories,
as it happens for orbifold models [45].
8 A different CFT was proposed for the case of Q1,1,1 in [46]; this different proposal does not seem to solve
the discrepancies with the KK expectations.
560
are totally symmetrized in flavor) and the supergravity prediction. We leave for future
work the clarification of the dynamical mechanism (or possible modification of the threedimensional gauge theories) for suppressing the non-symmetric operators as well as the
search for a RG flow from an orbifold model.
We already discussed the chiral operators of the two CFTs. We obtained two main
results from this analysis. The first one states that all chiral operators are symmetrized
in flavor indices. The second one, more quantitative, predicts the conformal dimension
of some composite objects. When appearing in gauge invariant chiral operators, the
symmetrized combinations ABC and U 3 V 2 have dimensions 1 and 2, respectively.
Having this information, there are two types of important and non-trivial checks that we
can make:
The full spectrum of KK excitations should match with composite operators in the
CFT. Specifically, besides the hypermultiplets, there are many other short multiplets
in the spectrum. All these multiplets should match with CFT operators with protected
dimension. This will be verified in Sections 6.3, 6.4.
We can determine the dimension of a baryon operator by computing the volume
of the cycle the M5-brane is wrapping, following [6]. From this, we can determine
the dimension of the fundamental fields of the CFT. This can be compared with
the expectations from the KK spectrum. The agreement of the two methods can
be considered as a non-trivial check of the AdS/CFT correspondence. This will be
discussed in Section 5.
Leaving the actual computation and detailed comparison of spectra for the second Part
of this paper, here we summarize the results of our analysis.
The spectrum of M 1,1,1 is completely known [39]. This allows a detailed comparison
of all the states in supergravity with CFT operators. Besides the hypermultiplets, which fit
the quantum field theory expectations in a straightforward manner, there are various series
of multiplets which are short and therefore protected. An highly non-trivial result is that
we will be able to identify all the KK short multiplets with candidate CFT operators of
requested quantum numbers and conformal dimension. Most of them can be obtained by
tensoring conserved currents with chiral operators. A similar analysis was done for T 1,1 in
[7]. In N = 2 supersymmetric compactifications, the KK spectrum contains both short and
long multiplets. We will notice that there is a common pattern in Q1,1,1 , M 1,1,1 as well as
in T 1,1 , of long multiplets which have rational and protected dimension. In particular, we
can identify in all these models rational long gravitons with products of the stress energy
tensor, conserved currents and chiral operators. We suspect the existence of some field
theoretical reason for the unexpected protected dimension of these operators.
The dimension of the fundamental fields A, B, C and U, V at the superconformal point
can be computed and compared with the KK spectrum prediction. In the KK spectrum,
these fields always appear in particular combinations. For example, we already know
that ABC has dimension 1 and U 3 V 2 has dimension 2. A, B, C have clearly the same
dimension 1/3 since there is a permutation symmetry. But, whats about U or V ? From
the CFT point of view, we expect the existence of several baryon operators: det A, det B,
det C for Q1,1,1 and det U , det V for M 1,1,1 . All of them should correspond to M5-branes
561
562
The rest of this paper will be devoted to an exhaustive comparison between quantum
field theory and supergravity and to a detailed description of the geometry involved in such
a comparison.
Part II
21 12
i1 |11 11
2N N2
1 N 1 N 2 2 2 2
1
iN |1N N1
2 2 2
2 2 2
(5.2)
1 1
N N
2 2 2 2 2 2
1
21
1
2N
3N
21 2N
3 3
2
|
1
N
N
N
(5.5)
31
2
1 |1
Bi
11
i1 |31
1
N |N
(5.3)
(5.4)
Ai
det B Bi
1 N 2 2
1
1 |1
det A Ai
det C C
1N
iN |3N
1 N 1 1 .
3
(5.6)
If these operators are truly chiral primary fields, then their conformal dimensions are
obviously given by
h[det U ] = h[U ] N;
h[det V ] = h[V ] N,
h[det A] = h[A] N;
h[det B] = h[B] N;
h[det C] = h[C] N
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
563
where the conventions for the flavor representation labeling are those explained later in
Eqs. (6.10), (6.11).
The interesting fact is that the conformal operators (5.2)(5.6) can be reinterpreted as
solitonic supergravity states obtained by wrapping a 5-brane on a non-trivial supersymmetric 5-cycle. This gives the possibility of calculating directly the mass of such states
and, as a byproduct, the conformal dimension of the individual supersingletons. All what
is involved is a geometrical information, namely the ratio of the volume of the 5-cycles to
the volume of the entire compact 7-manifold. In addition, studying the stability subgroup
of the supersymmetric 5-cycles, we can also verify that the gauge-theory predictions (5.8)
(5.12) for the flavor representations are the same one obtains in supergravity looking at the
state as a wrapped solitonic 5-brane.
To establish these results we need to derive a general mass-formula for baryonic states
corresponding to wrapped 5-branes. This formula is obtained by considering various
relative normalizations.
5.1. The M2-brane solution and normalizations of the 7-manifold metric and volume
Using the conventions and normalizations of [56,59] for D = 11 supergravity and for
its KaluzaKlein expansions, a FreundRubin solution on AdS4 M 7 is described by the
following three equations:
ab
= 24e2 a ,
R ab = 16e2E a E b Rcb
b
R = R a b B c B d
cd
F [4] = eabcd E a
a b
with R a b = 12 e2ca ,
cb
b
c
d
E E E ,
(5.13)
1 ba
E ,
2/9
[4]
b[4] ,
= 11/9 F
Fabcd
abcd
Ba =
1 ba
B ,
2/9
2 = 8G11 =
(2)8 9
`P .
2
(5.14)
After such a rescaling, the relations between the FreundRubin parameter and the curvature
scales for both AdS4 and M 7 become
RicciAdS
= 2g ,
Ricci
= g ,
(5.15)
(5.16)
564
e2
.
(5.17)
4/9
Note that in Eq. (5.17) we have used the normalization of the Ricci tensor which is standard
ab
in the general relativity literature and is twice the normalization of the Ricci tensor Rcb
appearing in Eq. (5.13). Furthermore Eqs. (5.13) were written in flat indices while Eqs.
(5.15), (5.16) are written in curved indices.
For our further reasoning, it is convenient to write the anti-de-Sitter metric in the solvable
coordinates [1,25]:
d 2
2
2
2
2
2
2
dsAdS4 = RAdS dt + dx1 + dx2 + 2 ,
3
(5.18)
RicciAdS
= 2 g ,
RAdS
def
= 24
R
2
b[4]
b[4] A
b[3]
b[4] F
(5.20)
+ 288 F
I11 = d 11 x g 2 3 F
(where the coupling constant for the last term is = ) and the exact M2-brane solution is
as follows:
k 2/3
k 1/3 2
2
= 1+ 6
dscone ,
dt 2 + dx12 + dx22 + 1 + 6
dsM2
r
r
2
2
= dr 2 + r 2 dsM
dscone
7,
6
k 1
,
(5.21)
A[3] = dt dx1 dx2 1 + 6
r
2 is the Einstein metric on M 7 , with Ricci tensor as in Eq. (5.17), and ds 2
where dsM
7
cone is
the corresponding Ricci flat metric on the associated cone. When we go near the horizon,
r 0, the metric (5.21) is approximated by
dr 2
2
2
r 4 dt 2 + dx12 + dx22 k 2/3 + k 1/3 2 + k 1/3 dsM
dsM2
7.
6
r
The FreundRubin solution AdS4 M 7 is obtained by setting
2
= r2
k
and by identifying
RAdS =
k 1/6
= 6k 1/3 .
2
(5.22)
(5.23)
(5.24)
565
1
=
.
(5.26)
6
Vol(S 7 )
lP (25 2 N)1/6
We can now consider the solitonic particles in AdS4 obtained by wrapping M2- and
M5-branes on the non-trivial 2- and 5-cycles of M 7 , respectively. They are associated with
boundary operators with conformal dimensions that diverge in the large N limit. The exact
dependence on N can be easily estimated. Without loss of generality, we can put = 1
using a conformal transformation; its only role in the game is to fix a reference scale and
it will eventually cancel in the final formulae. The mass of a p-brane wrapped on a p(p+1) p/2
(p+1)
lP
. Once the mass of the
cycle is given by Tp Vol(p-cycle) lP
non-perturbative states is known, the dimension E0 of the associated boundary operator is
given by the relation m2 = (2/3)(E0 1)(E0 2) 2E02 /3. From Eq. (5.26) we learn
N Vol(5-cycle)
,
Vol(M 7 )
(5.28)
where the volume is evaluated with the internal metric normalized so that (5.16) is true.
566
C 1 (M1 = 0, M2 = 0, J = N/2),
(5.30)
C (M1 = N, M2 = 0, J = 0),
(5.31)
(see Eqs. (7.51) and (7.52)). Comparing Eqs. (5.30), (5.31) with Eqs. (5.8), (5.9), we see
that the first cycle is a candidate to represent the operator det U , while the second cycle is
a candidate to represent the operator det V . The final check comes from the evaluation of
the cycle volumes. This is done in Eqs. (7.38) and (7.39). Inserting these results and the
formula (7.40) for the M 1,1,1 volume into the general formula (5.28), we obtain
4
N h[U ] =
9
1
E0 (det U ) = N h[V ] =
3
E0 (det U ) =
4
,
9
1
.
3
(5.32)
(5.33)
567
1
3
(5.34)
568
U
V
i|
i|
(x, ) = u
(x) +
A|
A|
(x, ) = v
(x) +
i|
(x)+ ,
A|
v
(x) + ,
(6.1)
where (i, A) are SU(3) SU(2) flavor indices, (, ) are SU(N) SU(N) color indices
while is a world volume spinorial index of SO(1, 2). The supersingletons are chiral
superfields, so they satisfy E0 = |y0 |.
U i is in the fundamental representation 3 of SU(3)flavor and in the (22, 22? ) of
(SU(N) SU(N))color . V A is in the fundamental representation 2 of SU(2)flavor and in the
(222? , 222) of (SU(N) SU(N))color . In Eqs. (6.1) we have followed the conventions
that lower SU(N) indices transform in the fundamental representation, while upper SU(N)
indices transform in the complex conjugate of the fundamental representation.
Studying the non-perturbative baryon state, obtained by wrapping the 5-brane on the
supersymmetric cycles of M 1,1,1 , we have unambiguously established the conformal
weights of the supersingletons (or, more precisely, the conformal weights of the Clifford
vacua u, v) that are:
4
1
E0 (v) = y0 (v) = .
(6.2)
E0 (u) = y0 (u) = ,
9
3
For the Q1,1,1 theory the singleton superfields are instead the following ones:
Ai1 |12 (x, ) = ai1 |12 (x) + a i |2 (x)+ ,
1 1
3
3
3
(6.5)
(6.6)
only the highest weight representation of SU(3) SU(2), that is the completely symmetric
in the SU(3) indices and completely symmetric in the SU(2) indices, survives. So, as
569
advocated in Eq. (7.23), the ring of the chiral superfields should be composed by superfields
of the form
U ik U jk U lk V Ak V B}k .
(i1 j1 l1 ik jk lk )(A1 B1 Ak Bk ) = |U i1 U j1 U l1 V A1 V B1 {z
(6.7)
First of all, we note that a product of supersingletons is always a chiral superfield, that is,
a field satisfying the equation (see [40])
D+ = 0,
(6.8)
(6.9)
Following the notations of [39], we identify the flavor representations with three nonnegative integers M1 , M2 , 2J , where M1 , M2 count the boxes of an SU(3) Young diagram
according to
{z
}|
M2
{z
(6.10)
M1
while J is the usual isospin quantum number and counts the boxes of an SU(2) Young
diagram as follows
{z
(6.11)
2J
The superfields (6.7) are in the same Osp(2|4) SU(3) SU(2) representations as the bulk
hypermultiplets that were determined in [39] through harmonic analysis:
M1 = 3k,
M = 0,
2
k > 0.
(6.12)
J
=
k,
E0 = y0 = 2k,
In particular, it is worth noticing that every block U U U V V is in the (222, 22)flavor and
has conformal weight
1
4
+2
= 2,
(6.13)
3
9
3
as in the KaluzaKlein spectrum. As a matter of fact, the conformal weight of a product
of chiral fields equals the sum of the weights of the single components, as in a free field
theory. This is due to the relation E0 = |y0 | satisfied by the chiral superfields and to the
additivity of the hypercharge.
When the gauge group is promoted to SU(N) SU(N), the coordinates become tensors
(see (6.1)). Our conclusion about the composite operators is that the only primary chiral
570
superfields are those which preserve the structure (6.7). So, for example, the lowest lying
operator is:
U i|( U j | U `| ) V
A|( V
B| ) ,
(6.14)
where the color indices of every SU(N) are symmetrized. The generic primary chiral
superfield has the form (6.7), with all the color indices symmetrized before being
contracted. The choice of symmetrizing the color indices is not arbitrary: if we impose
symmetrization on the flavor indices, it necessarily follows that also the color indices are
symmetrized (see Appendix C for a proof of this fact). Clearly, the Osp(2|4) SU(3)
SU(2) representations (6.12) of these fields are the same as in the abelian case, namely
those predicted by the AdS/CFT correspondence.
It should be noted that in the 4-dimensional analogue of these theories, namely in the
T 1,1 case [4,7], the restriction of the primary conformal fields to the geometrical chiral
ring occurs through the derivatives of the quartic superpotential. As we already noted, in
the D = 3 theories there is no superpotential of dimension 2 which can be introduced
and, accordingly, the embedding equations defining the vanishing ideal cannot be given
as derivatives of a single holomorphic function. It follows that there is some other nonperturbative and so far unclarified mechanism that suppresses the chiral superfields not
belonging to the highest weight representations.
Let us know consider the case of the Q1,1,1 theory. Here, as already pointed out, the
complete KaluzaKlein spectrum is still under construction [42]. Yet the information
available in the literature is sufficient to make a comparison between the KaluzaKlein
predictions and the gauge theory at the level of the chiral multiplets (and also of the
graviton multiplets as we show below). Looking at Table 7 of [39], we learn that, in a
generic AdS4 M 7 compactification, each hypermultiplet contains a scalar state S of
energy label E0 = |y0 |, which is actually the Clifford vacuum of the representation and
corresponds to the world-volume field S of Eq. (6.9). From the general bosonic massformulae of [35,36], we know that S is related to traceless deformations of the internal
metric and its mass is determined by the spectrum of the scalar Laplacian on M 7 . In the
notations of [35], we normalize the scalar harmonics as
(0)3 Y = H0 Y
and we have the mass-formula (see [35] or Eq. (B.3) of [39])
p
m2S = H0 + 176 24 H0 + 36
(6.15)
(6.16)
which, combined with the general AdS4 relation between scalar masses and energy labels
16(E0 2)(E0 1) = m2 , yields the formula
q
p
3 1
180 + H0 24 36 + H0
(6.17)
E0 = +
2 4
for the conformal weight of candidate hypermultiplets in terms of the scalar Laplacian
eigenvalues. These are already known for Q1,1,1 since they were calculated by Pope in [41].
In our normalizations, Popes result reads as follows:
(6.18)
H0 = 32 J1 (J1 + 1) + J2 (J2 + 1) + J3 (J3 + 1) 14 y 2 ,
571
where (J1 , J2 , J3 ) denotes the SU(2)3 flavor representation and y the R-symmetry U (1)
charge. From our knowledge of the geometrical chiral ring of Q1,1,1 (see Section 7.2.1)
and from our calculation of the conformal weights of the supersingletons, on the gauge
theory side we expect the following chiral operators:
i1 j1 `1 ,...,ik jk `k = Tr(Ai1 Bj1 C`1 Aik Bjk C`k )
(6.19)
J1 = J2 = J3 = 12 k,
k > 1.
(6.20)
(6.21)
(6.22)
In components, the -expansion of this superfield yields the stressenergy tensor T (x),
the N = 2 supercurrents jA (x) (A = 1, 2) and the U (1) R-symmetry current JR (x).
Obviously T is a singlet with respect to the flavor group Gflavor and it has
E0 = 2,
y0 = 0,
s0 = 1.
(6.23)
This corresponds to the massless graviton multiplet of the bulk and explains the first entry
in the above enumeration.
To each generator of the flavor symmetry group there corresponds, via Noether theorem,
a conserved vector supercurrent. This is a scalar superfield J I (x, ) transforming in the
adjoint representation of Gflavor and satisfying the conservation equations
D+ D+ J I = D D J I = 0.
(6.24)
572
y0 = 0,
S0 = 0
(6.25)
and correspond to the N = 2 massless vector multiplets of Gflavor that propagate in the
bulk. This explains the second item of the above enumeration.
In the specific theories under consideration, we can easily construct the flavor currents
in terms of the supersingletons:
i
i|
`|
=U
13 ji U
,
J
U j |
U `|
SU(3)|j
A|
A
1,1,1
(6.26a)
M
JSU(2)|B = V
V B|
1 A C|
2 B V
,
V C|
i1
i1 |1
` |
12 ji11 A 1 12 A`1 |1 2 ,
2 Aj1 |1
JSU(2)1 |j = A
(6.26b)
Q1,1,1 JSU(2) |j i2 = B i2 |2 B j1 |2 3 12 ji2 A`2 |2 A`2 |2 3 ,
3
2
3
1 2
1
1
i3
i1 |3
1 i3 `3 |3
2 j3 A
JSU(2)3 |j3 = C
1 C j3 |3
1 A`3 |3 .
These currents satisfy Eq. (6.24) and are in the right representations of SU(3) SU(2).
Their hypercharge is y0 = 0. The conformal weight is not the one obtained by a naive sum,
being the theory interacting. As shown in [40], the conserved currents satisfy E0 = |y0 |+1,
hence E0 = 1.
Let us finally identify the gauge theory superfields associated with the Betti multiplets.
As we stressed in the introduction, the non-abelian gauge theory has SU(N)p rather
than U (N)p as gauge group. The abelian gauge symmetries that were used to obtain the
toric description of the manifold M 1,1,1 and Q1,1,1 in the one-brane case N = 1 are not
promoted to gauge symmetries in the many brane regime N . Yet, they survive as
exact global symmetries of the gauge theory. The associated conserved currents provide
the superfields corresponding to the massless Betti multiplets found in the KaluzaKlein
spectrum of the bulk. As the reader can notice, the b2 Betti number of each manifold always
agrees with the number of independent U (1) groups needed to give a toric description of
the same manifold. It is therefore fairly easy to identify the Betti currents of our gauge
theories. For instance, for the M 1,1,1 case the Betti current is
JBetti = 2U
`|
U `|
3V
C|
.
V C|
(6.27)
The two Betti currents of Q1,1,1 are similarly written down from the toric description.
Since the Betti currents are conserved, according to what shown in [40], they satisfy E0 =
|y0 | + 1. Since the hypercharge is zero, we have E0 = 1 and the Betti currents provide the
gauge theory interpretation of the massless Betti multiplets.
6.3. Gauge theory interpretation of the short multiplets
Using the massless currents reviewed in the previous section and the chiral superfields,
one has all the building blocks necessary to construct the constrained superfields that
correspond to all the short multiplets found in the KaluzaKlein spectrum.
573
As originally discussed in [31] and applied to the explicitly worked out spectra in [39,
40], short Osp(2|4) multiplets correspond to the saturation of the unitarity bound that
relates the energy (or conformal dimension) E0 and hypercharge y0 of the Clifford vacuum
to the highest spin smax contained in the multiplet. Hence short multiplets occur when:
short graviton,
smax = 2
(6.28)
E0 = |y0 | + smax ,
smax = 3/2 short gravitino,
short vector.
smax = 1
In abstract representation theory condition (6.28) implies that a subset of states of the
Hilbert space have zero norm and decouple from the others. Hence the representation is
shortened. In superfield language, the -expansion of the superfield is shortened by imposing a suitable differential constraint, invariant with respect to Poincar supersymmetry
[40]. Then Eq. (6.28) is the necessary condition for such a constraint to be invariant also
under superconformal transformations. Using chiral superfields and conserved currents as
building blocks, we can construct candidate short superfields that satisfy the appropriate
differential constraint and Eq. (6.28). Then we can compare their flavor representations
with those of the short multiplets obtained in KaluzaKlein expansions. In the case of the
M 1,1,1 theory, where the KaluzaKlein spectrum is known, we find complete agreement
and hence we explicitly verify the AdS/CFT correspondence. For the Q1,1,1 manifold we
make instead a prediction in the reverse direction: the gauge theory realization predicts the
outcome of harmonic analysis. While we wait for the construction of the complete spectrum [42], we can partially verify the correspondence using the information available at the
moment, namely the spectrum of the scalar Laplacian [41].
6.3.1. Superfields corresponding to the short graviton multiplets
The gauge theory interpretation of these multiplets is quite simple. Consider the
superfield
(x, ) = T (x, ) chiral(x, ),
(6.29)
where T is the stressenergy tensor (6.22) and chiral (x, ) is a chiral superfield. By
construction, the superfield (6.29), at least in the abelian case, satisfies the equation
D+ = 0
(6.30)
and then, as shown in [40], it corresponds to a short graviton multiplet of the bulk. It is
natural to extend this identification to the non-abelian case.
Given the chiral multiplet spectrum (6.12) and the dimension of the stress energy current
(6.12), we immediately get the spectrum of superfields (6.29) for the case M 1,1,1 :
M1 = 3k,
M = 0,
2
k > 0.
(6.31)
J = k,
E0 = 2k + 2, y0 = 2k,
This exactly coincides with the spectrum of short graviton multiplets found in Kaluza
Klein theory through harmonic analysis [39].
574
For the Q1,1,1 case the same analysis gives the following prediction for the short graviton
multiplets:
J1 = J2 = J3 = 12 k,
k > 0.
(6.32)
E0 = k + 2, y0 = k,
We can make a consistency check on this prediction just relying on the spectrum of the
Laplacian (6.18). Indeed, looking at Table 4 of [39], we see that in a short graviton multiplet
the mass of the spin two particle is
m2h = 16y0(y0 + 3).
(6.33)
Looking instead at Eq. (B.3) of the same paper, we see that such a mass is equal to the
eigenvalue of the scalar Laplacian m2h = H0 . Therefore, for consistency of the prediction
(6.32), we should have H0 = 16k(k + 3) for the representation J1 = J2 = J3 = k/2; Y = k.
This is indeed the value provided by Eq. (6.18).
It should be noted that when we write the operator (6.29), it is understood that all color
indices are symmetrized before taking the contraction.
6.3.2. Superfields corresponding to the short vector multiplets
Consider next the superfields of the following type:
(x, ) = J (x, ) chiral(x, ),
(6.34)
where J is a conserved vector current of the type analyzed in Eq. (6.26b) and chiral is a
chiral superfield. By construction, the superfield (6.34), at least in the abelian case, satisfies
the constraint
D+ D+ = 0
(6.35)
and then, according to the analysis of [40], it can describe a short vector multiplet
propagating into the bulk.
In principle, the flavor irreducible representations occurring in the superfield (6.34) are
those originating from the tensor product decomposition
X
R ,
(6.36)
ad Rk = Rmax
<max
where ad is the adjoint representation, k is the flavor weight of the chiral field at level k,
max is the highest weight occurring in the product ad Rk and < max are the lower
weights occurring in the same decomposition.
Let us assume that the quantum mechanism that suppresses all the candidate chiral
superfields of subleading weight does the same suppression also on the short vector
superfields (6.34). Then in the sum appearing on the l.h.s. of Eq. (6.36) we keep only
the first term and, as we show in a moment, we reproduce the KaluzaKlein spectrum
of short vector multiplets. As we see, there is just a universal rule that presides at the
selection of the flavor representations in all sectors of the spectrum. It is the restriction to
the maximal weight. This is the group theoretical implementation of the ideal that defines
the conifold as an algebraic locus in Cp . We already pointed out that, differently from
575
the D = 4 analogue of these conformal gauge theories, the ideal cannot be implemented
through a superpotential. An equivalent way of imposing the result is to assume that the
color indices have to be completely symmetrized: such a symmetrization automatically
selects the highest weight flavor representations.
Let us now explicitly verify the matching with KaluzaKlein spectra. We begin with
the M 1,1,1 case. Here the highest weight representations occurring in the tensor product
of the adjoint (M1 = M2 = 1, J = 0) (M1 = M2 = 0, J = 1) with the chiral spectrum
(6.12) are M1 = 3k + 1, M2 = 1, J = k and M1 = k, M2 = 0, J = k + 1. Hence the
spectrum of vector fields (6.34) limited to highest weights is given by the following list
of Osp(2|4) SU(2) SU(3) irreps:
M1 = 3k + 1,
M = 1,
2
k > 0,
(6.37)
J = k,
M1 = 3k,
M = 0,
2
J = k + 1,
E0 = 2k + 1,
k > 0.
(6.38)
J1 = k/2 + 1,
J = k/2,
2
k > 0,
(6.39)
J3 = k/2,
E0 = k + 1, y0 = k,
and all the other are obtained from (6.39) by permuting the role of the three SU(2) groups.
Looking at Table 6 of [39], we see that in every N = 2 short multiplet emerging from Mtheory compactification on AdS4 M 7 the lowest energy state is a scalar S with squared
mass
m2S = 16y0(y0 1).
(6.40)
Hence, recalling Eq. (6.16) and combining it with (6.40), we see that for consistency of
our predictions we must have
p
(6.41)
H0 + 176 24 H0 + 36 = 16k(k 1)
for the representations (6.39). The quadratic equation (6.41) implies H0 = 16k 2 + 80k + 64
which is precisely the result obtained by inserting the values (6.32) into Popes formula
(6.18) for the Laplacian eigenvalues. Hence, also the short vector multiplets follow a
general pattern identical in all Sasakian compactifications.
We can finally wonder why there are no short vector multiplets obtained by multiplying
the Betti currents with chiral superfields. The answer might be the following. From the
576
flavor view point these would not be highest weight representations occurring in the tensor
product of the constituent supersingletons. Hence they are suppressed from the spectrum.
6.3.3. Superfields corresponding to the short gravitino multiplets
The spectrum of M 1,1,1 derived in [39] contains various series of short gravitino
multiplets. We can provide their gauge theory interpretation through the following
superfields. Consider:
(ii j ` i j ` )(AC D C D )
k k
1 1
0 j B1 1 1 k k k
iA
= U U D V V V V D U U
|U U U V
i1
j1
`1
C1
D1
jB
jk `k
{z
U U U V Ck V D}k
ik
(6.42)
and
00
= U i U j U ` V A D V B AB
i1 j1 `1 C1 D1
U ik U jk U `k V Ck V D}k ,
U
| U U V V {z
(6.43)
where all the color indices are symmetrized before being contracted. By construction the
superfields (6.42), (6.43), at least in the abelian case, satisfy the equation
D+ = 0
(6.44)
and then, as explained in [40], they correspond to short gravitino multiplets propagating in
the bulk. We can immediately check that their highest weight flavor representations yield
the spectrum of Osp(2|4) SU(2) SU(3) short gravitino multiplets found by means of
harmonic analysis in [39]. Indeed for (6.42), (6.43) we respectively have:
M1 = 3k + 1,
M = 1,
2
k > 0,
(6.45)
J = k + 1,
E0 = 2k + 3/2, y0 = 2k,
and
M1 = 3k + 3,
M = 0,
2
k > 0.
(6.46)
J
=
k,
E0 = 2k + 3/2, y0 = 2k,
We postpone the analysis of short gravitino multiplets on Q1,1,1 to [42] since this requires
a more extended knowledge of the spectrum.
6.4. Long multiplets with rational protected dimensions
Let us now observe that, in complete analogy to what happens for the T 1,1 conformal
spectrum one dimension above [5,7], also in the case of M 1,1,1 there is a large class of
577
long multiplets with rational conformal dimensions. Actually this seems to be a general
phenomenon in all KaluzaKlein compactifications on homogeneous spaces G/H . Indeed,
although the Q1,1,1 spectrum is not yet completed [42], we can already see from its
Laplacian spectrum (6.18) that a similar phenomenon occurs also there. More precisely,
while the short multiplets saturate the unitarity bound and have a conformal weight related
to the hypercharge and maximal spin by Eq. (6.28), the rational long multiplets satisfy a
quantization condition of the conformal dimension of the following form
E0 = |y0 | + smax + ,
Inspecting the
multiplets:
M 1,1,1
N.
(6.47)
(6.48)
(6.49)
corresponding to
= 1.
(6.50)
M2 = 3k + 1,
J =k+1
(6.51)
E0 = 2k +
9
7
= |y0 | + ,
2
2
(6.52)
M2 = 3k,
J =k1
(6.53)
E0 = 2k +
= 2.
(6.54)
(6.55)
M2 = 3k,
J =k
(6.56)
578
E0 = 2k + 4 = |y0 | + 4,
(6.57)
(6.58)
(6.59)
E0 = 2k + 10 = |y0 | + 10,
(6.60)
(6.61)
The generalized presence of these rational long multiplets hints at various still unexplored
quantum mechanisms that, in the conformal field theory, protect certain operators from
acquiring anomalous dimensions. At least for the long graviton multiplets, characterized by
= 1, the corresponding protected superfields are easy to guess. If we take the superfield
of a short vector multiplet J (x, )chiral(x, ) and we multiply it by a stressenergy
superfield T (x, ), namely if we consider a superfield of the form
conserved vector current stressenergy tensor chiral operator,
(6.62)
we reproduce the right Osp(2|4) SU(3) SU(2) representations of the long rational
graviton multiplets of M 1,1,1 . The soundness of such an interpretation can be checked by
looking at the graviton multiplet spectrum on Q1,1,1 . This is already available since it
is once again determined by the Laplacian spectrum. Applying formula Eq. (6.62) to the
Q1,1,1 gauge theory leads to predict the following spectrum of long rational multiplets:
J1 = k/2 + 1,
J = k/2,
2
k > 0,
(6.63)
= k/2,
J
3
E0 = k + 1, y0 = k,
and all the other are obtained from (6.63) by permuting the role of the three SU(2) groups.
Looking at Table 1 of [39], we see that in a graviton multiplet the spin two particle has
mass
m2h = 16(E0 + 1)(E0 2),
(6.64)
(6.65)
On the other hand, looking at Eq. (B.3) of [39] we see that the squared mass of the graviton
is just the eigenvalue of the scalar Laplacian m2h = H0 . Applying Popes formula (6.18) to
the representations of (6.63) we indeed find
H0 = 16k 2 + 80k + 64 = 16(k + 4)(k + 1).
(6.66)
579
It appears, therefore, that the generation of rational long multiplets is based on the
universal mechanism codified by the ansatz (6.62) applicable to all compactifications. Why
these superfields have protected conformal dimensions is still to be clarified within the
framework of the superconformal gauge theory. The superfields leading to rational long
multiplets with much higher values of , like the cases = 3 and = 9 that we have
found, are more difficult to guess. Yet their appearance seems to be a general phenomenon
and this, as we have already stressed, hints at general protection mechanisms that have still
to be investigated.
Part III
(7.1)
The rationale for this description is that there is a holomorphic version; one first
complexifies G to GC in the standard way, next one chooses an orientation of the roots
580
of Lie GC in such a way that the character is the exponential of an antidominant weight
eC by exponentiating the missing positive
and finally one completes the complexification H
e. Giving to Ma the
roots. This gives a parabolic subgroup P GC and GC /P ' G/H
complex structure of GC /P we get a compact complex 3-fold.
e determined above extends to a (holomorphic) character of the
The character of H
parabolic subgroup P and this induces a holomorphic line bundle L over Ma , which is
homogeneous for GC and has plenty holomorphic sections spanning the irrep with highest
weight log(). Restricting to the compact form G, L acquires a fibre metric and M 7 is
simply the unit circle bundle inside L. It turns out that L produces a Kodaira embedding of
Ma in P(V ), the linear space V = H 0 (Ma , L) being precisely the space of holomorphic
sections of L.
Embedding quadrics and representation theory
One can also write down the equations for the image of Ma in P(V ) by means of
representation theory. Being Ma a homogeneous variety, it is cut out by homogeneous
equations of degree at most two. To find them one proceeds as follows. The space of
quadrics in P(V ) is the symmetric tensor product Sym2 (V ). As a representation of GC
this is actually reducible (for a generic dominant character 1 ) and decomposes as
Sym2 (V ) = W 2 W ,
(7.2)
W being the irrep induced by the character of P . It turns out that the weight
vectors spanning the addenda W , 6= 2 , considered as quadratic relations among the
homogeneous coordinates of P(V ), generate the ideal I of Ma . Generically the image of
the embedding is not a complete intersection.
Coordinate ring versus chiral ring
Finally, the homogeneous coordinate ring of Ma P(V ) is
C[W 1 ]/I ' k>0 W k .
(7.3)
The physical interpretation of this coordinate ring in the context of 3D conformal field
theories emerging from an M2-brane compactification on a Sasakian MS7 is completely
analogous to the interpretation of the coordinate ring in the context of 2D conformal field
6 .
theories emerging from string compactification on an algebraic CalabiYau threefold MCY
4
In the second case let X be the projective coordinates of ambient P space and W (X) = 0
the algebraic equation cutting out the CalabiYau locus. Then the ring
C[X]
WCY (X)
(7.4)
is isomorphic to the ring of primary conformal chiral operators of the (2, 2) CFT with
c = 9 realized on the world sheet. These latter are characterized by being invariant under
one of the two world-sheet supercurrents (say G (z)) and by having their conformal
weight h = |y|/2 fixed in terms of their U (1) charge. Geometrically, this is also the ring of
Hodge structure deformations. In a completely analogous way the coordinate ring (7.3) is
isomorphic to the ring of conformal hypermultiplets of the N = 2, D = 3 superconformal
theory. The hypermultiplets are short representations [39,40] of the conformal group
581
Osp(2|4) and are characterized by E0 = |y0 |, where E0 is the conformal weight while
y0 is the R-symmetry charge.
Homology
For applications to brane geometry, it is also important to know the homology
(equivalently, cohomology) of M 7 . Being M 7 a circle bundle, we can use the Gysin
sequence in cohomology [62]. Since the base Ma is acyclic in odd dimensions, the Gysin
sequence splits into subsequences of the form
c1
(7.6)
7.1.1. Generalities
Let us call hi , i = 1, 2, h and Y the generators of the Lie algebras of the standard
maximal tori of SU(3), SU(2) and U (1), respectively, all normalized with periods 2 .
Then SU(2) is embedded in SU(3) as the stabilizer of the last basis vector of C3 , and the
U (1)s are generated by Z 0 = (h1 + 2h2 ) h 4Y , and Z 00 = (h1 + 2h2 ) + 3h.
To reconstruct the general structure described at the beginning of this section, notice that
the image of H 0 under the projection of G0 onto SU(3) is isomorphic to S(U (2) U (1)).
This projection gives an exact sequence 0 K H 0 H 0 and, since K is normal
in H 0 , we have an isomorphism G/H = (G0 /K)/(H 0 /K) ' G0 /H 0 . The elements of H 0
have the form
g 0
(7.7)
exp i( + )(h1 + 2h2 ) , exp i(3 )h , exp(i4 Y ) ,
0 1
where g SU(2) and , [0, 2]. So K is given by g = (1)l , + = l and its
generic element is (1, exp(4 i h), exp(4 i Y )). Taking the quotient by K, the third
factor of G0 is factored out and there is an extra Z2 acting on the maximal torus of SU(2).
Consequently, we find that G/K = SU(3) SO(3) and the image of H in G/K has a
component on the maximal torus of SO(3) (generated by exp 2 it ) corresponding to the
infinitesimal character (h1 ) = 0, (h2 ) = 3t.
e = S(U (2)
Summing up, we have G = SU(3) SO(3), H = S(U (2) U (1)) and H
U (1)) U (1). Accordingly
Ma = Gr(2, 3) P1 ' P2 P1 ,
(7.8)
582
Pn being the complex n-dimensional space and Gr(2, 3) being the Grassmannian of 2planes in C3 . 9
e onto H
e/H . We
Now we have to recognize the character. This comes by projecting H
get
exp(i h1 ) eH = eH,
exp(i h2 ) eH = exp(3 i t)H,
exp(i t) eH = exp(i t)H,
e/H . The character restricted to SO(3)
where t has been identified with the generator of H
is the fundamental one, which corresponds to the adjoint representation of SU(2) and
therefore is the square of the fundamental character of SU(2). We see then that M 1,1,1
is the circle bundle 10 inside L = O(3) O(2) over P2 P1 .
The fundamental group of the circle bundle associated to the infinitesimal character
(h1 ) = 0, (h2 ) = m, (h) = n is Zgcd(m,n) Applying the same analysis to M p,q,r , the
character corresponds to
r
2r
3p
lcm
,
,
m=
2r
gcd(r, q) gcd(2r, 3p)
r
2r
q
,
,
n = lcm
r
gcd(r, q) gcd(2r, 3p)
for r 6= 0 and
m=
n=
3p
,
gcd(3p, 2q)
2q
,
gcd(3p, 2q)
for r = 0. Notice that M 1,1,1 = M 1,1,r is simply connected. Although in [38] it is stated
that M 1,1,1 = M 1,1,0/Z4 , a closer analysis shows that the Z4 action is trivial. In particular
H1 (M 1,1,1, Z) is torsionless.
7.1.2. The algebraic embedding equations and the chiral ring of M 1,1,1
As for the algebraic embedding of M 1,1,1 , since dim W1 = 30, L embeds
Ma ' P2 P1 , P29
(7.9)
by
Xij k|AB = U i U j U k V A V B
(i, j, k = 0, 2, 3;
A, B = 1, 2),
(7.10)
583
image of Ma is cut out by dim Sym2 (W1 ) dim W2 = 465 140 = 325 equations.
Alternatively, the same 325 equations can be seen as the embedding of the cone C(M 1,1,1)
over the Sasakian U (1) bundle into C30 .
For further clarification we describe the explicit form of these embedding equations
in the language of Young tableaux. From Eq. (7.9) it follows that the 30 homogeneous
coordinates of P29 are assigned to the representation (10, 3) of SU(3) SU(2):
Xij k|AB 7 (10, 3)
(7.11)
This means that the quadric monomials X2 span the following symmetric tensor product:
h
i h
i
2
(7.12)
X =
sym
X2
is just
In general the number of independent components of
30 31
= 465,
(7.13)
dim X2 =
2
which corresponds to the sum of dimensions of all the irreducible representations of
SU(3) SU(2) contained in the symmetric product (7.12), but on the locus defined by
the explicit embedding (7.9) only 28 5 = 140 of these components are independent.
These components fill the representation of highest weight
.
(28, 5)
(7.14)
The remaining 325 components are the quadric equations of the locus. They are nothing
else but the statement that all the representations of SU(3) SU(2) contained in the
symmetric product (7.12) should vanish with the exception of the representation (7.14).
Let us work out the representations that should vanish. To this effect we begin by writing
the complete decomposition into irreducible representations of SU(3) of the tensor product
10 10:
| {z }
| {z }
10
{z
10
28
{z
35
{z
(7.15)
|
27
{z
10
| {z } | {z } sym |
{z
}
|
{z
10
10
28
(7.16)
}
27
and
| {z }
10
| {z }
10
antisym
{z
35
(7.17)
|
{z
10
584
As a next step we do the same decomposition for the tensor product 3 3 of SU(2)
representations. We have
| {z } | {z } | {z }
| {z } | {z }
3
3
5
3
(7.18)
| {z } | {z }
3
sym
=
| {z }
| {z }
5
(7.19)
and
| {z } | {z }
3
antisym
| {z }
(7.20)
(7.21)
(7.22)
325
Hence the equations can be arranged into 5 representations corresponding to the list
appearing in the second row of Eq. (7.22). Indeed, Eq. (7.22) is the explicit form, in the
case M 1,1,1 , of the general equation (7.2) and the addenda in its second line are what we
named W , 6= 2 , in the general discussion.
Coming now to the coordinate ring (7.3) it is obvious from the present discussion that,
in the M 1,1,1 case, it takes the following form:
X
... .
(7.23)
C[W 1 ]/I ' k>0 W k =
...
|
{z
} |
{z
}
k>0
3k
2k
In Eq. (7.23) we recognize the spectrum of SU(3) SU(2) representations of the Osp(2|4)
hypermultiplets as determined by harmonic analysis on M 1,1,1 . Indeed, recalling the results
of [39,40], the hypermultiplet of conformal weight (energy label) E0 = 2k and hypercharge
y0 = 2k is in the representation
M1 = 3k,
M2 = 0,
J = k.
(7.24)
585
H 0 (M 1,1,1 ) = H 7 (M 1,1,1) = Z,
c1
0 H 1 (M 1,1,1) Z Z Z H 2 (M 1,1,1) 0,
c1
0 H 3 (M 1,1,1) Z Z Z Z H 4 (M 1,1,1 ) 0,
c1
0 H 5 (M 1,1,1) Z Z Z H 6 (M 1,1,1) 0.
(7.25)
The first c1 sends 1 H 0 (Ma ) to c1 H 2 (Ma ). Its kernel is zero, and its image is Z.
Accordingly, H 2 (M 1,1,1) = Z (1 + 2 ). The second c1 sends (1 , 2 ) Z Z =
H 2 (Ma ) to (31 2 , 21 2 + 322 ) Z Z = H 4 (Ma ). Its kernel vanishes and therefore
H 3 (M 1,1,1) = 0. Its cokernel is Z9 = H 4 (M 1,1,1 ) generated by (1 2 + 22 ). Finally,
the last c1 sends 1 2 and 22 H 4 (Ma ) = Z Z respectively to 31 22 and 21 22
H 6 (Ma ). This map is surjective, so H 6 (M 1,1,1) = 0 and its kernel is generated by =
21 2 + 322 . Hence H 5 (M 1,1,1) = Z , with = .
7.1.4. Explicit description of the Sasakian fibration for M 1,1,1
We proceed next to an explicit description of the fibration structure of M 1,1,1 as a
U (1)-bundle over P2 P1 . We construct an atlas of local trivializations and we give the
appropriate transition functions. This is important for our discussion of the supersymmetric
cycles leading to the baryon states.
as local coordinates on
We take [0, 4) as a local coordinate on the fibre and ( , )
P1 ' S 2 . To describe P2 we have to be a little bit careful. P2 can be covered by the three
patches W ' C2 in which one of the three homogeneous coordinates, U , does not vanish.
The set not covered by one of these W is homeomorphic to S 2 . We choose to parametrize
W3 as in [55]:
1 = U1 /U3 = tan cos(/2) ei(+)/2
2 = U2 /U3 = tan sin(/2) ei()/2,
(7.26)
where
(0, /2),
(0, ),
0 6 ( + ) 6 4,
0 6 ( ) 6 4.
(7.27)
These coordinates cover the whole W3 ' C2 except for the trivial coordinate singularities
= 0 and = 0, . Furthermore, and can be extended to the complement of W3 .
Indeed, the ratio
z = 1 / 2 = tan1 (/2) ei
(7.28)
is well defined in the limit /2 and it constitutes the usual stereographic map of S 2
onto the complex plane (see the next discussion of Q1,1,1 and in particular Fig. 4).
Just as for the sphere, we must be careful in treating some one-forms near the coordinate
singularities. In particular, d and d are not well defined on the three S 2 which are not
covered by one of the patches W : { = /2}, { = 0} and { = /2} (see Fig. 3). Actually,
except for the three points of these spheres that are covered by only one patch ({ = 0}
586
Fig. 3. Schematic representation of the atlas on P2 . The three patches W cover the open ball and
part of the boundary circle, which constitutes the set of coordinate singularities. This latter is made of
three S 2 s: { = 0}, { = } and { = /2}, which touch each other at the three points marked with
a dot. Each W covers the whole P2 except for one of the spheres (for example, W3 does not cover
{ = /2}). The three most singular points are covered by only one patch (for example, { = 0} is
covered by the only W3 ).
Regular
1-form
d + d
d d
d
Singular
1-forms
d + d ( 6= )
d d( 6= )
d
(7.29)
The singular one-forms become well defined if we multiply them by a function having a
double zero at the coordinate singularities.
We come now to the description of the fibre bundle M 1,1,1 . We cover the base P2 P1
with six open charts U = W H ( = 1, 2, 3) on which we can define a local fibre
coordinate [0, 4). The transition functions are given by:
1 = 3 3( + ) + 2( ) (, = 1),
(7.30)
1 = 2 6 + 2( ).
On this principal fibre bundle we can easily introduce a U (1) Lie algebra valued connection
which, on the various patches of the base space, is described by the following one-forms:
3
3
A1 = (cos 2 + 1)(d + d) (cos 2 1)(cos 1) d
2
2
d,
+ 2(1 cos )
3
3
A2 = (cos 2 + 1)(d d) (cos 2 1)(cos + 1) d
2
2
d,
+ 2(1 cos )
3
(7.31)
587
Due to (7.30), the one-form (d A) is a global angular form [62]. It can then be taken as
the 7th vielbein of the following SU(3) SU(2) U (1) invariant metric on M 1,1,1 :
2
2
2
2
2
dsM
1,1,1 = c (d A) + dsP2 + dsP1 .
(7.32)
3
3 8 + 3 + 3 8 + 3 +
8
7
2
X
1 X A
A 3
+
m m,
+
8
4
A=4
(7.33)
m=1
2
3
d 3 sin2 (d + cos d) + 2 cos d
32
9
1
d2 + sin2 cos2 2 (d + cos d)2
+
2
4
1
3
d 2 + sin2 d 2 ,
+ sin2 d 2 + sin2 d 2 +
4
4
(7.34)
where the three addenda of (7.34) are one by one identified with the three addenda of
(7.33). The second and the third addenda are the P2 and S 2 metric on the base manifold
of the U (1) fibration, while the first term is the fibre metric. In other words, one recognizes
the structure of the metric anticipated in (7.32). The parameter appearing in the metric
(7.34) is the internal cosmological constant defined by Eq. (5.16).
7.1.5. The baryonic 5-cycles of M 1,1,1 and their volume
As we saw above, the relevant homology group of M 1,1,1 for the calculation of the
baryonic masses is
H5 (M 1,1,1 , R) = R.
Let us consider the following two 5-cycles, belonging to the same homology class:
(7.35)
588
= 0 = const,
= 0 = const,
= 0 = const,
C2:
= 0 = const.
C1:
(7.36)
(7.37)
The two representatives (7.36), (7.37) are distinguished by their different stability
subgroups which we calculate in the next subsection.
Volume of the 5-cycles
The volume of the cycles (7.36), (7.37) is easily computed by pulling back the metric
(7.34) on C 1 and C 2 , that have the topology of a U (1)-bundle over P2 and P1 P1
respectively:
I
Z
8 5/2
g1 = 9
sin3 cos sin d d d d d
Vol(C 1 ) =
3
C1
I
Vol(C 2 ) =
C2
9 3 3 5/2
,
=
2 2
(7.38)
Z
8 5/2
g2 = 6
sin cos sin d d d d d
3
= 6
3
2
5/2
.
(7.39)
g
Vol(M 1,1,1) =
M 1,1,1
Z
8 7/2
sin3 cos sin sin d d d d d d d
= 18
3
27 4 3 5/2
.
(7.40)
=
2 2
The results (7.38), (7.39), (7.40) can be inserted into the general formula (5.28) to calculate
the conformal weights (or energy labels) of 5-branes wrapped on the cycles C 1 and C 2 . We
obtain:
N
4N
E0 (C 2 ) =
.
(7.41)
E0 (C 1 ) = ,
3
9
As stated above, the result (7.41) is essential in proving that the conformal weight of the
elementary world-volume fields V A , U i are
h U i = 4/9,
(7.42)
h V 4 = 1/3,
respectively. To reach such a conclusion we need to identify the states obtained by
wrapping the 5-brane on C 1 , C 2 with operators in the flavor representations M1 = 0,
589
(7.43)
of the two cycles (7.36), (7.37). Let us begin with the first cycle defined by (7.36). As we
have previously said, this is the restriction of the U (1)-fibration to P2 {p}, p being a
point of P1 . Hence, the stability subgroup of the cycle C 1 is:
(7.44)
H C 1 = SU(3) U (1)R U (1)B,1,
where U (1)R is the R-symmetry U (1) appearing as a factor in SU(3) SU(2) U (1)R
while U (1)B,1 SU(2) is a maximal torus.
Turning to the case of the second cycle (7.37), which is the restriction of the U (1)-bundle
to the product of a hyperplane of P2 and P1 , its stabilizer is
(7.45)
H C 2 = S U (1)B,2 U (2) SU(2) U (1)R ,
where SU(2) U (1)R is the group appearing as a factor in SU(3) SU(2) U (1)R ,
U (1)B,2 SU(3) is the subgroup generated by h1 = diag(1, 1, 0) and S(U (1)B,2
U (2)) SU(3) is the stabilizer of the first basis vector of C3 .
Following the procedure introduced by Witten in [43] we should now quantize the
collective coordinates of the non-perturbative baryon state obtained by wrapping the 5brane on the 5-cycles we have been discussing. As explained in Wittens paper this leads
to quantum mechanics on the homogeneous manifold G/H (C). In our case the collective
coordinates of the baryon live on the following spaces:
SU(2)
for C 1 ,
U (1)B,1 ' P1
G
=
(7.46)
space of collective coordinates
SU(3)
H (C)
' P2 for C 2 .
S(U (1)B,2 U (2))
The wave function (collect. coord.) is in Wittens phrasing a section of a line bundle of
degree N . This happens because the baryon has baryon number N , namely it has charge
N under the additional massless vector multiplet that is associated with a harmonic 2form and appears in the KaluzaKlein spectrum since dimH2 (M 1,1,1 ) = 1 6= 0. These
are the Betti multiplets mentioned in Section 4.3. Following Wittens reasoning there is
a morphism
i : U (1)Baryon , H (C i ),
i = 1, 2,
(7.47)
of the non-perturbative baryon number group into the stability subgroup of the 5-cycle.
Clearly the image of such a morphism must be a U (1)-factor in H (C) that has a non-trivial
590
action on the collective coordinates of the baryons. Clearly in the case of our two baryons
we have:
Im i = U (1)B,i ,
i = 1, 2.
(7.48)
The name given to these groups anticipated the conclusions of such an argument.
Translated into the language of harmonic analysis, Wittens statement that the baryon
wave function should be a section of a line bundle with degree N means that we
are supposed to consider harmonics on G/H (C) which, rather than being scalars of
H (C), are in the 1-dimensional representation of U (1)B with charge N . According to
the general rules of harmonic analysis (see [29,35,39]) we are supposed to collect all
the representations of G whose reduction with respect to H (C) contains the prescribed
representation of H (C). In the case of the first cycle, in view of Eq. (7.44) we want all
representations of SU(2) that contain the state 2J3 = N . Indeed the generator of U (1)B,1
can always be regarded as the third component of angular momentum by means of a change
of basis. The representations with this property are those characterized by:
2J = N + 2k,
k > 0.
(7.49)
Since the Laplacian on G/H (C) has eigenvalues proportional to the Casimir
2SU(2)/U (1) = const J (J + 1),
(7.50)
the harmonic satisfying the constraint (7.49) and with minimal energy is just that with
2J = N.
(7.51)
This shows that under the flavor group the baryon associated with the first cycle is
neutral with respect to SU(3) and transforms in the N -times symmetric representation of
SU(2). This perfectly matches, on the superconformal field theory side, with our candidate
operator (5.3).
Equivalently the choice of the representation 2J = N corresponds with the identification
of the baryon wave-function with a holomorphic section (= zero mode) of the U (1)-bundle
under consideration, i.e., with a section of the corresponding line bundle. Indeed such a
line bundle is, by definition, constructed over P1 and declared to be of degree N , hence it
is OP1 (N). Representation-wise a section of OP1 (N) is just an element of the J = N/2
representation, namely it is the N -times symmetric of SU(2).
Let us now consider the case of the second cycle. Here the same reasoning instructs us
to consider all representations of SU(3) which, reduced with respect to U (1)B,2 , contain
a state of charge N . Moreover, directly aiming at zero mode, we can assign the baryon
wave-function to a holomorphic sections of a line bundle on P2 , which must correspond
to characters of the parabolic subgroup S(U (1)B,2 U (2)). As before the degree N of
this line bundle uniquely characterizes it as O(N). In the language of Young tableaux, the
corresponding SU(3) representation is
M1 = 0,
M2 = N,
(7.52)
i.e., the representation of this baryon state is the N -time symmetric of the dual of SU(3)
and this perfectly matches with the complex conjugate of the candidate conformal operator
591
(5.2). In other words we have constructed the antichiral baryon state. The chiral one
obviously has the same conformal dimension.
7.1.7. These 5-cycles are supersymmetric
The 5-cycles we have been considering in the above subsections have to be supersymmetric in order for the conclusions we have been drawing to be correct. Indeed all our
arguments have been based on the assumption that the 5-brane wrapped on such cycles is
a BPS-state. This is true if the 5-brane action localized on the cycle is supersymmetric.
The -symmetry projection operator for a 5-brane is
1
1
1 i XM XN XP XQ XR MNP QR ,
(7.53)
P =
2
5! g
where the functions XM ( ) define the embedding of the 5-brane into the 11-dimensional
spacetime, and g is the square root of the determinant of the induced metric on the
brane. The gamma matrices MNP QR , defining the spacetime spinorial structure, are the
pullback through the vielbeins of the constant gamma matrices ABCDE satisfying the
standard Clifford algebra:
A B C D E
eN eP eQ eR ABCDE .
MNP QR = eM
(7.54)
A possible choice of vielbeins for C(M 1,1,1) M3 , namely the product of the metric cone
over M 1,1,1 times three-dimensional Minkowski space is the following one:
1
1
e2 = r sin d,
e1 = r d,
2 2
2 2
1
e3 = r d + 3 sin2 (d + cos d) + 2 cos d ,
8
3
4
r d,
e =
2
3
r sin cos (d + cos d),
e5 =
4
3
r sin (sin d cos sin d),
e6 =
4
3
r sin (cos d + sin sin d),
e7 =
4
e8 = dr,
e9 = dx 1 ,
e10 = dx 2 ,
e0 = dt.
(7.55)
In these coordinates the embedding equations of the two cycles (7.36), (7.37) are very
simple, so we have
,
1
M
N
P
Q
R
X X X X X MNP QR =
(7.56)
,
5!
for C 1 and C 2 , respectively. By means of the vielbeins (7.55) these gamma-matrices are
immediately computed:
592
3 2 5 3
=
r sin cos sin 34567,
32
3 5
r sin cos sin 31245,
(7.57)
=
512
while the square root of the determinant of the metric on the two cycles is easily seen to be
2
3
g1 =
r 5 sin3 cos sin ,
32
3 5
The important thing to check is that the projectors (7.59) are non-zero on the two
Killing spinors of the space C(M 1,1,1) M3 . Indeed, this latter has not 32 preserved
supersymmetries, rather it has only 8 of them. In order to avoid long and useless
calculations we just argue as follows. Using the gamma-matrix basis of [28], the Killing
spinors are already known. We have:
0 = 0 188 ,
9 = 2 188 ,
i = 5 i
8 = 1 188 ,
10 = 3 188 ,
(i = 1, . . . , 7)
(7.60)
0
u
(7.61)
Killing spinors = (x) , =
0 ,
u?
where
u=
a + ib
,
0
u? =
0
1
1
0
u? =
0
a + ib
(7.62)
122
0
0
0
0
122
0
0
,
34567 = 5 U8 U4 U5 U6 U7 3 = i5
0
0
122
0
0
0
0
122
593
3 0 0 0
0 3 0 0
31245 = 5 iU8 U4 U5 1 = i5
0 0 3 0 .
0 0 0 3
(7.63)
As we see, by comparing Eq. (7.59) with Eqs. (7.61) and (7.63), the -supersymmetry
projector reduces for both cycles to a chirality projector on the 4-component space
time part (x). As such, the -supersymmetry projector always admits non-vanishing
eigenstates implying that the cycle is supersymmetric. The only flaw in the above argument
is that the Killing spinor (7.61) was determined in [28] using as vielbein basis the suitably
rescaled MaurerCartan forms 3 , m (m = 1, 2) and A (A = 4, 5, 6, 7). Our choice
(7.55) does not correspond to the same vielbein basis. However, a little inspection shows
that it differs only by some SO(4) rotation in the space of P2 vielbein 4, 5, 6, 7. Hence
we can turn matters around and ask what happens to the Killing spinor (7.61) if we apply
an SO(4) rotation in the directions 4, 5, 6, 7. It suffices to check the form of the gammamatrices [A , B ] which are the generators of such rotations. Using again the Appendix of
[28] we see that such SO(4) generators are of the form
0
0
0
i
i 0 0 0
0 i 0
0 i 0 0
0
,
(7.64)
i
0 0 i 0 or i 0
0
i
0
0
so that the SO(4) rotated Killing spinor is of the same form as in Eq. (7.61) with,
however, u replaced by u0 = Au where A SU(2). It is obvious that such an SU(2)
transformation does not alter our conclusions. We can always decompose u0 into 3
eigenstates and associate the 3 -eigenvalue with the chirality eigenvalue, so as to satisfy
the -supersymmetry projection. Hence, our 5-cycles are indeed supersymmetric.
7.2. The manifold Q1,1,1
This is defined by G = SU(2) SU(2) SU(2), H = U (1) U (1). If we call hi
the generators of the maximal tori of the SU(2)s, normalized with periods 2 , H is
e=
generated by h1 h2 and h1 h3 , i.e., the complement of Z = h1 + h2 + h3 , while H
U (1) U (1) U (1) is the product of the three maximal tori. So the base is
Ma = P1 P1 P1 .
(7.65)
(7.66)
showing that Q1,1,1 is the circle bundle inside O(1) O(1) O(1) over P1 P1 P1 .
7.2.1. The algebraic embedding equations and the chiral ring
Since dim H 0 (Ma , L) = 8, L embeds
Ma ' P1 P1 P1 , P7
(7.67)
594
(i, j, k = 1, 2).
(7.68)
Xij k 7 (2, 2, 2)
(7.69)
(7.70)
and it is easy to find the structure of the embedding equations. Here we have
1 1 1
1 1 1
1, 1) (1, 0, 0) + (0, 1, 0) + (0, 0, 1) .
2 , 2 , 2 2 , 2 , 2 sym = (1,
{z
}
| {z } |
27
(7.71)
The 9 embedding equations i are given by the vanishing of the irreducible representations
not of highest weight, namely:
0 = A ij Xi`p Xj mq `m pq ,
0 = A `m Xi`p Xj mq ij pq ,
(7.72)
0 = A pq Xi`p Xj mq ij `m .
Coming now to the coordinate ring (7.3), it follows that in the Q1,1,1 case it takes the
following form:
X
... ... .
(7.73)
C[W 1 ]/I ' k>0 W k =
...
| {z } | {z } | {z }
k>0
k
In Eq. (7.73) we predict the spectrum of SU(2) SU(2) SU(2) representations of the
Osp(2|4) hypermultiplets as determined by harmonic analysis on Q1,1,1 . We find that the
hypermultiplet of conformal weight (energy label) E0 = k and hypercharge y0 = k should
be in the representation:
J1 = k/2,
J2 = k/2,
J3 = k/2.
(7.74)
(7.75)
595
Fig. 4. Two coordinate patches for the sphere. They constitute the base for a local trivialization of a
fibre bundle on S 2 . Each patch covers only one of the poles, where the coordinates (, ) are singular.
(7.76)
To describe the total space we have to specify the transition maps for on the
intersections of the patches. These maps for the generic Qp,q,r space are
1 1 1 = 2 2 2 + p(1 2 )1 + q(1 2 )2 + r(1 2 )3 .
(7.77)
(7.78)
We note that these maps are well defined, being all the s and s defined modulo 4 and
2 , respectively.
It is important to note that and are clearly not good coordinates for the whole S 2 .
The most important consequence of this fact is that the one-form d is not extensible to
the poles. To extend it to one of the poles, d has to be multiplied by a function which has
a double zero on that pole, such as sin2 2 d.
We can define a U (1)-connection A on the base S 2 S 2 S 2 by specifying it on each
patch H 11 :
A = ( cos 1 ) d1 + ( cos 2 ) d2 + ( cos 3 ) d3.
(7.79)
596
(7.80)
2
dsQ
1,1,1 =
(7.81)
i=1
where is the compact space cosmological constant defined in Eq. (5.16). The Einstein
metric (7.81) was originally found by DAuria, Fr and van Nieuwenhuizen [33], who
introduced the family Qp,q,r of D = 11 compactifications and found that N = 2
supersymmetry is preserved in the case p = q = r. All the other cosets in the family break
supersymmetry to N = 0, namely, in mathematical language, are not Sasakian. In [33] the
Einstein metric was constructed using the intrinsic geometry of coset manifolds and using
MaurerCartan forms. An explicit form was also given using stereographic coordinates on
the three S 2 . In the coordinate form of Eq. (7.81) the Einstein metric of Q1,1,1 was later
written by Page and Pope [34].
7.2.4. The baryonic 5-cycles of Q1,1,1 and their volume
The relevant homology group of Q1,1,1 for the calculation of the baryonic masses is
H5 (Q1,1,1 , R) = R2 .
(7.82)
(Q1,1,1 )
3 6 5/2
.
=
4
(7.83)
(7.84)
i=1
Just as in the M 1,1,1 case, inserting the above results (7.83), (7.84) into the general formula
(5.28) we obtain the conformal weight of the baryon operator corresponding to the fivebrane wrapped on this cycle:
E0 =
N
.
3
(7.85)
The other two cycles can be obtained from this by permuting the role of the three P1 s
and their volume is the same. This fact agrees with the symmetry which exchanges the
597
fundamental fields A, B and C of the conformal theory, or the three gauge groups SU(N).
Indeed, naming SU(2)i (i = 1, 2, 3) the three SU(2) factors appearing in the isometry
group of Q1,1,1 , the stability subgroup of the first of the cycles described above is
H (C 1 ) = SU(2)1 SU(2)2 U (1)B,3 ,
U (1)B,3 SU(2)3 ,
(7.86)
1
(7.87)
h[Ci ] = .
3
The stability subgroup of the permuted cycles is obtained permuting the indices 1, 2, 3 in
Eq. (7.86) and we reach the obvious conclusion
1
(7.88)
h[Ai ] = h[Bj ] = h[C` ] = .
3
This matches with the previous result (7.74) on the spectrum of chiral operators, which are
predicted of the form
chiral operators = Tr(Ai1 Bj1 C`1 Aik Bjk C`k )
and should have conformal weight E = k. Indeed, we have k ( 13 +
(7.89)
1
3
+ 13 ) = k!
8. Conclusions
We saw, using geometrical intuition, that there is a set of supersingletons fields which are
likely to be the fundamental degrees of freedom of the CFTs corresponding to Q1,1,1 and
M 1,1,1 . The entire KK spectrum and the existence of baryons of given quantum numbers
can be explained in terms of these singletons.
We also proposed candidate three-dimensional gauge theories which should flow in the
IR to the superconformal fixed points dual to the AdS4 compactifications. The singletons
are the elementary chiral multiplets of these gauge theories. The main problem we did not
solve is the existence of chiral operators in the gauge theory that have no counterpart in
the KK spectrum. These are the non-completely flavor symmetric chiral operators. Their
existence is due to the fact that, differently from the case of T 1,1 , we are not able to
write any superpotential of dimension two. If the proposed gauge theories are correct, the
dynamical mechanism responsible for the disappearing of the non-symmetric operators in
the IR has still to be clarified.
It would be quite helpful to have a description of the conifold as a deformation of
an orbifold singularity [4,14]. It would provide an holographic description of the RG
598
flow between two different CFT theories and it would also help in checking whether the
proposed gauge theories are correct or require to be slightly modified by the introduction
of new fields. In general, different orbifold theories can flow to the same conifold CFT in
the IR. In the case of T 1,1 , one can deform a Z2 orbifold theory with a mass term [4] or
a Z2 Z2 orbifold theory with a FI parameter [14]. The mass deformation approach for a
Z2 Z2 singularity was attempted for the case of Q1,1,1 in [46], where a candidate conifold
CFT was written. This theory is deeply different from our proposal. It is not obvious to us
whether this theory is compatible or not with the KK expectations. It also seems that, in the
approach followed in [46], the singletons degrees of freedom needed for constructing the
KK spectrum are not the elementary chiral fields of the gauge theory but are rather obtained
with some change of variables which should make sense only in the IR. The FI approach
was pursued in [47], were Q1,1,1 was identified as a deformation of orbifold singularities
whose associated CFTs can be explicitly written. Unfortunately, the order of the requested
orbifold group and, consequently, the number of requested gauge factors, make difficult an
explicit analysis of these models and the identification of the conifold CFT. It would be
quite interesting to investigate the relation between the results in [47] and our proposal or
to find simpler orbifold singularities related to Q1,1,1 and M 1,1,1 . For the latter one, for the
moment, no candidate orbifold has been proposed.
We did not discuss at all the CFT associated to V5,2 and N 0,1,0 . The absence of a toric
description makes more difficult to guess a gauge theory with the right properties and
also to find associated orbifold models. We leave for the future the investigation of these
interesting models.
Acknowledgements
We thank S. Ferrara for many important and enlightening discussions throughout all
the completion of the present work, and I. Klebanov for very useful and interesting
conversations. We are also indebted to C. Procesi for deep hints and valuable discussions
on the geometric side of representation theory.
+ g I J zi (TI )i j zj zk (TJ )k l zl
2
599
+ zi M I (TI )i j j k M J (TJ )k l zl
(A.1)
where:
(1) zi are the complex scalar fields belonging to the chiral multiplets;
(2) W (z) is the holomorphic superpotential;
(3) The Hermitian matrices (TI )i j (I = 1, . . . , dim G) are the generators of the gauge
group G in the (in general reducible) representation R supported by the chiral
multiplets;
?
(4) ij is the G invariant metric;
(5) g I J is the Killing metric of G;
(6) MI are the real scalar fields belonging to the vector multiplets that obviously
transform in ad(G);
(7) is the coefficient of the ChernSimons term, if present;
(8) J are the coefficients of the FayetIliopoulos terms that take values in the center of
If we put the ChernSimons and the FayetIliopoulos terms to zero = J = 0, the scalar
potential becomes the sum of three quadratic forms:
1
U (z, z, M) = |W (z)|2 + g I J DI (z, z)DJ (z, z) + M I M J KI J (z, z),
2
where the real functions
D I (z, z) = zi (TI )i j zj
(A.2)
(A.3)
are the D-terms, namely the on-shell values of the vector multiplet auxiliary fields, while
by definition we have put
def
(A.4)
If the quadratic form MI MJ KI J (z, z) is positive definite, then the vacua of the gauge
theory are singled out by the three conditions
W
= 0,
zi
D I (z, z) = 0,
(A.5)
(A.6)
MI MJ KI J (z, z) = 0.
(A.7)
The basic relation between the candidate superconformal gauge theory CFT 3 and the
compactifying 7-manifold M 7 that we have used in Eqs. (3.2), (3.5) is that, in the Higgs
branch (hMI i = 0), the space of vacua of CFT 3 , described by Eqs. (A.5)(A.7), should be
equal to the product of N copies of M 7 :
7
. . M }7 /N .
vacua of gauge theory = M
| .{z
N
(A.8)
600
Indeed, if there are N M2-branes in the game, each of them can be placed somewhere in
M 7 and the vacuum is described by giving all such locations. In order for this to make
sense it is necessary that
The Higgs branch should be distinct from the Coulomb branch;
The vanishing of the D-terms should indeed be a geometric description of (A.8).
Let us apply our general formula to the two cases under consideration and see that these
conditions are indeed verified.
A.1. The scalar potential in the Q1,1,1 case
Here the gauge group is
G = SU(N)1 SU(N)2 SU(N)3
(A.9)
(A.10)
in the abelian case N = 1. The chiral fields Ai , Bj , C` are in the SU(2)3 flavor
representations (1/2, 0, 0), (0, 1/2, 0), (0, 0, 1/2) and in the color SU(N)3 representations
(N, N, 1), (1, N, N), (N, 1, N), respectively (see Fig. 1). We can arrange the chiral fields
into a column vector:
Ai
(A.11)
zE = Bj .
C`
Naming (tI ) the N N Hermitian matrices such that itI span the SU(N) Lie algebra
(I = 1, . . . , N 2 1), the generators of the gauge group acting on the chiral fields can be
written as follows:
tI 1 0
1 tI
0
0
0
[1]
[2]
0
0 , TI = 0
tI 1 0 ,
TI = 0
0
0
0
0
0 1 tI
0
0
0
0 .
(A.12)
TI[3] = 0 1 tI
0
0
tI 1
Then the D 2 -terms appearing in the scalar potential take the following form:
"N 2 1
X
2
1
Ai (tI 1)Ai C i (1 tI )Ci
D 2 -terms =
2
I =1
2 1
NX
2
I =1
2 1
NX
2
#
.
(A.13)
I =1
The part of the scalar potential involving the gauge multiplet scalars is instead given by:
601
(A.14)
(A.15)
|A1 |2 + |A2 |2 (M1 M2 )2
+ |B1 |2 + |B2 |2 (M2 M3 )2
+ |C1 |2 + |C2 |2 (M3 M1 )2 .
(A.16)
Eqs. (A.15) and (A.16) are what we have used in our toric description of Q1,1,1 as the
manifold of gauge-theory vacua in the Higgs branch. Indeed it is evident from Eq. (A.16)
that if we give non-vanishing vev to the chiral fields, then we are forced to put hM1 i =
hM2 i = hM3 i = m. Alternatively, if we give non-trivial vevs to the vector multiplet scalars
Mi , then we are forced to put hAi i = hBj i = hC` i = 0 which confirms that the Coulomb
branch is separated from the Higgs branch.
Finally, from Eqs. (A.13), (A.14) we can retrieve the vacua describing N separated
branes. Each chiral field has two color indices and is actually a matrix. Setting
ai ,
Ai| =
bi ,
Bi| =
(A.17)
Ci| = ci ,
a little work shows that the potential (A.13) vanishes if each of the N -triplets ai , bj , c`
separately satisfies the D-term equations, yielding the toric description of a Q1,1,1
manifold (3.2). Similarly, for each abelian generator belonging to the Cartan subalgebra of
Ui (N) and having a non-trivial action on ai , bj , c` we have hM1 i = hM2 i = hM3 i =
m .
A.2. The scalar potential in the M 1,1,1 case
Here the gauge group is
G = SU(N)1 SU(N)2
(A.18)
(A.19)
602
in the abelian case N = 1. The chiral fields Ui , VA are in the SU(3) SU(2) flavor representations (3, 1), (1, 2), respectively. As for color, they are in the SU(N)2 representations
Sym2 (CN ) Sym2 (CN ), Sym3 (CN ) Sym3 (CN ), respectively (see Fig. 2). As before,
we can arrange the chiral fields into a column vector:
Ui
.
(A.20)
zE =
VA
Naming (tI[3] ) the Hermitian matrices generating SU(N) in the three-times
symmetric representation and (tI[2] ) the same generators in the two-times symmetric
representation, the generators of the gauge group acting on the chiral fields can be written
as follows:
[2]
tI 1
1 tI[2]
0
0
[1]
[2]
.
(A.21)
, TI =
TI =
0
1 tI[3]
0
tI[3] 1
Then the D 2 -terms appearing in the scalar potential take the following form:
"N 2 1
X
2
1
U i tI[2] 1 Ui V A 1 tI[3] VA
D 2 -terms =
2
I =1
#
2 1
NX
2
[2]
i
A [3]
+
U 1 tI Ui V tI 1 VA
,
(A.22)
I =1
while the part of the scalar potential involving the gauge multiplet scalars is given by
M 2 -terms = M1I M1J U i tI[2] tJ[2] 1 Ui + V A 1 tI[3] tJ[3] VA
+ M2I M2J U i 1 tI[2] tJ[2] Ui + V A tI[3] tJ[3] 1 VA
(A.23)
2M1I M2J U i tI[2] tJ[2] Ui 2M2I M1J V A tI[3] tJ[3] VA .
In the abelian case we simply get
2
1 n
2 |U1 |2 + |U2 |2 + |U3 |2 3 |V1 |2 + |V1 |2
D 2 -terms =
2
2 o
+ 2 |U1 |2 + |U2 |2 + |U3 |2 3 |V1 |2 + |V2 |2
,
M 2 -terms = 4 |U1 |2 + |U2 |2 + |U3 |2 + 9 |V1 |2 + |V2 |2 (M1 M2 )2 .
(A.24)
(A.25)
Once again from Eqs. (A.24) and (A.25) we see that the Higgs and Coulomb branches are
separated. Furthermore, in Eq. (A.24) we recognize the toric description of M 1,1,1 as the
manifold of gauge-theory vacua in the Higgs branch (see Eq. (3.5)).
As before, from Eqs. (A.13), (A.14) we can retrieve the vacua describing N separated
branes. In this case the color index structure is more involved and we must set
= u
Ui|
i ,
i = vA
.
(A.26)
VA|
A little work shows that the potential (A.13) vanishes if each of the N -doublets u
i , vA
1,1,1
separately satisfies the D-term equations yielding the toric description of a M
manifold
603
(3.5). Similarly, for each abelian generator belonging to the Cartan subalgebra of Ui (N)
(B.1)
is the complete flag variety of lines inside planes in C3 . A realization of this variety is
given by parametrizing separately the lines and the planes by P2 P2 and then imposing
the incidence relation
k k zk = 0,
(B.2)
where zi and i are homogeneous coordinates on P2 and P2 . Notice that this relation is
the singleton in the tensor product C3 C3 .
The generator of the fibre is h2 , so
exp(i h1 ) eH = exp(2i h2 )H,
exp(i h2 ) eH = exp(i h2 )H,
showing that N 0,1,0 is the circle bundle inside O(1, 1) over the flag variety F(1, 2; 3).
This time dim H 0 (Ma , L) = 8 and the embedding space is P7 ; the ideal of the image is
generated by 36 27 = 9 equations.
We now list the cohomology groups. Since F(1, 2; 3) is a P1 -bundle over P2 , we can
again apply the Gysin sequence to this S 2 fibration to compute its cohomology. This turns
out to be Z[1 , 2 ]/h13 , 22 i; the Chern class of L is c1 = 1 + 2 . We can now apply the
Gysin sequence to the Sasakian fibration, getting
H 1 (N 0,1,0 , Z) = H 6 (N 0,1,0 , Z) = 0,
H 3 (N 0,1,0 , Z) = H 4 (N 0,1,0 , Z) = 0,
H 2 (N 0,1,0 , Z) = Z 1 ,
H 5 (N 0,1,0 , Z) = Z ,
where = 12 1 2 , and the pullbacks are left implicit.
(B.3)
604
i
0
0
e
0 U
(B.5)
0 ,
i
0 0 e
where U is in SU(2). As such it stabilizes a line in a 3-plane in C4 . If we look at the
fibration p : F(1, 3; 4) P3 given by forgetting the second element of the flags V1 V3 ,
605
we see that the map V1 7 V1 Ker M(V1 , ) is a section of p which is Sp(2, H) invariant.
Ma is the image of this section and hence
Sp(2, H)/U (1) SU(2) ' P3 .
(B.6)
It is clear that the Sasakian fibration is M 7 = Sp(2, H)/SU(2) ' S 7 and obviously
S 7 /U (1) = P3 . Notice that S 7 /SU(2) = P1 (H) = S 4 and we have a commutative diagram
S7
id
U (1)
P3
S7
SU(2)
S2
(B.7)
S4
where the action of Sp(2, H) on P3 preserves the fibration given by the bottom line. If
we forget about this fibration, P3 can be considered a homogeneous space of SU(4) and
again the Sasakian fibration over it is S 7 . There are two more ways of getting S 7 as a
homogeneous space (namely SO(8)/SO(7) and SO(7)/G2 ), but the action of the group
does not preserve the U (1) fibration over P3 .
606
links. Thus, we have the action of the full symmetric group of the letters belonging to every
connected component of the graph.
If there are two disconnected components, we can permute a letter in one component
with a letter in the other as follows. First permute the nodes to which they belong by the
action of 2k ; then use the symmetric group of each component to put the extra two letters
of each involved triple at the endpoints of a link. Next exchange the couple of links got in
this way by an action of 3k . Finally use again the symmetric group of each component
to restore the sequence of letters we started from, except for the two which have been
exchanged.
Summing up this proves that, if we consider combinations of the U s and V s completely
symmetric on flavor indices, the structure of the saturation of the color indices is unique:
the totally symmetric one saturated with its dual.
References
[1] J. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor.
Math. Phys. 2 (1998) 231; hep-th/9711200.
[2] S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from non-critical string
theory, Phys. Lett. B 428 (1998) 105; hep-th/9802109.
[3] E. Witten, Anti-de-Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253; hepth/9802150.
[4] I. Klebanov, E. Witten, Superconformal field theory on three-branes at a CalabiYau singularity,
Nucl. Phys. B 536 (1998) 199; hep-th/9807080.
[5] S.S. Gubser, Einstein manifolds and conformal field theories, Phys. Rev. D 59 (1999) 025006;
hep-th/9807164.
[6] S.S. Gubser, I. Klebanov, Baryons and domain walls in an N = 1 superconformal gauge theory,
Phys. Rev. D 58 (1998) 125025; hep-th/9808075.
[7] A. Ceresole, G. DallAgata, R. DAuria, S. Ferrara, Spectrum of type IIB supergravity on
AdS5 T 11 : Predictions on N = 1 SCFTs, hep-th/9905226.
[8] S. Ferrara, A. Kehagias, H. Partouche, A. Zaffaroni, Membranes and fivebranes with lower
supersymmetry and their AdS supergravity duals, Phys. Lett. B 431 (1998) 42; hep-th/9803109.
[9] J. Gomis, Anti-de-Sitter geometry and strongly coupled gauge theories, Phys. Lett. B 435 (1998)
299; hep-th/9803119.
[10] M.R. Douglas, G. Moore, D-branes, quivers, and ALE instantons, hep-th/9603167.
[11] J.M. Figueroa-OFarrill, Near-horizon geometries of supersymmetric branes, hep-th/9807149.
[12] B.S. Acharya, J.M. Figueroa-OFarrill, C.M. Hull, B. Spence, Branes at conical singularities
and holography, Adv. Theor. Math. Phys. 2 (1999) 1249; hep-th/9808014.
[13] J.M. Figueroa-OFarrill, On the supersymmetries of anti-de-Sitter vacua, Class. Quant. Grav. 16
(1999) 2043; hep-th/9902066.
[14] D.R. Morrison, M.R. Plesser, Non-spherical horizons I, hep-th/9810201.
[15] P.G.O. Freund, M.A. Rubin, Dynamics of dimensional reduction, Phys. Lett. B 97 (1980) 233.
[16] M.J. Duff, B.E.W. Nilsson, C.N. Pope, KaluzaKlein supergravity, Phys. Rep. 130 (1986) 1.
[17] M.J. Duff, C.N. Pope, KaluzaKlein supergravity and the seven-sphere, ICTP/82/83-7, in:
Trieste Workshop, September School on Supergravity and Supersymmetry, Trieste, Italy,
September 618, 1982, QC178:T7:1982.
[18] M.A. Awada, M.J. Duff, C.N. Pope, N = 8 supergravity breaks down to N = 1, Phys. Rev.
Lett. 50 (1983) 294.
607
608
[46] K. Oh, R. Tatar, Three-dimensional SCFT from M2-branes at conifold singularities, JHEP 9902
(1999) 025; hep-th/9810244.
[47] G. DallAgata, N = 2 conformal field theories from M2-branes at conifold singularities, hepth/9904198.
[48] N. Itzhaki, J.M. Maldacena, J. Sonnenschein, S. Yankielowicz, Supergravity and the large N
limit of theories with sixteen supercharges, Phys. Rev. D 58 (1998) 046004; hep-th/9802042.
[49] M. Porrati, A. Zaffaroni, M-theory origin of mirror symmetry in three dimensional gauge
theories, Nucl. Phys. B 490 (1997) 107; hep-th/9611201.
[50] C. Ahn, K. Oh, R. Tatar, Branes, orbifolds and the three dimensional N = 2 SCFT in the large
N limit, JHEP 9811 (1998) 024; hep-th/9806041.
[51] A. Kehagias, New type IIB vacua and their F-theory interpretation, Phys. Lett. B 435 (1998)
337; hep-th/9805131.
[52] L. Castellani, A. Ceresole, R. DAuria, S. Ferrara, P. Fr, M. Trigiante, G/H M-branes and
AdSp+2 geometries, Nucl. Phys. B 527 (1998) 142; hep-th/9803039.
[53] G. DallAgata, D. Fabbri, C. Fraser, P. Fr, P. Termonia, M. Trigiante, The Osp(8/4) singleton
action from the supermembrane, Nucl. Phys. B 542 (1999) 157; hep-th/9807115.
[54] S. Gubser, N. Nekrasov, S.L. Shatashvili, Generalized conifolds and 4d N = 1 SCFT,
JHEP 9905 (1999) 003; hep-th/9811230.
[55] G.W. Gibbons, C.N. Pope, CP2 as a gravitational instanton, Commun. Math. Phys. 61 (1978)
239.
[56] R. DAuria, P. Fr, Geometric d = 11 supergravity and its hidden supergroup, Nucl. Phys. B 201
(1982) 101.
[57] O. Aharony, E. Witten, Anti-de-Sitter space and the center of the gauge group, JHEP 9811
(1998); hep-th/9807205.
[58] L. Castellani, A. Ceresole, R. DAuria, S. Ferrara, P. Fre, E. Maina, The complete N = 3 matter
coupled supergravity, Nucl. Phys. B 268 (1986) 317.
[59] L. Castellani, R. DAuria, P. Fr, Supergravity and Superstring Theory: A Geometric
Perspective, World Scientific, Singapore, 1990.
[60] S.P. de Alwis, Coupling of branes and normalization of effective actions in string/M-theory,
Phys. Rev. D 56 (1997) 7963; hep-th/9705139.
[61] C. Ahn, H. Kim, B. Lee, H.S. Yang, N = 8 SCFT and M-theory on AdS4 RP7 , hepth/9811010.
[62] R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, Springer-Verlag, 1982.
[63] I.R. Klebanov, E. Witten, AdS/CFT correspondence and symmetry breaking, hep-th/9905104.
[64] P. Boyer, K. Galicki, 3-Sasakian manifolds (see in particular Corollary 3.1.3; Remark 3.1.4),
hep-th/9810250.
[65] K. Becker, M. Becker, A. Strominger, Fivebranes, membranes and non-perturbative string
theory, Nucl. Phys. B 456 (1995) 130; hep-th/9507158.
[66] W. Fulton, J. Harris, Representation Theory: A First Course, Springer-Verlag, 1991.
[67] G. Lancaster, J. Towber, Representation-functors and flag-algebras for the classical groups I, II,
J. Algebra 59 (1979) 1638; J. Algebra 94 (1985) 265316.
[68] A. Borel, F. Hirzebruch, Characteristic classes and homogeneous spaces I, II, III, Am. J.
Math. 80 (1958) 458538; Am. J. Math. 81 (1959) 315382; Am. J. Math. 82 (1960) 491504.
Abstract
Using the collective field theory approach of large-N generalized two-dimensional YangMills
theory on cylinder, it is shown that the classical equation of motion of collective field is a generalized
Hopf equation. Then, using the ItzyksonZuber integral at the large-N limit, it is found that the
classical Young tableau density, which satisfies the saddle-point equation and determines the largeN limit of free energy, is the inverse of the solution of this generalized Hopf equation, at a certain
point. 2000 Elsevier Science B.V. All rights reserved.
PACS: 11.10.Kk; 11.15.Pg; 11.15.Tk
Keywords: Large-N , YangMills theory; Generalized Hopf equation; Density function
1. Introduction
The 2D YangMills theory (YM2 ) is a theoretical tool for understanding one of the
most important theories of particle physics, i.e., QCD4 . It is known that the YM2 theory
is a solvable model, and in the recent years there have been much efforts to analyze the
different aspects of this theory. The lattice formulation of YM2 has been known for a long
time [1], and many of the physical quantities of this model, e.g., the partition function and
the expectation values of the Wilson loops, have been calculated in this context [2,3]. The
continuum (path integral) approach of YM2 has also been studied in [4] and, using this
approaches, besides the above mentioned quantities, the Green functions of field strengths
have also been calculated [5].
It is known that the YM2 theory is defined by the Lagrangian tr(F 2 ) on a Riemann
surface. In an equivalent formulation, one can use i tr(BF ) + tr(B 2 ) as the Lagrangian of
1 mamwad@theory.ipm.ac.ir
2 alimohmd@theory.ipm.ac.ir
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 1 1 - 5
610
this model, where B is an auxiliary pseudo-scalar field in the adjoint representation of the
gauge group. Path integration over the field B leaves an effective Lagrangian of the form
tr(F 2 ).
Now the YM2 theory is essentially characterized by two important properties: invariance
under area-preserving diffeomorphisms and the lack of propagating degrees of freedom.
These properties are not unique to the i tr(BF ) + tr(B 2 ) Lagrangian, but rather are shared
by a wide class of theories, called the generalized 2D YangMills theories (gYM2 s). These
theories are defined by replacing the tr(B 2 ) term by an arbitrary class function f (B) [6].
Several properties of gYM2 theories have been studied in recent years, for example the
partition function [7,8], and the Green functions on arbitrary Riemann surface [9].
One of the important features of YM2 , and also gYM2 s, is its behaviour in the case of
large gauge groups, i.e., the large-N behaviour of SU (N) (or U (N)) gauge theories. Study
of the large-N limit of these theories is motivated on one hand by an attempt to find a string
representation of QCD in four dimension [10]. It was shown that the coefficients of 1/N
expansion of the partition function of SU (N) gauge theories are determined by a sum over
maps from a two-dimensional surface onto the two-dimensional target space. These kinds
of calculations have been done in [11] and [12] for YM2 and in [8] for gYM2 .
On the other hand, the study of the large-N limits is useful in exploring more general
properties of large-N QCD. To do this, one must calculate, for example, the large-N
behaviour of the free energy of these theories. This is done by replacing the sum over
irreducible representations of SU (N) (or U (N)), appearing in the expressions of partition
function, by a path integral over continuous Young tableaus, and calculating the areadependence of the free energy from the saddle-point configuration. In [13], the logarithmic
behaviour of the free energy of U (N) YM2 on a sphere with area A < Ac = 2 has been
obtained, and in [14] the authors have considered the case A > Ac and proved the existence
of a third-order phase transition in YM2 . A fact that has been known earlier in the context
of lattice formulation [15]. In the case of gYM2 models, the same transition has been shown
for f (B) = tr(B 4 ) in [16] and for f (B) = tr(B 6 ) and f (B) = tr(B 2 ) + g tr(B 4 ) in [17],
all on the sphere.
Such kinds of investigations are much more involved in the cases of surfaces with boundaries. This is because in these cases, the characters of the group elements, which specify
the boundary conditions, enter in the expressions of the partition functions and this makes
the saddle-point equations too complicated. In [18] (see also [21]), the authors considered the YM2 theory on cylinder and investigated its large-N behaviour. If we denote
the two circles forming the boundaries of the cylinder by C1 and C2H, then the boundary conditions
by fixed holonomy matrices UC1 = P exp C1 A (x)dx and
H are specified
UC2 = P exp C2 A (x)dx . In the large-N limit, in which the eigenvalues of these matrices become continuous, the eigenvalue densities of UC1 and UC2 are denoted by 1 ( )
and 2 ( ), respectively, where [0, 2]. Then it was shown that the free energy of YM2
on cylinder, minus some known functions, satisfies a HamiltonJacobi equation with a
Hamiltonian describing a fluid of a certain negative pressure [18]. The time coordinate
of this system is the area of the cylinder between one end and a loop (0 6 t 6 A), and
its position coordinate is , and there are two boundary conditions ( )|t =0 = 1 ( ) and
611
( )|t =A = 2 ( ). It is found that the classical equation of motion of this fluid is the Hopf
(or Burgers) equation. Further, it was shown that the Young tableau density c , satisfying
the saddle-point equation (and therefore specifying the representation which has the dominant contribution in the partition function at large-N ), satisfies c (0 ( )) = . 0 ( )
is (, t) at a time (area) t at which the fluid is at rest. When UC1 = UC2 , 0 ( ) is the
solution of Hopf equation at t = A/2. In the case of a disc, 2 ( ) = ( ), the authors have
calculated the critical area Ac by using the results of the ItzyksonZuber integral [19] at
large-N limit.
Studying the same problem for gYM2 has begun in [20], in the context of master
field formalism. In this paper we study this problem, gYM2 on cylinder, using the above
described technique. The plan of the paper is as following. In Section 2, by calculating
the classical HamiltonJacobi equation, we obtain the corresponding Hamiltonian for the
eigenvalue density for almost general f (B) and find the classical equations of motion. It
is found that these equations are the generalized Hopf equation. In Section 3, we show that
the Young tableau density c is the inverse function of the solution of the generalized Hopf
equation at some certain time (area).
(1)
where U1 and U2 are Wilson loops corresponding to the boundaries of the cylinder, the
summation is over all irreducible representations of the gauge group, R is the trace
of the representation R of the group element, and C is a certain Casimir of the group,
characterizing the particular gYM2 theory we are working with. For the gauge group U(N ),
the group we are working with, the representation R is labeled by N integers l1 to lN ,
satisfying
li < lj ,
i < j.
(2)
ei1
det{eilj k }
,
J {eik }
to
eiN .
The
(3)
where
Y i
e j eik .
J eik =
(4)
j <k
N
X
i=1
c(li ),
(5)
612
( ) :=
N
1 X
( j ),
N
(7)
N
1 X
(y yj ).
N
(8)
j =1
(y) :=
j =1
Inserting (3), (4), and (5) in (1), using (6), and making some obvious redefinition of the
function c, one arrives at
Z=K
X det{eiNyj k1 } det{eiNyj k2 }
D{k1 }D{k2 }
eNA
k g(yk )
(9)
(10)
j <k
(11)
X det{eNyj k1 } det{eNyj k2 }
D{k1 }D{k2 }
where
D{k } :=
sinh
j <k
eNA
j k
.
2
k g(yk )
(12)
(13)
2F
(14)
F
= 1
g
(D 1 Z),
A D Z N
Nk
(15)
D1
D{k1 }.
is
The function g is assumed to be a polynomial. So, to calculate the
where
right-hand side of (15), lets first calculate it for a monomial. We have
n
1 X
(D 1 Z)
ND 1 Z
Nk
k
m
(nm)
1 X n
1
D
Z
=
Nk
Nk
m
ND 1 Z
k,m
m
1
1
F nm
1 X n
1
D
+
O
N
.
=
N
k
m D 1 Nk
N2
613
(16)
k,m
In the large-N limit, one can of course omit the O(1/N 2 ) term (which contains higher
derivatives of F ). In this limit, one must also note the limiting behaviours
Z
1 X
bk d ( )b( ),
(17)
N
k
and
N
b
b
.
k
( ) =k
(18)
(20)
Here, the subscript s denotes that part of expression which contains only the derivatives
of Dk , not Dk itself. The first equality simply comes from (D 1 )1 N 1 (/k )D 1 =
N 1 (/k ) + Dk . To obtain the second equality, one may consider a term with l factors
of Dk . There are m!/[l!(m l)!] ways to choose l factors of Dk from m factors persent.
The other Dk s, either are differentiated or are not present.
It may seem that this s part vanishes at the large-N limit, since it contains factors of 1/N .
This is, however, not the case, since there are singular terms in Dk at j k. To calculate
the non-vanishing part of this expression, let us define a generating function:
m
X um 1
u N1 +Dk
k
+ Dk
=e
.
(21)
qk (u) :=
m! N k
m
This is easily seen to be
k j +u/N
Y sinh
2
1 Nu 1 D 1 k + Nu
.
=
qk (u) = 1 e k D =
D
D 1 (k )
sinh k 2 j
j 6=k
(22)
614
The s-part of this expression is contained in terms with small values for k j . To obtain
this, we use
kj
,
(23)
k j
N (k )
let j run from to (but j 6= k), and keep only the leading terms in k j . It
is easily seen that if one uses this prescription for Dk itself, Dk vanishes. So, using the
above-mentioned prescription in (22) gives exactly qks (u), the s-part of qk (u). That is,
"
#
(
)
u
Y Nkj
Y
u (k ) 2
(k ) + N
=
1
qks (u) =
kj
kj
j 6=k
j <k
N (k )
(
)
Y
u (k ) 2
1
.
(24)
=
n
n=1
So,
sin[u (k )]
.
u (k )
Having found this, we return to (20) and arrive at
1 1 m 1 X m
D
=
Dkml al [ (k )]l ,
D 1 N k
l
qks (u) =
(25)
(26)
l=0
l=0
X al
X 1 cos(l/2)
sin x
=:
xl =
xl.
x
l!
l! l + 1
Inserting (26) in (16), one obtains
1 X 1 n 1
(D Z)
ND 1 Z
N k
k
F nm
1 X n m
l ml
al [ (k )] Dk
N
=
N
k
m
l
k,m,l
X
1 X
n
nl
F nm
=
al [ (k )]l
Dkml N
N
k
l m ml
k,l
nl
1 X
n
F
=
al [ (k )]l
+ Dk
N
N
k
l
k,l
nl
Z
X
S
n
l
=
al
.
d ( )[ ( )]
( )
l
(27)
(28)
(29)
S X n
=
,
al gn d ( )[ ( )]l
A
( )
l
615
(30)
n,l
l
l,n
1
d G i ( ) +
G i ( ) +
,
H=
2i
where G is an integral of g:
dG
= g(x).
dx
e:
From (32), one can obtain the equations of motion for and
e
e
1
H
g i ( ) +
g i ( ) +
,
=
=
e
2i
(32)
(33)
(34)
and
e
e
i
e
=
g i ( ) +
+ g i ( ) +
.
(35)
=
2i
Defining
e
,
(36)
v :=
as a velocity field, in correspondence with what defined in [18,19], one can combine (34)
and (35) into a generalized Hopf equation:
(37)
v i = [g(v i )].
In the case of YM2 , where g(yk ) = 12 yk2 , this equation reduces to Hopf equation found in
[18]. In this case, when one of the boundaries shrinks, so that one has a disc instead of a
cylinder, that is 2 ( ) = ( ), the ItzyksonZuber integral can be used to obtain a solution
for the Hopf equation and from that the critical area of the disc has been obtained [18]. In
our problem, gYM2 , we do not know such an integral representation.
2 [, ]
(38)
616
Z
dy y 2 (y) + B[ ].
=
(x) ,
dx (x)
t
2
x
3
(39)
(40)
(41)
(x, t = 1) = (x).
(42)
and
= 0,
t
x
(43)
:= V + i
(44)
where
and V = y
.
Using (39) in (1) and (12), we see that in the large-N limit, the dominant representation
satisfies the following saddle-point equation
X i
= Ag 0 (y),
(45)
y (y)
i
or
X
X i
= Ag 0 (y)
y,
y (y)
i
(46)
g(f) = 0,
f +
where
f := v i ,
one can write an implicit solution to (47) as
f(, b) = f (b a)g 0 [f(, b)], a ,
(47)
(48)
(49)
where a and b are two particular values of the time variable (here, actually the area). The
same thing can be done to solve (43). In fact, one can define two functions H + and H as
H + (x) := x (t T )(x, T ),
(50)
617
and
H (x) := x + (t T )(x, t),
(51)
and see that is a solution to (43) if these two functions are inverses of each other, i.e.,
H [H + (x)] = x [19].
As an ansatz for H (with t = 1 and T = 0), we take
Hi := Ai g 0 + i,
(52)
Hi+ := vi i i = fi
(53)
and
where
f(x) = f0 x Ai g 0 [fi (x)] .
(54)
Here Ai is the area between a curve for which f is f0 and the ith boundary. The meaning
of (52) and (53) is that we are seeking a solution to (43) with boundary conditions
Vi (x, 1) = Ai g 0 (x) x,
(55)
(x, 0) = (x).
(56)
and
(57)
(58)
then we have
Hi [Hi+ (x)] = x.
(59)
(60)
(61)
where
f := v i.
(62)
The argument of in (61) is real if v0 is zero. So, if there exits a loop on the surface, for
which the velocity field is zero, then there exists a dominant representation , satisfying
[0 ( )] = .
(63)
This is the same as that obtained in [18]. Note, however, that the equation governing the
evolution of , (37), is different from the corresponding equation in [18].
618
Acknowledgement
M. Alimohammadi would like to thank the Institute for Studies in Theoretical Physics
and Mathematics and also the Research Council of the University of Tehran, for their
partial financial supports.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
A. Migdal, Zh. Eskp. Teor. Fiz. 69 (1975) 810, Sov. Phys. JETP 42 413.
B. Rusakov, Mod. Phys. Lett. A 5 (1990) 693.
E. Witten, Commun. Math. Phys. 141 (1991) 153.
M. Blau, G. Thompson, Int. J. Mod. Phys. A 7 (1992) 3781.
M. Alimohammadi, M. Khorrami, Int. J. Mod. Phys. A 12 (1997) 1959.
E. Witten, J. Geom. Phys. 9 (1992) 303.
M.R. Douglas, K. Li, M. Staudacher, Nucl. Phys. B 240 (1994) 140.
O. Ganor, J. Sonnenschein, S. Yankielowicz, Nucl. Phys. B 434 (1995) 139.
M. Khorrami, M. Alimohammadi, Mod. Phys. Lett. A 12 (1997) 2265.
M.R. Douglas, Conformal field theory technique in large-N YangMills theory, hepth/9311130.
D.J. Gross, Nucl. Phys. B 400 (1993) 161.
D.J. Gross, W. Taylor, Nucl. Phys. B 400 (1993) 181.
B. Rusakov, Phys. Lett. B 303 (1993) 95.
M.R. Douglas, V.A. Kazakov, Phys. Lett. B 319 (1993) 219.
D.J. Gross, E. Witten, Phys. Rev. D 21 (1980) 446.
M. Alimohammadi, M. Khorrami, A. Aghamohammadi, Nucl. Phys. B 510 (1998) 313.
M. Alimohammadi, A. Tofighi, Eur. Phys. J. C 8 (1999) 711.
D.J. Gross, A. Matytsin, Nucl. Phys. B 437 (1995) 541.
A. Matytsin, Nucl. Phys. B 411 (1994) 805.
R. Gopakumar, Nucl. Phys. B 471 (1996) 246.
M. Caselle, A. DAdda, L. Magnea, S. Panzeri, Two-dimensional QCD on the sphere and on
the cylinder, hep-th/9309107.
Abstract
We introduce a class of cellular automata associated with crystals of irreducible finite dimensional representations of quantum affine algebras Uq0 (gn ). They have solitons labeled by crystals of the smaller algebra Uq0 (gn1 ). We prove stable propagation of one soliton for g n =
(2)
(2)
(1)
(1)
(1)
(2)
(1)
A2n1 , A2n , Bn , Cn , Dn and Dn+1 . For g n = Cn , we also prove that the scattering matrices
(1)
of two solitons coincide with the combinatorial R matrices of Uq0 (Cn1 )-crystals. 2000 Elsevier
Science B.V. All rights reserved.
PACS: 02.10.Eb; 02.20.Sv; 05.45.-a; 05.50.+q
Keywords: Quantum affine algebras; Crystal bases; Integrable systems
1. Introduction
Cellular automata are the dynamical systems in which the dependent variables assigned
to a space lattice take discrete values and evolve under a certain rule. They exhibit rich
behavior, which have been widely investigated in physics, chemistry, biology and computer
sciences [31]. When the space lattice is one-dimensional, there are several examples known
as the soliton cellular automata [5,2325,27,28]. They possess analogous features to the
solitons in integrable non-linear partial differential equations. For example, some patterns
propagate with fixed velocity and they undergo collisions retaining their identity and only
changing their phases.
There is a notable progress recently in understanding the integrable structure in the
soliton cellular automata. In the papers [20,29,30] it was shown that a class of soliton
cellular automata can be derived from the known soliton equations such as LotkaVolterra
and Toda equations through a limiting procedure called ultra-discretization. The method
enables one to construct the explicit solutions and the conserved quantities of the former
from that of the latter.
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 0 5 - X
620
+0
+0
In a sense they change + into max and into +. This is a transformation of the continuous
operations into piecewise linear ones preserving the distributive law:
(A + B) C = (A C) + (B C) max(a, b) + c = max(a + c, b + c).
The non-uniqueness of the distributive structure is noted by Schtzenberger in combinatorics, where the procedure corresponding to the inverse of the ultra-discretization is called
tropical variable change [17].
There is yet further intriguing aspect in the soliton cellular automata (called box and
ball systems) in [28,30]. There the scattering of two solitons is described by the rule
which turns out to be identical with the Uq0 (A(1)
n ) combinatorial R matrix [22] from the
crystal base theory. The latter has an origin in the quantum affine algebras at q = 0, where
the representation theory is piecewise linear in a certain sense.
Motivated by these observations we formulate in this paper and [7] a class of cellular
automata directly in terms of crystals and link the subject to the 1 + 1 dimensional
quantum integrable systems. The theory of crystals is invented by Kashiwara [12] as a
representation theory of the quantized KacMoody algebras at q = 0. It is a powerful
tool that reduces many essential problems into combinatorial questions on the associated
crystals. Irreducible decomposition of tensor products and the RobinsonSchenstedKnuth
correspondence are typical such problems [4,16,21]. By connecting the classical and affine
crystals, it also explains [14,15] the appearance of the affine Lie algebra characters [3] in
Baxters corner transfer matrix method in solvable lattice models [1].
Here we shall introduce a cellular automaton associated with crystals of irreducible
finite dimensional representations of quantum affine algebras Uq0 (gn ). The basic idea is
to regard the time evolution in the automaton as the action of a row-to-row transfer matrix
of integrable Uq0 (gn ) vertex models at q = 0. The essential point is to consider the tensor
product of crystals not around the anti-ferromagnetic vacuum as in [14,15], but rather in
the vicinity of the ferromagnetic vacuum.
Let B be a classical crystal of irreducible finite dimensional representations of the
quantum affine algebra Uq0 (gn ). It is a finite set having a weight decomposition and
equipped with the maps ei , fi : B B t {0} and i , i : B Z>0 (i {0, 1, . . . , n})
satisfying certain axioms (cf. Definition 2.1 in [13]). For two crystals B and B 0 the tensor
products B 0 B and B B 0 are again crystals which are canonically isomorphic. The
u \ 1 ' 1 u\ ,
1 More precisely, the isomorphism combined with the data on the energy function is called the combinatorial
R matrix [14].
(II)
621
z }| {
u 1 1 ' b 1 b k u\ ,
bi B.
We take dynamical variables of our automaton from the crystal B and regard their array
t
, b0t , b1t , . . . at time t as an element of the tensor product of crystals
. . . , b1
t
b0t b1t B B B ,
b1
where we assume the boundary condition bjt = 1 B for |j | 1. See Section 2.2 for a
precise treatment. The ferromagnetic state bit = 1 is understood as the vacuum of the
automaton. The infinite tensor product with such a boundary condition does not admit a
crystal structure. Nevertheless one can make sense of the construction below thanks to the
properties (I) and (II). The time evolution is induced by sending u\ from left to right via
the repeated application of the combinatorial R matrix as
B 0 ( B B B ) ' ( B B B ) B 0 ,
t +1
t
b0t b1t ' b1
b0t +1 b1t +1 u\ ,
u\ b1
which is well-defined as long as the above properties and the boundary conditions are
fulfilled. In the language of the quantum inverse scattering method [18,26], this is the action
of the q = 0 row-to-row transfer matrix whose auxiliary and quantum spaces are labeled by
B 0 and B B , respectively. Note that the transfer matrix has effectively reduced
to the (u\ , u\ )-component of the monodromy matrix since its action is considered under
the ferromagnetic boundary condition. The fundamental case g n = A(1)
n will be studied in
a more general setting in [7]. In this paper we concentrate on the other non-exceptional
series
(2)
(2)
(2)
g n = A2n1 , A2n , Bn(1) , Cn(1) , Dn(1) , Dn+1 ,
B 0 = B\ 3 u\ .
Here B1 is the crystal associated with the vector representation of the classical subalgebra
(2)
(2)
of g n except for A2n and Dn+1 . Their cardinalities are ]B = 2n, 2n + 1, 2n + 1, 2n, 2n and
2n + 2, respectively. The element 1 B1 is the highest weight one. 2 To explain B\ and u\ ,
recall the coherent family {Bl | l Z>1 } of the perfect crystals obtained in [13]. It contains
the B1 as its first member. 3 The Bl with higher l corresponds to an l-fold symmetric
fusion of B1 . Then B\ in question is an infinite set corresponding to a certain l limit
of Bl and u\ is its highest weight element. 4 We shall call the resulting dynamical system
Uq0 (gn ) automaton. They are essentially solvable trigonometric vertex models at q = 0 in
the vicinity of the ferromagnetic vacuum. A peculiarity here is the extreme anisotropy with
2 For (g , B ) treated in this paper, a parallel construction seems possible also with the choice 1 = lowest
n 1
weight element.
3 For C (1) , the family in [13] does not contain B . See [8].
n
1
4 They are different from the limits B and b in [13].
622
respect to the relevant fusion degrees; B1 is the simplest one, while B 0 = B\ corresponds
to an infinite fusion. 5
Once the automata are constructed the first question will be if they are solitonic. We
prove a theorem that:
The Uq0 (gn ) automaton has the patterns labeled by the crystals {Bl } of the algebra
Uq0 (gn1 ) that propagate stably with velocity l.
Computer experiments indicate that they indeed behave like solitons. For instance, the
initially separated patterns labeled by the Uq0 (gn1 )-crystal elements b Bl and c Bk
(l > k) undergo a scattering into two patterns labeled again by some c0 Bk and b0 Bl .
Let S : Bl Bk Bk Bl be the two-body scattering matrix of such collisions, namely,
S(b c) = c0 b0 . Let R : Bl Bk Bk Bl denote the combinatorial R matrix of
Uq0 (gn1 ). Then we prove
S =R
(1)
for g n = Cn and conjecture it for all the other g n . Similarly the scattering of multi-solitons
labeled by Bl1 , . . . , BlN (l1 > > lN ) is given by the isomorphism Bl1 BlN '
BlN Bl1 experimentally. Thus the solitonic nature is guaranteed by the YangBaxter
equation obeyed by S = R. A precise formulation of these claims is done through an
injection
l
l : Uq0 (gn1 )-crystal Bl Uq0 (gn )-crystal B1 ,
which will be described in Section 3.
Admitting that they are soliton cellular automata, the second question is if there exist
classical integrable equations governing them, possibly via the ultra-discretization. Here
(2)
(1)
we only confirm this for A2 case by relating the associated automaton to the known A1
example [27]. This observation is due to [10].
The layout of the paper is as follows. In Section 2 we first explain our construction of the
0
0 g ) and
Uq (gn ) automata concretely along the g n = A(2)
n
2 example. It is valid for any Uq (
any finite crystals having the properties (I) and (II). In Section 3, we formulate the theorem
(1)
and the conjecture for g n precisely. We sketch a proof of S = R for Cn case. In principle
the idea used in the proof can also be used for the other g n . We will specify B\ as an
infinite set with the actions ei , fi : B\ B\ t {0} but without the maps i , i . In Section 4
concluding remarks are given. Appendix A is devoted to an explanation of what is meant by
B\ B1 ' B1 B\ , which shows up when the infinite set B\ is substituted into the finite
crystal B 0 . This is actually abuse of notation meaning an invertible map R 0 : B\ B1
B1 B\ between the sets. We state a conjecture on a stability of the combinatorial R matrix
Bl B1 ' B1 Bl when l gets large, which ensures the well-definedness of the map R 0 .
It assures that we may regard B\ as a finite crystal Bl with a sufficiently large l to define
our automata.
Our construction here and in [7] is a crystal interpretation of the L-operator approach
(1)
(1)
[11] for a g n = An case. The Uq0 (An ) automaton in this sense coincides with the ones in
[27,28,30]. As in the Cn(1) case in this paper, the properties stated in the above can actually
5 To take B 0 = B with finite l is an interesting generalization. See [7] for A(1) case.
l
n
623
be proved by means of the crystal theory. The detail will appear elsewhere along with the
results on a more general choice of the crystals B and B 0 [7].
2. A(2)
2 example
(2)
Let us explain our automata concretely along the case g n = A2 . This simple example
is helpful to gain the idea for the general g n case treated in the next section.
As a peculiarity in the rank 1 situation, the Uq0 (A(2)
2 ) automaton turns out to be an even
time sector of [27].
(2)
2.1. A2 crystals
For l Z>1 set
Bl = {(x, y) | x, y Z>0 , x + y 6 l}.
(2.1)
f0 (x, y) =
(x, y 1) if x < y,
f1 (x, y) = (x 1, y + 1).
In the above, the right hand sides are to be understood as 0 if they are not in Bl . Setting
i (b) = maxk {eik b 6= 0 | k > 0} and i (b) = maxk {fik b 6= 0 | k > 0}, one has
0 (b) = l x y + 2(x y)+ ,
1 (b) = y,
1 (b) = x,
These results are obtained by extrapolating the A2n result [13] to n = 1. For l = 1 we use
a simpler notation as
B = B1 ,
(2.2)
Given two crystals B and B 0 , one can form another crystal (tensor product) B B 0 [12].
The crystal Bl B1 is connected and so is B1 Bl . Calculating the map Bl B1 B1 Bl
commuting with ei and fi , one has:
624
1 (l, 0)
if (x, y) = (l, 0),
3 (x + 1, y 1) if x + y = l, y > 1,
(x, y) 1 '
2 (x + 1, y)
if x + y = l 1,
1 (x + 1, y + 1) otherwise,
1 (l 1, 0)
(x, y) 2 ' 3 (x, y 1)
2 (x, y)
3 (0, l)
2 (0, l)
1 (0, 1)
(x, y) 3 '
1 (x 2, y)
1 (x, y + 2)
3 (x 1, y 1)
In Section 2.2 we shall use formal l limits of Bl and the combinatorial R matrix
Bl B ' B Bl . In the present case the prescription is to simply shift the coordinate
(x, y) to (x l, y) and to consider
B\ = {(x, y) | x Z60 , y Z>0 , x + y 6 0},
without specifying a crystal structure. The map R 0 : B\ B ' B B\ in the sense of
Appendix A is deduced from Proposition 2.1 by concentrating on those (x, y) in the
vicinity of (0, 0). Thus it reads
1 (0, 0)
3 (x + 1, y 1)
(x, y) 1 '
2 (x + 1, y)
1 (x + 1, y + 1)
1 (1, 0)
(x, y) 2 ' 3 (x, y 1)
2 (x, y)
1 (x 2, y)
(x, y) 3 '
3 (x 1, y 1)
To depict this in a figure we put
b
b0
(2.3)
(2.4)
if y = 0,
otherwise.
(2.5)
R 0 : (x, y) b b0 (x 0 , y 0 ).
We call b and b 0 the upper index and the lower index, respectively. Now (2.3)(2.5) are
summarized in the semi-infinite triangle in Fig. 1.
625
Fig. 1. The semi-infinite triangle representing R 0 . There are 6 different patterns depicted by circles,
squares and diagonal squares which are filled or empty.
z
}|
{ z
}|
{
B\ B B ' B B B\ ,
u bi bj ' bi0 bj0 u0
(2.6)
626
are distinct elements in P. The set P (2.7) is not equipped with a crystal structure.
Nevertheless the properties (I) and (II) enable us to define an invertible map T : P P
that formally corresponds to an L limit of (2.6). To describe it precisely, note that
any element in P has the form
p = 1 1 bi bj 1 1
(i 6 j ),
(2.8)
where bi , bi+1 , . . . , bj B. Owing to the properties (I) and (II) there exists k0 Z>0 such
that
k
z }| {
u\ bi bj 1 1 ' bi0 bj0 bj0 +1 bj0 +k u\
for all k > k0 . Then T (p) P is defined by
T (p) = 1 1 bi0 bj0 bj0 +1 bj0 +k 1 1 ,
which is k-independent as long as k > k0 . The inverse T 1 can be described similarly.
The map T plays the role of the time evolution operator. It is a q = 0 analogue of the
row-to-row transfer matrix of a solvable lattice model in the vicinity of the ferromagnetic
vacuum.
Given p P in (2.8) define um B\ for all m Z by the recursion relation and the
boundary condition
0
um
um1 bm ' bm
u m = u\
for all m Z,
for m 6 i 1.
0
0 u b
bm
Plainly, u\ ( bm1 bm bm+1 ) ' bm1
m
m+1 . Due to
the properties (I) and (II) the sequence um , um+1 , . . . tends to u\ = (0, 0) B\ . In this way
any element p P specifies a trajectory {um }
m= in the semi-infinite triangle (Fig. 1)
that starts at the origin (0, 0) and returns to it finally. This picture is useful in calculating
T (p). Namely, the trajectory is determined by following the arrows with the upper indices
. . . , bm1 , bm , bm+1 , . . . appearing in p = bm1 bm bm+1 . Then T (p) is
0
0 b0
bm
constructed by tracing their lower indices as T (p) = bm1
m+1 .
t
For p = bj bj +1 P and t Z define bj B by T t (p) = bjt
bjt +1 . Then the time evolution of the cellular automaton is displayed with the arrays
0
0
b1
b00 b10 b20 . . .
. . . b2
1
1
b1
b01 b11 b21 . . .
. . . b2
2
2
b1
b02 b12 b22 . . .
. . . b2
627
Example 2.3.
t = 0 : 11333311111111111111111111111111
t = 1 : 11111111113333111111111111111111
t = 2 : 11111111111111111133331111111111
t = 3 : 11111111111111111111111111333311
Example 2.4.
t
t
t
t
t
= 0 : 1133311111123111111111111111111111
= 1 : 1111111133311123111111111111111111
= 2 : 1111111111111133123311111111111111
= 3 : 1111111111111111123111133311111111
= 4 : 1111111111111111111123111111133311
Example 2.5.
t
t
t
t
t
t
t
t
= 0 : 11233111331111111111111211111111111111111111111
= 1 : 11111112331133111111111121111111111111111111111
= 2 : 11111111111133112331111112111111111111111111111
= 3 : 11111111111111113311123311211111111111111111111
= 4 : 11111111111111111111331111223311111111111111111
= 5 : 11111111111111111111111133121111233111111111111
= 6 : 11111111111111111111111111121331111112331111111
= 7 : 11111111111111111111111111112111133111111123311
Example 2.6.
t
t
t
t
t
t
t
t
= 0 : 11233111111133111121111111111111111111111111111
= 1 : 11111112331111113312111111111111111111111111111
= 2 : 11111111111123311112133111111111111111111111111
= 3 : 11111111111111111233211113311111111111111111111
= 4 : 11111111111111111111211233111331111111111111111
= 5 : 11111111111111111111121111112331133111111111111
= 6 : 11111111111111111111112111111111133112331111111
= 7 : 11111111111111111111111211111111111113311123311
The last two show the independence of the order of collisions. These examples suggest that
the following patterns are stable (Q Z>1 , R = 0, 1):
Q
z
}|
{
2 3 3
R = 1,
z
}|
{
3 3 3
R = 0.
The both patterns should not be followed by 3. The former pattern can be preceded by any
element in B while the latter should only be preceded by 1. Q is the size of the soliton and
628
R is the number of occurrences of 2 in its front. They move to the right with the velocity
2Q R when separated sufficiently. These features are consistent with g n = A(2)
2 case of
Theorem 3.1. See also Section 3.2.
In fact the Uq0 (A(2)
2 ) automaton described above can be interpreted [10] as an even time
sector of the automaton in [27]. Replace the array of {1, 2, 3} by that of {1, 2} with double
length via the rule 1 11, 2 12 and 3 22. In the resulting array, play the box and
ball game as in [27]. Namely, we regard the array as a sequence of cells which contains
a ball or not according to the array variable is 2 or 1, respectively. In each time step, we
move each ball once to the nearest right empty box starting from the leftmost ball. Then
(2)
the 2 time steps in the box and ball system yield the 1 time step in our Uq0 (A2 ) automaton.
In terms of crystals, this can be explained as follows. First, the box and ball game in [27]
(1)
is known [7] to be equivalent to the Uq0 (A1 ) automaton. Let
bk = {m1 . . . mk | mi {1, 2}, m1 6 6 mk }
B
denote the Uq0 (A1 )-crystal corresponding to the k-fold symmetric tensor representation.
(We have omitted the frame of the usual semistandard tableaux.) Consider the maps h\ and
h1 defined by
(1)
bk B
bk ,
h\ : B\ B
k2y
(k 1),
k+x+y xy
2y
z }| { z }| { z }| { z }| {
(x, y) 7 1 . . . 1 2 . . . 2 1 . . . 1 2 . . . 2,
b1 B
b1 ,
h1 : B1 B
1 7 1 1,
2 7 1 2,
3 7 2 2.
Then for k large enough, we have the commutative diagram:
B\ B1
h\ h1
bk B
b1 B
b1 ,
bk B
B
h1 h\
b1 B
bk B
bk ,
b1 B
B
R0
B1 B\
where the down arrow in the right column is the crystal isomorphism. This asserts that the
(2)
0
square of the T in the Uq0 (A(1)
1 ) automaton coincides with the T of the Uq (A2 ) automaton.
3. Uq0 (gn ) automaton
3.1. Theorem and conjecture
(2)
(2)
(1)
Let us proceed to the Uq0 (gn ) automata for g n = A2n1 (n > 3), A2n (n > 1), Bn
(2)
(n > 3), Cn(1) (n > 2), Dn(1) (n > 4) and Dn+1
(n > 2). Our aim here is to formulate
the theorem and the conjecture stated in Section 1 precisely. In principle construction of
the automata is the same as the A(2)
2 case explained in Section 2.2. The time evolution
629
Theorem 3.2. Let g n = Cn (n > 3). For fixed positive integers l > k, let Bl Bk 3
b1 b2 ' c2 c1 Bk Bl be an isomorphism under the combinatorial R matrix of
the Uq0 (gn1 )-crystals. Then for m l, there exists t0 > 0 such that for any t > t0 and
some m0
7 For C (1) we have a concrete description of the combinatorial R matrix B B ' B B in terms of an
l
k
k
l
n
630
T t : 1 1 l (b1 ) 1
k (b2 ) 1 1 1
7 1 1 1 k (c2 ) 1
m0
l (c1 ) 1 1 .
In other words, we have the combinatorial R matrix as the scattering matrix of the ultradiscrete solitons. Compared with l (b1 ), k (c2 ) is shifted to the right, but we do not concern
the precise distance. A sketch of a proof of Theorem 3.2 will be given in Section 3.4.
In fact we have a conjecture on N -soliton case for general g n .
Conjecture 3.3. Let N Z>2 . Fix positive integers k1 , . . . , kN (k1 > > kN ) and the
elements bi Bki (i = 1, . . . , N) of the Uq0 (gn1 )-crystals. Define ci Bki by b1
bN ' cN c1 under the isomorphism Bk1 Bk2 BkN ' BkN Bk2 Bk1 .
For m1 , . . . , mN1 k1 , there exists t0 > 0 such that for any t > t0 ,
T t : 1 k1 (b1 ) 1
m1
7 1 kN (cN ) 1
1
m01
mN1
kN (bN ) 1
m0N1
k1 (c1 ) 1 ,
= 0 : 112 23211111
33111111111111111111111
= 1 : 1111112232111331111111111111111111
= 2 : 111111111122321
3311111111111111111
= 3 : 111111111111112333 3311111111111111
= 4 : 1111111111111111133123 331111111111
= 5 : 11111111111111111113311123 33111111
= 6 : 111111111111111111111331111123 3311
631
Example 3.5.
t = 0 : 111 2 2311111
2 32111111111111111111111111
t = 1 : 11111111223111232111111111111111111111
t = 2 : 11111111111112311
2 2 3211111111111111111
t = 3 : 111111111111111113111
2 2 2 32111111111111
t = 4 : 11111111111111111111311111
2 2 2 321111111
t = 5 : 1111111111111111111111131111111
2 2 2 3211
t = 0 : 111 3 33311111
1 2311111
2 3111111111111111111111111111111
t = 1 : 1111111113333111123111231111111111111111111111111111
2 311111111111111111111111111
t = 2 : 1111111111111113 3311
1 233
t = 3 : 111111111111111111113 311331
1 2 2 2211111111111111111111
3 333111
t = 4 : 11111111111111111111111111
1 2 2 2211111111111111
t = 5 : 1111111111111111111111111111113333111111222211111111
t = 6 : 1111111111111111111111111111111111
3 3331111111
12 2 2211
ei ,fi
gl
632
)
n
X
2n
Bl = (x1 , . . . , xn , xn , . . . , x1 ) Z xi , xi > 0,
(xi + xi ) = l ,
i=1
(
B\ = (x1 , . . . , xn , xn , . . . , x1 ) Z2n x1 6 0, xi > 0 (i 6= 1), x i > 0,
)
n
X
(xi + xi ) = 0 .
(
i=1
if xi = 1, others = 0,
i
i
if xi = 1, others = 0.
x 1
x n1
xn1
x1
)
n
X
(xi + xi ) 6 l ,
Bl = (x1 , . . . , xn , xn , . . . , x1 ) Z xi , xi > 0,
i=1
(
2n
B\ = (x1 , . . . , xn , xn , . . . , x1 ) Z x1 6 0, xi > 0 (i 6= 1), x i > 0,
)
n
X
(xi + xi ) 6 0 .
(
2n
i=1
i
(x1 , . . . , xn , xn , . . . , x1 ) = i
if xi = 1, others = 0,
if xi = 1, others = 0,
if xi = 0, x i = 0 for all i.
n1
X
(xi + xi ), s 0 (b) = l s(b) /2 .
i=1
If l s(b) is odd,
l (b) = 1
s 0 (b)
x 1
x n1
xn1
x1
s 0 (b)
otherwise
l (b) = 1
s 0 (b)
x 1
x n1
xn1
x1
s 0 (b)
When n = 1, the above notation for B1 and (2.2) are related by 1 = 1 , 2 = and 3 = 1 .
The solitons and their velocity mentioned in the beginning of Section 2.2 agree with the
633
n = 1 case here. One interprets s(b) = 0 and s 0 (b) = [l/2] for b from Uq0 (A(2)
0 )-crystal Bl .
Then, under the identification R = (1 (1)l )/2 and Q = R + s 0 (b), the velocity is indeed
l = 2Q R.
(1)
g n = Bn :
x = 0 or 1, xi , xi > 0,
,
Bl = (x1 , . . . , xn , x0 , xn , . . . , x1 ) Z2n {0, 1} 0 Pn
x0 + i=1 (xi + xi ) = l
x = 0 or 1, x 6 0,
0
1
.
B\ = (x1 , . . . , xn , x0 , xn , . . . , x1 ) Z2n {0, 1} xi > 0 (i 6= 1), xi > 0,
Pn
x0 + i=1 (xi + xi ) = 0
For B1 we use a simpler notation
(x1 , . . . , xn , x0 , xn , . . . , x1 ) =
if xi = 1, others = 0,
i
i
if xi = 1, others = 0.
l (b) = 2
g n = Cn(1) :
x 1
x n1
x0
Bl = (x1 , . . . , xn , xn , . . . , x1 ) Z
B\ = (x1 , . . . , xn , xn , . . . , x 1 ) Z2n
(
2n
i
i
xn1
x1
xi , xi > 0,
P
l > ni=1 (xi + xi ) l 2Z
)
,
x1 6 0, xi > 0 (i 6= 1), xi > 0,
Pn
.
0 > i=1 (xi + xi ) 2Z
if xi = 1, others = 0,
if xi = 1, others = 0.
(1)
)-crystal Bl , we define
For b = (x1 , . . . , xn1 , xn1 , . . . , x1 ) in Uq0 (Cn1
s(b) =
n1
X
(xi + xi ), s 0 (b) = l s(b) /2,
i=1
l (b) = 1
g n = Dn(1) :
s 0 (b)
x 1
x n1
xn1
x1
)
= 0 or xn = 0, xi , xi > 0,
Pn
,
Bl = (x1 , . . . , xn , xn , . . . , x1 ) Z
i=1 (xi + x i ) = l
x = 0 or x = 0, x 6 0,
n
n
1
.
B\ = (x1 , . . . , xn , xn , . . . , x1 ) Z2n xi > 0 (i 6= 1), xi > 0,
Pn
(x
+
x
)
=
0
i
i=1 i
(
2n xn
s 0 (b)
634
if xi = 1, others = 0,
i
i
(x1 , . . . , xn , xn , . . . , x1 ) =
if xi = 1, others = 0.
(1)
)-crystal Bl ,
For b = (x1 , . . . , xn1 , xn1 , . . . , x1 ) in Uq0 (Dn1
l (b) = 2
x 1
x n1
xn1
x1
(2)
g n = Dn+1 :
(
Bl = (x1 , . . . , xn , x0 , x n , . . . , x1 ) Z
2n
)
0 or 1, xi , xi > 0,
x0 = P
,
{0, 1}
x0 + ni=1 (xi + xi ) 6 l
x = 0 or 1, x 6 0,
0
1
.
B\ = (x1 , . . . , xn , x0 , xn , . . . , x1 ) Z2n {0, 1} xi > 0 (i 6= 1), x i > 0,
P
x0 + ni=1 (xi + xi ) 6 0
For B1 we use a simpler notation
i
B1 3 (x1 , . . . , xn , x0 , xn , . . . , x1 ) = i
if xi = 1, others = 0,
if x i = 1, others = 0,
if xi = 0, x i = 0 for all i.
s(b) =
n1
X
(xi + xi ), s 0 (b) = (l s(b))/2 .
i=1
If l s(b) is odd,
l (b) = 1
n
s 0 (b)
xn1
x 1
n
x1
x n1
s 0 (b)
x0
otherwise
l (b) = 1
s 0 (b)
xn1
x 1
x1
x n1
s 0 (b)
x0
Our proof below uses the crystal structure of Bl given in [8] for g n = Cn and in [13]
for the other types.
635
ei+1
ei
eip +1 ei1 +1
l (b) uL
eip +1 ei1 +1
uL l (b0 ) 1
(L, 0, . . . , 0) 2
l (b0 ) uL
(L, 0, . . . , 0) 2
f
l
e0L f1L+2l
(0, . . . , 0, L) 1
(L, 0, . . . , 0)
e0L f1L+2l
l
2l
(0, l, 0, . . . , 0, L l)
(0, . . . , 0, L) 1
(L, 0, . . . , 0)
f
l
2l
e1L+l f0L+2l
(L, 0, . . . , 0) 1
(0, . . . , 0, l, L l)
e1L+l f0L+2l
2l
2l
(L, 0, . . . , 0).
636
uL l b0(l) 1
(L, 0, . . . , 0) 2
l (b0(l)) uL
(L, 0, . . . , 0) 2
f
l
(L, 0, . . . , 0) 1
(L, 0, . . . , 0)
2l
(L l, l, 0, . . . , 0)
(L, 0, . . . , 0) 1
(L, 0, . . . , 0)
f
l
2l
(L l, 0, . . . , 0, l, 0)
(L, 0, . . . , 0) 1
2l
2l
(L, 0, . . . , 0).
(L, 0, . . . , 0) 2
' 1
(L, 0, . . . , 0).
to 1
(3.1)
(m)
k k 1
2
1
according to Section 3.2.
k 0 = [ lm+1
2 ] so that l (b0 ) =
(2)
0
Since Bl of Uq (A2n2 )-crystal is isomorphic to B(l31 ) B((l 1)31 ) B(0) as
Uq (Cn1 )-crystals, for any b Bl there exists a sequence i1 , . . . , ip {1, . . . , n 1} such
that b0(m) = eip ei1 b for some 0 6 m 6 l. From Lemma 3.7 one has
(m)
uL l (b) 1
l+k 0
eip +1 ei1 +1
l (b) 1
k 0
uL
eip +1 ei1 +1
(m)
uL l b 0
l+k 0
0
0
0
e1L+k f0k+k f1L+k+2k
(L, 0, . . . , 0) 1
637
l (b0(m)) 1
k 0
uL
0
0
0
e1L+k f0k+k f1L+k+2k
k 0
k 0
(L, 0, . . . , 0),
where in the bottom we have used (3.1) and uL 1 ' 1 uL . Thus the isomorphism
l
l
12l BL sends uL l (b) 1
to 1
l (b) uL , proving
BL B12l B
Theorem 3.1.
g n = Bn(1) :
l
l 2l l
l
The proof is similar to the case g n = A(2)
2n1 with f = f2 fn1 fn fn1 f2 .
g n = Cn(1) :
The proof is similar to the case g n = A(2)
2n .
(1)
g n = Dn :
(2)
l
l
f2l .
fnl fn1
The proof is similar to the case g n = A2n1 with f = f2l fn2
(2)
g n = Dn+1 :
(2)
l
l
fn2l fn1
f2l .
The proof is similar to the case g n = A2n with f = f2l fn1
Since the Cn(1) case has the multiplicity, we need Part II, which relies on the explicit result
(1)
on the combinatorial R matrices for Uq0 (Cn ) in [9]. In the rest of Section 3 we shall write
0
0
el for distinction.
Uq (gn )-crystals as Bl and Uq (gn1 )-crystals as B
L1
L01
l (b1 ) 1
k (c2 ) 1
L2
L02
k (b2 ) 1
l (c1 ) 1
L3
L03
ut
L
(3.2)
holds for any t > t0 for some L01 , L02 , L3 and L03 under the isomorphism BLt B1
el B
ek B
ek B
el through this relation
B1 ' B1 B1 BLt . Define S : B
638
ek
el B
B
(B1 )l+k
ei+1
ei
el B
ek ) t {0}
(B
l k
where
aj0
=
t
0
0
= ut
L a 1 a L uL ,
pk0
if j = ik ,
otherwise,
t 0
t
t 0
0
0
2. fi+1 (ut
L a1 aL uL ) = uL a1 aL uL fi+1 (p1
0
0
0
0
pm ) = p1 pm , where pk = aik ,
0
3. fi+1 (ut a1 aL ut ) = 0 fi+1 (p1 pm ) = 0, for each i =
L
Setting t 0 = 0, a1 aL = 1 1 l (b1 ) 1
pm = l (b1 ) k (b2 ) in Lemma 3.10, one has
fi+1 ut
L 1
= ut
L 1
L1
L1
l (b1 ) 1
l (b10 ) 1
L2
L2
L2
k (b2 ) 1
k (b2 ) 1
k (b20 ) 1
L3
and p1
L3
L3
fi+1 1
= 1
L01
L01
k (c2 ) 1
k (c20 ) 1
L02
L02
l (c1 ) 1
l (c10 ) 1
L03
L03
ut
L
639
ut
L
in terms of the (c20 , c10 ) specified above. With the help of Lemma 3.10 and 3.9 we can
now go backwards to see that this implies fi+1 (k (c2 ) l (c1 )) = k (c20 ) l (c10 ) hence
fi (c2 c1 ) = c20 c10 . Thus we have R(b10 b20 ) = R(fi (b1 b2 )) = fi R(b1 b2 ) =
fi (c2 c1 ) = c20 c10 . It is very similar to verify that ei (b1 b2 ) = 0 implies ei S(b1 b2 ) =
0 for any i = 1, . . . , n 1. 2
(1)
The above proof is also valid in An case [7]. Viewed as Uq (gn1 )-crystals with
el B
ek and B
ek B
el decompose
Kashiwara operators fi , ei (1 6 i 6 n 1), both crystals B
into connected components. Note that (3.2) obviously tells that Uq (gn1 )-weights of
b1 b2 and S(b1 b2 ) = c2 c1 are equal. Therefore apart from the separation into two
solitons in the final state, Proposition 3.8 reduces the proof of Theorem 3.2 to showing
R(b1 b2 ) = S(b1 b2 ) only for the Uq (gn1 )-highest weight elements b1 b2 . In
ek is multiplicity-free, it only
el B
particular, if the tensor product decomposition of B
remains to check the separation.
Part II. Here we concentrate on the Uq0 (Cn ) automaton and prove the separation and
R(b1 b2 ) = S(b1 b2 ) directly for Uq (Cn1 )-highest weight elements b1 b2 (n > 3).
ek or B
el , they have the form
In the notation (x1 , . . . , xn1 , xn1 , . . . , x1 ) B
(1)
ek ,
el B
(f, 0, . . . , 0) (d, c, 0, . . . , 0, b) B
f > b + c.
(3.4)
In what follows we always assume l > k and use the non-negative integers e and a defined
by l = f + 2e and k = 2a + b + c + d.
ek ' B
ek B
el of Uq0 (C (1) )el B
Proposition 3.11 [9]. Under the isomorphism R : B
n1
crystals, the image of (3.4) is given by
(k 2e, 0, . . . , 0) (d + l k y, c, 0, . . ., 0, b y),
y = min l k, (b d)+ ,
(3.5)
640
and always assume L s, t, u. To save the space, the tensor product of Uq0 (Cn(1) )-crystals
such as
[s, t, u]j 1
[s 00 , t 00 , u00 ] 3
[s 0 , t 0 , u0 ]j
1[s, t + 1, u 1]
1[s, t 1, 0]
[s, t, u]1 '
2[s 1, 0, 0]
1[0, 0, 0]
u > 0,
u = 0, t > 0,
u = t = 0, s > 0,
u = t = s = 0.
if (I),
if (II),
if (III),
if (IV),
if (V),
if (VI),
if (VII),
if (VIII),
641
(3.6)
j
(B1 B1 ) ' (B1 B1 ) BL
for an interval m.
This calculation can be done by using Lemma 3.12 and 3.13 only. It branches into
numerous sectors depending on the seven parameters a, b, c, d, f, e and m. (We use the
dependent variables l = f + 2e and k = 2a + b + c + d simultaneously.) We have checked
case by case that the collision always completes within two time steps (j = 2) ending
with the result 1m1 k (c2 )1m2 l (c1 )11 . . . [0, 0, 0]j for some m1 and m2 . This establishes
R = S hence Theorem 3.2. Below we illustrate a typical such calculation in the sector
a > 2e,
b > d,
a e 6 m 6 f + e a 2b c.
(3.7)
ek .
c2 = (k 2e, 0, . . . , 0) B
(3.8)
to show up after a collision and separate from each other. Starting from (3.6) with j = 3,
one can rewrite it by means of Lemma 3.12 and (3.7) as
[0, 0, 0]211 1 e 2me [s1 , 0, 0]1 a 2 b 3c 2d 11 . . .,
with s1 = f + e m and 1 = f + 2e. From (3.7) one can apply Lemma 3.13 (VIII) to
transform this into
0
(3.9)
(3.10)
(3.11)
642
Compared with (3.9), the distance of the two patterns here has increased by l k in
accordance with the velocity of solitons in Theorem 3.1. In the other case k > 3 one
rewrites (3.10) as
14 1 e 23 2e [k 3 , 0, 0]1 a+bd 2 d 3c 2lkb+2d 11 . . . [0, 0, 0]2,
for which Lemma 3.13 (V) can be applied. The result coincides with (3.11). 2
4. Discussion
In this paper we have proposed a class of cellular automata associated with Uq0 (gn ).
They are essentially solvable vertex models at q = 0 in the vicinity of the ferromagnetic
vacuum. They exhibit soliton behavior stated in Theorem 3.1, 3.2 and Conjecture 3.3.
Behavior of tensor products of crystals in the vicinity of ferromagnetic vacuum has
not been investigated in detail so far. To understand it better will be a key to prove
Conjecture 3.3 and to clarify the relevance to the subalgebra g n1 , which has also been
observed in the RSOS models in the ferromagnetic-like regime II [2,19].
The automata considered in this paper are associated with an l limit R 0 : B\
B1 B1 B\ of the ordinary combinatorial R matrices R : Bl B1 ' B1 Bl . To vary
B1 from site to site and to replace R 0 with the corresponding combinatorial R matrices is
a natural generalization. See [7].
We close by raising a few questions. What kind of soliton equations can possibly be
related to the Uq0 (gn ) automata in general? Is it possible to derive them conceptually
q0
from the associated vertex models, bypassing the route; vertex models automata
tropical variable change
Acknowledgements
The authors thank K. Hikami, R. Inoue, T. Nakashima, M. Okado, T. Takebe, T. Tokihiro,
Z. Tsuboi and Y. Yamada for valuable discussions.
Appendix A. The map R 0 :B\ B1 B1 B\
Let gl : Bl B\ be the embedding of Bl as a set as mentioned in Section 3. By the
definition, for any u B\ there exists the unique element ul Bl such that gl (ul ) = u for
each l which is sufficiently large. Consider the composition:
gl1 id
ei ,fi
gl id
643
where of course either u0l = ul or b0 = b, and we interpret (gl id)(0) = 0. For all the Bl
considered in this paper it is easy to see that this is independent of l if l is sufficiently large.
We let B\ B1 denote the set B\ B1 equipped with the actions ei0 , fi0 determined as above
with sufficiently large l. We shall simply write it as ei0 , fi0 : B\ B1 (B\ B1 ) t {0}.
Similarly we define B1 B\ and ei0 , fi0 : B1 B\ (B1 B\ ) t {0}.
Given u B\ and b B1 , let c = c(l, u, b) B1 and v = v(l, u, b) Bl be the elements
determined by
ul b ' c v
under the combinatorial R matrix R : Bl B1 ' B1 Bl . Then we have
Conjecture A.1. For any fixed u B\ and b B1 , the elements c(l, u, b) B1 and
gl (v(l, u, b)) B\ are independent of l for l sufficiently large.
Assuming the conjecture we define the map R 0 : B\ B1 B1 B\ to be the composition:
gl1 id
idgl
R 0 : B\ B1 Bl B1 ' B1 Bl B1 B\ ,
7 c gl (v),
ub
7 ul b ' c v
with l sufficiently large. Obviously this is invertible. Moreover it commutes with ei0 and fi0 ,
although this fact is not used in the main text. For ei0 this can be seen from the commutative
diagram (fi0 case is completely parallel).
B\ B1
ei0
B\ B1
gl1 id
Bl B1
ei
gl1 id
Bl B1
B1 Bl
idgl
ei0
ei
R
B1 Bl
B B\
idgl
B B\ ,
where the left and the right squares are the definitions of ei0 (for elements not annihilated
by ei0 ). The map R 0 indeed has the property (I) in Section 1. The property (II) is also valid
(1)
for A(2)
2 and Cn . We conjecture it for all the cases considered in this paper.
Note. While writing the paper the authors learned from [6] that the energy of combinatorial R matrices for Uq0 (A(1)
n ) is encoded in the phase shift of soliton scattering in the
automaton. We thank Yasuhiko Yamada for communicating their result.
References
[1] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982.
[2] V.V. Bazhanov, N.Yu. Reshetikhin, Restricted solid-on-solid models connected with simply
laced algebras and conformal field theory, J. Phys. A Math. Gen. 23 (1990) 14771492.
[3] E. Date, M. Jimbo, A. Kuniba, T. Miwa, M. Okado, One-dimensional configuration sums in
vertex models and affine Lie algebra characters, Lett. Math. Phys. 17 (1989) 6977.
644
[4] E. Date, M. Jimbo, T. Miwa, Representations of Uq (gl(n, C)) at q = 0 and the Robinson
Schensted correspondence, in: L. Brink, D. Friedan, A.M. Polyakov (Eds.), Memorial Volume
for Vadim Kniznik, Physics and Mathematics of Strings, World Scientific, Singapore, 1990,
pp. 185211.
[5] A.S. Fokas, E.P. Papadopoulou, Y.G. Saridakis, Soliton cellular automata, Physica D 41 (1990)
287321.
[6] Fukuda, M. Okado, Y. Yamada, private communication.
(1)
[7] G. Hatayama, K. Hikami, R. Inoue, A. Kuniba, T. Takagi, T. Tokihiro, The AM automata related
to crystals of symmetric tensors, preprint, math.QA/9912209.
[8] G. Hatayama, Y. Koga, A. Kuniba, M. Okado, T. Takagi, Finite crystals and paths, preprint,
math.QA/9901082.
[9] G. Hatayama, A. Kuniba, M. Okado, T. Takagi, Combinatorial R matrices for a family of
(1)
(2)
crystals: Cn and A2n1 cases, preprint, math.QA/9909068.
[10] K. Hikami, R. Inoue, private communication.
[11] K. Hikami, R. Inoue, Y. Komori, Crystallization of the Bogoyavlensky lattice, J. Phys. Soc.
Jpn. 68 (1999) 22342240.
[12] M. Kashiwara, Crystalizing the q-analogue of universal enveloping algebras, Commun. Math.
Phys. 133 (1990) 249260.
[13] S-J. Kang, M. Kashiwara, K.C. Misra, Crystal bases of Verma modules for quantum affine Lie
algebras, Compositio Math. 92 (1994) 299325.
[14] S-J. Kang, M. Kashiwara, K.C. Misra, T. Miwa, T. Nakashima, A. Nakayashiki, Affine crystals
and vertex models, Int. J. Mod. Phys. (suppl. 1A) A 7 (1992) 449484.
[15] S-J. Kang, M. Kashiwara, K.C. Misra, T. Miwa, T. Nakashima, A. Nakayashiki, Perfect crystals
of quantum affine Lie algebras, Duke Math. J. 68 (1992) 499607.
[16] M. Kashiwara, T. Nakashima, Crystal graphs for representations of the q-analogue of classical
Lie algebras, RIMS preprint 1991 to appear in J. Alg.
[17] A.N. Kirillov, Discrete Hirota equations, Schtzenberger involution and rational representations
of extended affine symmetric group, talk at Developments in Infinite Analysis II, Okinawa,
December 1998.
[18] P.P. Kulish, E.K. Sklyanin, Quantum spectral transform method. Recent developments, Lect.
Note. Phys. 151 (1982) 61119.
(1)
[19] A. Kuniba, Thermodynamics of Uq (Xr ) Bethe ansatz system with q a root of unity, Nucl.
Phys. B 389 (1993) 209244.
[20] J. Matsukidaira, J. Satsuma, D. Takahashi, T. Tokihiro, M. Torii, Toda-type cellular automaton
and its N -soliton solution, Phys. Lett. A 225 (1997) 287295.
[21] T. Nakashima, Crystal base and a generalization of the LittlewoodRichardson rule for the
classical Lie algebras, Commun. Math. Phys. 154 (1993) 215243.
[22] A. Nakayashiki, Y. Yamada, Kostka polynomials and energy functions in solvable lattice
models, Selecta Mathematica, New Ser. 3 (1997) 547599.
[23] T.S. Papatheodorou, A.S. Fokas, Evolution theory, periodic particles, and solitons in cellular
automata, Stud. Appl. Math. 80 (1989) 165182.
[24] T.S. Papatheodorou, M.J. Ablowitz, Y.G. Saridakis, A rule for fast computation and analysis of
soliton automata, Stud. Appl. Math. 79 (1988) 173184.
[25] J.K. Park, K. Steiglitz, W.P. Thurston, Soliton-like behavior in automata, Physica D 19 (1986)
423432.
[26] E.K. Sklyanin, L.A. Takhatajan, L.D. Faddeev, Quantum inverse problem method I, Theor.
Math. Phys. 40 (1980) 688706.
[27] D. Takahashi, J. Satsuma, A soliton cellular automaton, J. Phys. Soc. Jpn. 59 (1990) 35143519.
[28] D. Takahashi, One some soliton systems defined by using boxes and balls, in: Proceedings of
the International Symposium on Nonlinear Theory and Its Applications (NOLTA 93), 1993,
pp. 555558.
645
Abstract
The modular transformation properties of admissible characters of the affine superalgebra
b
sl(2|1; C) at fractional level k = 1/u 1, u N \ {1} are presented. All modular invariants for
u = 2 and u = 3 are calculated explicitly and an A-series and D-series of modular invariants emerge.
2000 Elsevier Science B.V. All rights reserved.
PACS: 11.25.Hf; 11.25.Pm; 02.20.Tw
Keywords: Modular transformations; Affine superalgebras
1. Introduction
In a series of papers [13], the properties of the affine superalgebra b
sl(2|1; C)k
at fractional level were extensively investigated. As well as the abstract interest, the
motivation for this work was in the potential relevance to the N = 2 non-critical string. The
generally held notion that a non-critical (super)string is described in terms of a topological
G/G WessZuminoNovikovWitten model, with G a Lie (super)group [46], provides
the link: in order to describe the spectrum in the G = SL(2|1; R) case, believed to be
that related to the N = 2 string, a good understanding of b
sl(2|1; C)k at fractional level is
required. Indeed, when the matter, which is coupled to supergravity in the N = 2 noncritical string, is minimal, i.e., taken in an N = 2 super Coulomb gas representation with
central charge
2p
, p, u N, gcd(p, u) = 1,
(1.1)
cmatter = 3 1
u
it can be shown that the level of the matter affine superalgebra b
sl(2|1; C)k appearing in
the SL(2|1; R)/SL(2|1; R) model is of the form
g.b.johnstone@durham.ac.uk
0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 9 9 ) 0 0 8 2 3 - 8
k=
p
1.
u
647
(1.2)
648
2. Branching b
sl(2|1; C)k characters
In [3], branching formulae for the NeveuSchwarz class IV and class V characters
of the affine superalgebra b
sl(2|1; C)k (k = 1/u 1) were conjectured. The characters
were branched into products of b
sl(2; C)k characters, generalised theta functions and string
functions, the modular transformation properties of which are known.
By definition, b
sl(2|1; C)k characters are given by
b
d
sl(2|1;C)
(2.1)
h ,h+ k (, , ) = tr exp 2i J03 + U0 + L0 ,
sl(2|1; C)k . We label the
where J03 and U0 are the zero-mode Cartan generators of b
characters by the isospin and charge quantum numbers which characterise the b
sl(2|1; C)k
highest weight states |i of the associated representations 2 :
J03 |i = 12 h |i,
U0 |i = 12 h+ |i.
(2.2)
sl(2;C)k
(, ) =
n,n
0
v+ ,w (/u, ) v ,w (/u, )
,
1,2 (, ) 1,2 (, )
(2.3)
v = u((n + 1) n0 (k + 2)),
d
(2.4)
w = u2 (k + 2).
(2.5)
In the above, the generalised theta functions m,m0 [9] are defined as
2
m
d X m0 n+ m 0
m0 n+ 2m
0
2m
0
q
z
.
m,m (, ) =
(2.6)
nZ
q = exp(2i ),
z = exp(2i ).
(2.7)
m,m0
(/, 1/ ) = eik
2 /
u1
2u+t
X2 X
n=0
n0 =0
b
sl(2;C)k
(, ),
(2.8)
where
2 See Refs. [1,2] for more details. It is conditions on these quantum numbers that split the representations into
the classes mentioned earlier.
649
s
Smm0 ,nn0 =
2
0
0
0 0
(1)m (n+1)+(m+1)n ei(k+2)m n
+ 2)
(m + 1)(n + 1)
sin
.
k+2
u2 (k
(2.9)
(2.10)
r=0
u1
X
u1
X
a0 = 0
a 0 b0
where
0
s(a, b, a , b ) = e
ibb0 /(u1)
b0 = u + 2
s(a, b, a 0, b0 ) ca(u1)
0 ,b0 ( ),
(2.11)
mod 2
(a + 1)(a 0 + 1)
sin
.
u+1
(2.12)
(u1)
(u1)
(u1)
(2.13)
NS,IV,b
sl(2|1;C)k
(, , )
NS
hNS
,h+
u1
X
b
sl(2;C)
k
a,um1
(, )
a=0
u2
X
(2.14)
NS,V,b
sl(2|1;C)k
(, , )
NS
hNS
,h+
u1
X
sl(2;C)k
a,M+M
0 +1 (, )
u2
X
650
1 NS
For the NeveuSchwarz sector, the isospin ( 12 hNS
) and charge ( 2 h+ ) quantum numbers
are given by
1
hNS
= (u m 1),
u
hNS
+ =
1
(2m0 m),
u
0 6 m0 6 m 6 u 1
(2.16)
in class IV and by
1
0
hNS
= (M + M + 1 + u),
u
hNS
+ =
1
(M M 0 ),
u
0 6 M + M0 6 u 2
(2.17)
in class V.
As we will come to consider the characters and supercharacters of the Ramond sector
and the supercharacters of the NeveuSchwarz sector, we mention that the NeveuSchwarz
supercharacters have the same quantum numbers as the characters while the Ramond
characters and supercharacters have
hR
=
m
,
u
NS
hR
+ = h+ ,
0 6 m0 6 m 6 u 1
(2.18)
in class IV and
hR
=
1
(M + M 0 + 2),
u
NS
hR
+ = h+ ,
0 6 M + M0 6 u 2
(2.19)
2
u R 2
h hR
+
4
(2.20)
1u
.
4u
(2.21)
We can use spectral flow [2] to obtain the branching formulae for Ramond characters
from (2.14) and (2.15):
NS,b
sl(2|1;C)k
R,b
sl(2|1;C)k
(, , ) = q k/4zk/2 hR ,hR
(
NS
hNS
+
,h+
, , ),
(2.22)
giving
R,b
sl(2|1;C)k
hR ,hR
(, , ) = q (1u)/4uz(1u)/2u
NS,b
sl(2|1;C)k
(
NS
hNS
,h+
, , )
(2.23)
n,n0
and thus
b
sl(2;C)
k
( , ) = q (1u)/4uz(1u)/2uu1n,u1n
0 (, )
(2.24)
hR ,hR
u1
X
651
(, , )
b
sl(2;C)
k
u1a,m
(, )
a=0
u2
X
(2.25)
and in class V
R,V,b
sl(2|1;C)k
hR ,hR
u1
X
(, , )
sl(2;C)k
u1a,uMM
0 2 (, )
a=0
u2
X
(2.26)
NS,IV,b
sl(2|1;C)k
(, , )
NS
hNS
,h+
u1
X
sl(2;C)k
ei(a(um1))a,um1
(, )
a=0
u2
X
(2.27)
NS,V,b
sl(2|1;C)k
(, , )
NS
hNS
,h+
u1
X
b
sl(2;C)
ei(a(M+M )) a,M+Mk 0 +1 (, )
a=0
u2
X
b=0
(u1)
ca,a(u1)+2a( u [ u ])2b ( ).
2
2
(2.28)
652
The Ramond sector supercharacters are obtained in similar fashion from (2.25) and
R
(2.26), shifting + 1 and dividing by eih :
R,IV,b
sl(2|1;C)k
ShR ,hR
u1
X
(, , )
b
sl(2;C)k
ei(u1am)u1a,m
(, )
a=0
u2
X
(2.29)
and
R,V,b
sl(2|1;C)k
ShR ,hR
u1
X
(, , )
0
sl(2;C)k
ei(M+M a) u1a,uMM
0 2 (, )
a=0
u2
X
(2.30)
m,m0
(, , )
u1
X
653
b
sl(2;C)
k
a,um1
(, )
a=0
u2
X
m,m0
=
u1
X
(2.31)
(, , )
sl(2;C)k
u1a,m
(, )
a=0
u2
X
b=0
(u1)
( );
ca,a(u1)+2a(
u
u
2 [ 2 ])2b
NS,b
sl(2|1;C)k
Sm,m0
=
(2.32)
(, , )
u1
X
b
sl(2;C)k
(1)G+a(um1)a,um1
(, )
a=0
u2
X
(2.33)
and
R,b
sl(2|1;C)k
Sm,m0
=
(, , )
u1
X
b
sl(2;C)k
(1)G+u1am u1a,m
(, )
a=0
u2
X
0
1
if m > m0 ,
if m < m0
(2.34)
654
3. Modular S transformation of b
sl(2|1; C)k characters
The action of S : (, , ) (/, /, 1/ ) on the branched b
sl(2|1; C)k characters
(2.31), (2.32), (2.33) and (2.34) may now be obtained by use of (2.8), (2.10) and (2.11).
For example, in the case of the NeveuSchwarz characters (2.31) we find
NS,b
sl(2|1;C)k
m,m0
(/, /, 1/ )
2
2
u1 u1 u1
ei(u1)( )/u X X X
b
sl(2;C)
Sa(um1),nn0 n,n0 k (, )
(u 1) 2u(u + 1) a=0 n=0 n0 =0
u2 2u(u1)1
X
X
b=0
r=0
r,u(u1)(/u, )
u1
X
u1
X
a0 = 0
a 0 b0
b0 = u + 2
s(a, l, a 0 , b0 )ca(u1)
0 ,b0 ( ),
(3.1)
mod 2
n,n0
b
sl(2;C)
k
(, , ) 0,un1
(, )(u1)(n2n0 +u),u(u1)
(u1)
(/u, )c0,0
( ).
(3.2)
m,m0
(/, /, 1/ )
= ei(u1)(
2 2 )/u
u1
u1 X
X
n=0 n0 =0
NS,b
sl(2|1;C)k
NS
Smm
0 ,nn0 n,n0
(, , ),
(3.3)
where
NS
Smm
0 ,nn0 =
1
u(u 1)
e
XX
u
Sa(um1),0(un1)
2(u + 1)
u1 u2
a=0 b=0
i(n2n0 +u)((u1)(m2m0)+u(u1)(a+1)+2au( 2u [ u2 ])2ub)/u
(3.4)
655
1
u(u 1)(u + 1)
u2
u1 X
X
(1)(um1)+(a+1)(un1)ei(u+1)(um1)(un1)/u
a=0 b=0
(a + 1)u
sin
u+1
(3.5)
n,n0
b
sl(2;C)
(u1)
( );
(3.7)
Sn,n0
sl(2;C)k
(, , ) (1)G(un1) 0,un1
(, )(u1)(n2n0 +u),u(u1)
(u1)
(/u, )c0,0
( );
(3.8)
n,n0
sl(2;C)k
(, , ) (1)G+u1n u1,n
(, )(u1)(n2n0 +u),u(u1)
(u1)
(/u, )c0,0
( ),
(3.9)
m,m0
(/, /, 1/ )
= ei(u1)(
2 2 )/u
u1
u1 X
X
n=0 n0 =0
NS,b
sl(2|1;C)k
R
Smm
0 ,nn0 Sn,n0
(, , )
(3.10)
656
where
1
0
R
(1)G +m+n+u(un1) ei(u+1)m(un1)/u
Smm
0 ,nn0 =
u
0
0
ei(u1)(m2m +u)(n2n +u)/u ;
NS,b
sl(2|1;C)k
Sm,m0
(3.11)
(/, /, 1/ )
= ei(u1)(
2 2 )/u
u1
u1 X
X
n=0 n0 =0
R,b
sl(2|1;C)k
SNS
Smm
0 ,nn0 n,n0
(, , ),
(3.12)
where
1
SNS
(1)G+m+n+u(um1) ei(u+1)(um1)n/u
Smm
0 ,nn0 =
u
0
0
ei(u1)(m2m +u)(n2n +u)/u ;
(3.13)
and
R,b
sl(2|1;C)k
Sm,m0
=e
(/, /, 1/ )
i(u1)( 2 2 )/u
u1
u1 X
X
n=0 n0 =0
R,b
sl(2|1;C)k
SR
Smm
0 ,nn0 Sn,n0
(, , ),
(3.14)
where
1
0
0
0
SR
(1)G+G +(u1)(m+n) ei(u+1)mn/u ei(u1)(m2m +u)(n2n +u)/u ,
Smm
0 ,nn0 =
u
(3.15)
with
G=
and
G0 =
0
1
0
1
if m > m0 ,
if m < m0
if n > n0 ,
if n < n0 .
We note that S NS and S SR are symmetric; that S R = (S SNS )T ; that all of these matrices are
unitary; and that the matrices as calculated by brute force for u = 2 (as found in [2]) and
u = 3 (also calculated in [12] for the NeveuSchwarz characters) given in the appendices
agree with the above results.
In order to consider modular invariant combinations of b
sl(2|1; C)k characters, we
must also know how they transform under the modular T transformation T : (, , )
(, , + 1). It can be shown that the action of T is as follows [2,12]:
R,b
sl(2|1;C)k
m,m0
NS,b
sl(2|1;C)k
m,m0
R,b
sl(2|1;C)k
(, , + 1) = e2ih m,m0
NS
(, , ),
NS,b
sl(2|1;C)k
(, , + 1) = e2ih Sm,m0
(, , ),
Sm,m0
R,b
sl(2|1;C)k
Sm,m0
NS
NS,b
sl(2|1;C)k
(, , + 1) = e2ih m,m0
R
R,b
sl(2|1;C)k
(, , + 1) = e2ih Sm,m0
657
(, , ),
(, , ),
(3.16)
recalling that for class V characters (m < m0 ) we must use the appropriate M and M 0 and
the class V formulae to calculate the conformal weights.
4. Modular invariants
With the behaviour of b
sl(2|1; C)k characters under the modular S and T transformations
now established, we proceed by looking for modular invariant combinations of characters.
These could be taken as starting points in the building of partition functions for rational
conformal field theories. The canonical example of this is of course the classification of
sbu(2) modular invariants, implying the classification of the minimal models, by Cappelli,
Itzykson and Zuber [13,14] (see also Ref. [15]), with minimal superconformal models
also considered in [16]. It was found that these partition functions fell into an ADE
pattern. This work was extended to the case of admissible b
sl(2) representations by Koh
and Sorba [17] and Lu [18], who obtained a complete classification of modular invariants.
Although we do not attempt to obtain a full classification of fractional level b
sl(2|1; C)k
modular invariants here, we do find all invariants for the cases u = 2 and u = 3, special
cases of which are analogous to the A- and D-series obtained in the sbu(2) case. It is
straightforward to show that such modular invariants exist for all u > 2.
Modular invariant combinations of characters take the form
u1
X
Z=
m,m0 ,n,n0 =0
R
R
NS
NS
R
NS 0
Nmm
0 ,nn0 m,m0 n,n0 + Nmm,nn0 m,m0
n,n
SNS
NS
NS 0
+ Nmm
0 ,nn0 Sm,m0 S
n,n
u1
X
a,a 0 ,b,b0 =0
SR
R
R
Naa
0 ,bb0 Sa,a 0 S b,b0 ,
(4.1)
written in this way to emphasise the fact that the Ramond supercharacters form a closed set
under modular transformations, whereas the remaining sectors mix as detailed previously.
For physical modular invariants, the Nmm0 ,nn0 must be non-negative integers. In addition,
R
must be equal to 1 (the identity character in this
the identity should be unique so N00,00
R
context is 0,0 ).
For the case of u = 2, we find two possibilities (see Appendix A):
1
0
0
0
a
0 0 a1
0
0
a
a 1 0
1 0
0
NS
,
(4.2)
=
N
(i) N R =
0 a 1
0
a
0
0 1
0
0
or
a1
0 0
658
1
0
0
0
a
0 0 a1
0 a 1
0
a
0
0 1
0
,
. (4.3)
(ii) N R =
N NS =
0
a
a1 0
0
1 0
0
0
0
0
1
a1 0 0
a
Clearly for non-negative integer entries, a N. There are thus an infinite number of
modular invariants, a phenomenon also observed in [18]. For u = 3 we find a similar
situation, with an additional parameter:
(i)
a+b
0
0
0
a
b
0
b
a
0
0
0
0
0
0
N NS =
0
0
0
a + b 1
0
0
0
a1
b
0
b
a1
or
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
(ii)
a
b
b
0
a1
0
0
0
b
a
a 1
0
b
0
0
b
a
1
a
0
b
0
0
0
0
0
a
+
b
0
a
+
b1
NR =
0 a 1
b
b
0
a
0
0
0
0
0
a +b1
0
a+b
0
0
0
0
0
0
0
0
a+b
0
0
0
b
a
0
a
b
0
0
0
0
0
0
N NS =
0
0
0
a +b 1
0
0
0
b
a1
0
a1
b
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
,
0
0
1
0 a +b1
0
0
0
0
a1
b
0
0
b
a 1
0
0
0
0
0
0
0
0
1
0
0
0
0
a+b
0
0
0
0
a
b
0
0
b
a
1
0
0
0
0
0
0
b
a
a 1
0
b
0
0
0
a
b
b
0
a1
0
0 a 1
b
b
0
a
0
0
0
0
a+b
0
a+b1
NR =
0
0
b
a1
a
0
b
0
0
0
0
0
a+b1
0
a +b
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
(4.4)
0
0
0
0
,
0
1
0
0 a +b1
0
0
0
0
b
a 1
0
0
a1
b
1
0
0
0
0
0
0
0
. (4.5)
0
0
0
0
0
a+b
0
0
0
0
b
a
0
0
a
b
659
Table 1
Class IV b
sl(2|1; C)1/2 characters
m
m0
hR
hR
+
hR
hNS
hNS
+
hNS
12
18
12
12
12
12
1
2
1
2
1
8
1
8
Table 2
Class V b
sl(2|1; C)1/2 characters
M(m)
M 0 (m0 )
hR
hR
+
hR
hNS
hNS
+
hNS
0(0)
0(1)
1
2
32
18
Table 3
Class IV b
sl(2|1; C)2/3 characters
m
m0
hR
hR
+
hR
hNS
hNS
+
hNS
23
16
13
13
13
13
13
13
1
3
23
23
23
23
1
3
23
1
3
2
3
2
3
1
6
1
2
1
6
Table 4
Class V b
sl(2|1; C)2/3 characters
M(m)
M 0 (m0 )
hR
0(1)
0(2)
2
3
0(0)
1(1)
13
1
3
1(0)
0(2)
hR
+
hR
hNS
hNS
+
hNS
1
3
2
3
2
3
43
16
53
13
53
1
3
660
In the cases (4.2) and (4.4), when a = 1 and b = 0, we find that N R = N NS = I . This
diagonal invariant we find at all levels, since the matrices S and T are unitary. With a = 1
and b = 0 in (4.3) and (4.5), the resulting expressions are permutation invariants of the
form
X
(4.6)
m,m0 (m,m0 ) ,
where
(m, m0 ) = (m, (m m0 ) mod u).
(4.7)
5. Conclusion
We have found expressions for the modular S transformation of b
sl(2|1; C)k characters
at fractional level k = 1/u 1. This has allowed us to calculate all modular invariants for
the cases u = 2 and u = 3, leading to the discovery of an A-series and D-series of modular
invariants. The derivation of the general S transformation of b
sl(2|1; C)k characters neatly
rounds off the work of [3] and enables us to look at modular invariants in this framework
of affine superalgebras at fractional level, a subject little studied. It would certainly be
interesting to have a complete classification of these invariants, generally a non-trivial
problem; however, it would not be unlikely that an ADE type classification along the
lines of [18] might appear, given the underlying presence of b
sl(2). We make no claims
that these invariants constitute fully-fledged partition functions, given the complications
entering at fractional level: in these situations considering fusion rules requires the
inclusion of fields not corresponding to highest or lowest weight representations, as
originally discovered for fractional level b
sl(2) by Awata and Yamada [21]. Work has
also been done on the case of fractional level b
sl(3) see for example Ref. [22], which
also contains some discussion of the b
sl(2) case. As to superalgebras, the fusion rules of
admissible representations of osp(1|2)
c
have been studied in [23]. It would be interesting to
consider fusion rules in the present context and attempt to realise a rational conformal field
theory based on fractional level b
sl(2|1; C)k .
661
Acknowledgements
G.B.J. thanks Peter Bowcock and Anne Taormina for useful comments and discussions,
and acknowledges the award of an EPSRC research studentship.
Appendix A. u = 2
Here we list the explicit forms of the matrices S for each sector at u = 2. We then list the
possible modular invariants satisfying the condition that the matrices N have non-negative
integer entries. In what follows, we understand a sum over the repeated index :
NS,b
sl(2|1;C)1/2
(/, /, 1/ ) = ei(
2 2 )/2
NS,b
sl(2|1;C)1/2
NS
S
(, , ),
, = 1, 2, 3, 4,
where
(A.1)
i
1
1
1
1
i
i
NS
=
S
1
i
i
2
i 1 1
R,b
sl(2|1;C)1/2
i
1
;
1
(A.2)
(/, /, 1/ ) = ei(
2 2 )/2
NS,b
sl(2|1;C)1/2
R
S
S
(, , ),
, = 1, 2, 3, 4,
where
R
S
1
1
i
=
2 i
1
(A.3)
1
1 1
i
i
i
;
i
i
i
1 1 1
NS,b
sl(2|1;C)1/2
(A.4)
(/, /, 1/ ) = ei(
2 2 )/2
R,b
sl(2|1;C)1/2
SNS
S
, = 1, 2, 3, 4,
where
1
i
1
1
i
SNS
=
S
i
2 1
1 i
R,b
sl(2|1;C)1/2
where
(A.5)
i
i
i
i
1
1
;
1
(A.6)
(/, /, 1/ ) = ei(
, = 1, 2, 3, 4,
(, , ),
2 2 )/2
R,b
sl(2|1;C)1/2
SR
S
S
(, , ),
(A.7)
662
1
1 1
1 1 1
.
1 1
1
1
1
1
1
1
SR
S
=
2 1
1
(A.8)
Z=
R
R
NS
NS
R
NS 0
Nmm
0 ,nn0 m,m0 n,n0 + Nmm,nn0 m,m0
n,n
m,m0 ,n,n0 =0
SNS
NS
NS 0
+ Nmm
0 ,nn0 Sm,m0 S
n,n
u1
X
a,a 0 ,b,b0 =0
SR
R
R
Naa
0 ,bb0 Sa,a 0 S b,b0 ,
(A.9)
that is to say, N such that [S, N] = [T , N] = 0, using the appropriate matrices S and T .
We find that the general form of these N is
ab
0
0
0
0
c+b ac
0
,
NR =
0
ac c+b
0
0
0
N NS = N SNS =
0
b
ab
0
0
c
abc
0
0
abc
c
0
b
0
.
0
a
(A.10)
R
= 1, we
With the requirements that all the Nmm0 ,nn0 are non-negative integers and N00,00
find two possible cases:
1
0
0
0
a
0 0 a1
0
0
a
a 1 0
1 0
0
,
(A.11)
N NS =
(i) N R =
0 a 1
a
0
0
0 1
0
0
0
0
1
a1 0 0
a
or
1
0
0
0
a
0 0 a1
0 a 1
0
a
0
0 1
0
,
, (A.12)
N NS =
(ii) N R =
0
0
a
a 1 0
1 0
0
a1
0 0
663
d +eh
d
e
0
.
N SR =
f +gh
f
g
0
0
0
0
h
(A.13)
Appendix B. u = 3
NS,b
sl(2|1;C)2/3
(/, /, 1/ ) = e2i(
2 2 )/3
NS,b
sl(2|1;C)2/3
NS
S
(, , ),
, = 1, 2, . . . , 9,
(B.1)
where
e2 i/3
1
NS
=
S
3
e i/3
1
e i/3
e i/3
e2 i/3
e i/3
1
e i/3
e2 i/3
e i/3
1
1
1
1
e i/3
e2 i/3
e2 i/3
e i/3
e2 i/3
1
e i/3
1
e i/3
e2 i/3
1
e i/3
1
1
1
1
1
1
1
1
1
e i/3
e i/3
e2 i/3
1
e2 i/3
1
e2 i/3
e2 i/3
1
e i/3
e i/3
e2 i/3
1
e2 i/3
1
e2 i/3
e2 i/3
e i/3
e2 i/3
e2 i/3
1
1
1
e2 i/3
e i/3
e i/3
e2 i/3
e i/3
1
e2 i/3
1
e2 i/3
e i/3
1
e2 i/3
e2 i/3
1
e i/3
e2 i/3
;
1
e2 i/3
e i/3
2
i/3
e
1
(B.2)
R,b
sl(2|1;C)2/3
(/, /, 1/ ) = e2i(
2 2 )/3
NS,b
sl(2|1;C)2/3
R
S
S
(, , ),
, = 1, 2, . . . , 9,
(B.3)
where
R
=
S
i/3
e i/3
e
e2 i/3
1
e2 i/3
3
e2 i/3
e i/3
1
1
1
1
e i/3
1
e2 i/3
e2 i/3
e i/3
e2 i/3
e2 i/3
1
e i/3
1
e2 i/3
e2 i/3
1
e i/3
e2 i/3
e2 i/3
e i/3
e i/3
2
i/3
e
1
e2 i/3
1
e2 i/3
e2 i/3
1
1
1
1
1
1
1
1
1
1
e i/3
e i/3
e2 i/3
1
e2 i/3
1
e2 i/3
e2 i/3
e2 i/3
e2 i/3
e i/3
e i/3
e i/3
e2 i/3
1
1
e2 i/3
1
e i/3
e i/3
1
e2 i/3
e i/3
e i/3
1
1
e2 i/3
1
e i/3
;
e i/3
e2 i/3
i/3
e
e i/3
(B.4)
NS,b
sl(2|1;C)2/3
(/, /, 1/ ) = e2i(
, = 1, 2, . . . , 9,
where
2 2 )/3
R,b
sl(2|1;C)2/3
SNS
S
(, , ),
(B.5)
664
SNS
S
=
1
e i/3
1
1
1 e i/3
1 e i/3
1
1
1
3
1 e i/3
1 e2 i/3
1 e2 i/3
1
1
e i/3
e i/3
1
e i/3
1
e i/3
e2 i/3
1
e2 i/3
e2 i/3
1
e2 i/3
e2 i/3
1
e2 i/3
e i/3
e i/3
1
e2 i/3
e2 i/3
e2 i/3
1
1
1
e i/3
e i/3
e i/3
e2 i/3
e2 i/3
1
e2 i/3
1
e2 i/3
e i/3
1
e i/3
e i/3
e i/3
e i/3
1
1
1
e2 i/3
e2 i/3
e2 i/3
1
e2 i/3
e2 i/3
e2 i/3
;
1
e2 i/3
1
e i/3
e i/3
1
e2 i/3
e2 i/3
e2 i/3
1
e2 i/3
1
e i/3
e i/3
(B.6)
R,b
sl(2|1;C)2/3
(/, /, 1/ ) = e2i(
2 2 )/3
R,b
sl(2|1;C)2/3
SR
S
S
(, , ),
, = 1, 2, . . . , 9,
where
SR
=
S
1
1
1 21 i/3
1 e
1
1
1
1 e2 i/3
3
1 e2 i/3
1 e i/3
1
e i/3
1 e i/3
(B.7)
1
e2 i/3
1
e2 i/3
e2 i/3
1
e i/3
e i/3
e i/3
1
1
e2 i/3
1
e2 i/3
e2 i/3
e i/3
e i/3
e i/3
1
e2 i/3
e2 i/3
e2 i/3
e2 i/3
e2 i/3
e i/3
1
1
1
e2 i/3
1
e2 i/3
e2 i/3
1
e i/3
e i/3
e i/3
1
e i/3
e i/3
e i/3
e i/3
e i/3
e2 i/3
1
1
1
e i/3
e i/3
e i/3
1
e i/3
1
e2 i/3
e2 i/3
1
e i/3
e i/3
e i/3
1
.
e i/3
2
i/3
e
e2 i/3
(B.8)
In the above we used the definitions from Tables 3 and 4. For the u = 3 modular
invariants we find
R
N =
a b+cd
0
0
0
0
0
0
0
0
a+c
0
0
NS
SNS
=
N =N
0
0
b +d
0
0
0
a
c
b
0
d
0
0
0
0
c
a
d
0
b
0
0
0
0
a
c
0
0
0
0
d
b
0
b
d
a
0
c
0
0
0
0
c
a
0
0
0
0
b
d
0
0
0
0
a+c
0
b+d
0
0
0
0
0
ad
0
cb
0
0
0
0
d
b
c
0
a
0
0
0
0
0
0
0
b+d
0
a+c
0
0
0
0
0
0
0
0
0
ad
cb
0
0
0
0
ab+cd
0
0
0
0
0
0
0
0
,
0
cb
a d
0
0
0
cb
0
ad
0
0
0
b+d
0
0
0
0
0
a+c
0
0
0
d
b
0
0
0
0
a
c
0
b
d
0
. (B.9)
0
c
a
R
= 1 leads to the following
Requiring that entries be non-negative integers and that N00,00
invariants:
(i)
1
0
0
0
0
0
0
a
b
b
0
a1
0
0
0
b
a
a 1
0
b
0
0
b
a
1
a
0
b
0
0
0
0
0
a
+
b
0
a
+
b1
NR =
0 a 1
b
b
0
a
0
0
0
0
0
a +b1
0
a+b
0
0
0
0
0
0
0
0
0
0
0
0
0
0
a+b
0
0
0
a
b
0
b
a
0
0
0
0
0
0
N NS =
0
0
0
a + b 1
0
0
0
a1
b
0
b
a1
or
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
a+c
0
0
0
a
c
0
c
a
0
0
0
NS
0
0
0
N =
0
0
0
a +c 1
0
0
0
a
c1
0
c1
a
SR
N =
0
e
0
0
j
g2
f4
0
f
0
0
0
f3
g3
h
0
l
0
0
0
f4
f
j
0
g2
0
0
0
0
0
0
k
0
j7
0
0
h
l
f3
0
g3
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
j7
0
k
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
m8
m9
0
0
0
0
,
0
0
1
1
0
0
0
0
0
a
c
c1
0
a
0
0
c
a
a
0
c1
0 c 1
a
a
0
c
0
0
0
a+c
0
NR =
0
0
a
c1
c
0
a
0
0
0
0
a +c1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0 a +b1
0
0
0
0
a1
b
0
0
b
a 1
0
0
0
0
0
0
0
0
(B.10)
1
0
0
0
0
a+b
0
0
0
0
a
b
0
0
b
a
(ii)
665
0
0
0
0
a+c1
0
a+c
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
,
0
1
0
0 a+c1
0
0
0
0
a
c 1
0
0
c1
a
1
0
0
0
0
0
0
0
. (B.11)
0
0
0
0
0
a+c
0
0
0
0
a
c
0
0
c
a
0
0
m9
m8
(B.12)
666
where e1 = f + g + k + l, e2 = f + g h, e3 = e f + h, f3 = g j + k, f4 =
e + f + g h + l, g2 = f + g h j + k, g3 = e f g + h + j , j7 = e f l, m8 =
g + h + j , m9 = e + f + g h j + k + l. Setting j = k = 1, e = f = g = h = l = 0
gives us the identity and k = 1, e = f = g = h = j = l = 0 the permutation invariant.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
Aguilar-Saavedra, J.A.
Ahn, C.
Akemann, G.
Akutsu, Y.
Alexandrou, C.
Alford, M.
Alimohammadi, M.
Alls, B.
ALPHA Collaborations
Altarelli, G.
lvarez, E.
Anastasiou, C.
Anastasiou, C.
Angelantonj, C.
Antoniadis, I.
Ardalan, F.
Arfaei, H.
Arnsdorf, M.
Aschieri, P.
Azaria, P.
B576 (2000) 56
B572 (2000) 188
B576 (2000) 597
B575 (2000) 504
B571 (2000) 257
B571 (2000) 269
B577 (2000) 609
B576 (2000) 658
B571 (2000) 237
B575 (2000) 313
B574 (2000) 153
B572 (2000) 307
B575 (2000) 416
B572 (2000) 36
B572 (2000) 36
B576 (2000) 578
B576 (2000) 578
B577 (2000) 529
B574 (2000) 551
B575 (2000) 439
Bak, D.
Bakas, I.
Ball, P.
Ball, R.D.
Bandelloni, G.
Barbieri, R.
Barnes, K.J.
Bastero-Gil, M.
Bastianelli, F.
Becker, K.
Becker, M.
Beenakker, W.
Behrndt, K.
Blanger, G.
Belitsky, A.V.
Belitsky, A.V.
Berche, B.
Berends, F.A.
Berges, J.
Bialas, P.
Bialas, P.
Bianchi, M.
Bilal, A.
Bill, M.
Binoth, T.
Blas Achic, H.S.
Bobeth, C.
Bode, A.
Bogacz, L.
Bordag, M.
Boudjema, F.
Bouwknegt, P.
Brace, D.
Braden, H.W.
Brahmachari, B.
Brower, R.C.
Buchbinder, I.L.
Buchmller, W.
Bueno, A.
Burda, Z.
Cacciari, M.
Campanelli, M.
Carena, M.
Casas, J.A.
Castro-Alvaredo, O.A.
Chandrasekharan, S.
Chapovsky, A.P.
Chatelain, C.
Chattaraputi, A.
Chen, B.
Chen, B.
Cheng, H.-C.
Chetyrkin, K.G.
Chim, L.
Cho, G.-C.
Chu, C.-S.
Ciuchini, M.
Controzzi, D.
Cox, J.
Creminelli, P.
Cvetic, M.
Cvetic, M.
Cvetic, M.
Czakon, M.
668
DAdda, A.
DAppollonio, G.
Daflon Barrozo, M.C.
Damgaard, P.H.
Damgaard, P.H.
Dasgupta, T.
Dedes, A.
de Forcrand, Ph.
Deger, N.S.
De Holanda, P.C.
del Aguila, F.
Delduc, F.
Delduc, F.
Del Duca, V.
Del Duca, V.
Delepine, D.
De Pietri, R.
Derendinger, J.-P.
Dhar, A.
Diaconescu, D.-E.
Diakonov, D.
Daz, M.A.
Dixon, L.
Dobrescu, B.A.
Donini, A.
Dorey, P.
Dreiner, H.
Dudas, E.
Dudas, E.
Edelstein, J.D.
Eguchi, T.
Emparan, R.
Engels, J.
Engels, J.
Ennes, I.P.
Ermolaev, B.I.
Espinosa, J.R.
Evans, D.H.
Evans, T.S.
Eyras, E.
Fabbri, D.
Fendley, P.
Ferrandis, J.
Ferreira, L.A.
Fioravanti, D.
Fleischer, J.
Forte, S.
Frahm, H.
Franco, E.
Fr, P.
Freidel, L.
Frre, J.-M.
Freund, A.
Fring, A.
Frixione, S.
Fujii, A.
Fujii, Y.
Fujikawa, K.
Fukae, M.
Fukahori, T.
Fukui, T.
Gaillard, M.K.
Ganchev, A.Ch.
Garcia, D.
Garca Prez, M.
Garden, J.
Gavela, M.B.
Girlanda, L.
Giusti, L.
Glover, E.W.N.
Glover, E.W.N.
Gluza, J.
Gmez, C.
Gmez-Reino, M.
Gonzalez-Garcia, M.C.
Gorsky, A.
Greco, M.
Grinstein, B.
Groves, M.
Gualtieri, L.
Guillet, J.Ph.
Gukov, S.
Gnaydin, M.
Gupta, S.
Gyulassy, M.
B571 (2000) 3
B571 (2000) 457
B577 (2000) 88
B577 (2000) 263
B571 (2000) 237
B574 (2000) 23
B575 (2000) 285
B573 (2000) 201
B572 (2000) 307
B575 (2000) 416
B573 (2000) 57
B574 (2000) 153
B574 (2000) 587
B573 (2000) 3
B571 (2000) 120
B571 (2000) 137
B577 (2000) 240
B573 (2000) 449
B577 (2000) 547
B572 (2000) 361
B574 (2000) 169
B572 (2000) 131
B577 (2000) 529
B571 (2000) 197
Haack, M.
Hagiwara, K.
Hammou, A.B.
Handoko, L.T.
Hara, Y.
Hart, A.
Hashimoto, T.
Hastings, M.B.
Hatayama, G.
Hatsuda, M.
Hebecker, A.
Heinrich, G.
Heinrich, G.
Heitger, J.
Hernndez, P.
Hidaka, H.
Hieida, Y.
Hill, C.T.
Hioki, S.
Ho, P.-M.
Ho, P.-M.
Holland, K.
B575 (1999) 78
B571 (2000) 479
B575 (2000) 561
B575 (1999) 195
B571 (2000) 71
B572 (2000) 211
B575 (2000) 401
B575 (1999) 231
Ibez, L.E.
Ibarra, A.
Ichie, H.
Ichinose, I.
Ichinose, I.
Ichinose, S.
Iida, S.
Ishibashi, M.
Ishibashi, N.
Iso, S.
Iso, S.
Itoyama, H.
Itoyama, H.
Ivanov, E.
Ivanov, E.
Ivin, M.
Iwamoto, A.
Jegerlehner, F.
Jevicki, A.
Johnston, D.
Johnstone, G.
Jonsson, T.
Kabat, D.
Kac, V.G.
Kaczmarek, O.
Kar, S.
Karsch, F.
Kataev, A.L.
Kawai, H.
Kawai, H.
Kawamoto, N.
Kaya, A.
Kazakov, D.
Kazakov, K.A.
Kazakov, V.
Kemmoku, R.
Khorrami, M.
Khoze, V.V.
Khuri, R.R.
Kihara, H.
Kikukawa, Y.
Kilgore, W.B.
King, S.F.
Kitazawa, Y.
Kitazawa, Y.
669
Klebanov, I.R.
Kneipp, M.A.C.
Kniehl, B.A.
Komori, Y.
Korff, C.
Kostov, I.K.
Kostov, I.K.
Kovchegov, Y.V.
Krmer, M.
Krasnov, K.
Krivonos, S.
Kubo, J.
Kuniba, A.
Laenen, E.
Laermann, E.
Lalak, Z.
Lalak, Z.
Langfeld, K.
Larsen, F.
Lavignac, S.
Lazzarini, S.
Lecheminant, P.
Leibbrandt, G.
Lvai, P.
Levin, E.
Levin, E.
Li, M.
Lifschytz, G.
Lima, E.
Louis, J.
Lozano, C.
Lozano, Y.
Lozano, Y.
L, H.
L, H.
L, H.
Lubicz, V.
Lunghi, E.
Ma, B.-Q.
Maeshima, N.
Maillet, J.M.
Maltoni, F.
Maltoni, F.
Mandal, G.
Mario, M.
Marshakov, A.
Martn, C.P.
Martinelli, G.
Mas, J.
Massar, S.
Mateos, D.
Mathur, S.D.
Matias, J.
670
Matsufuru, H.
Matsuo, T.
Maul, M.
Maxwell, C.J.
McArthur, I.N.
McInnes, B.
Meggiolaro, E.
Mendes, T.
Mihailescu, M.
Minasian, R.
Minces, P.
Miramontes, J.L.
Mirjalili, A.
Mironov, A.
Misiak, M.
Miyamura, O.
Mizoguchi, S.
Mbius, M.
Moch, S.
Mohapatra, R.N.
Mller, P.
Morales, J.F.
Morales, J.F.
Morariu, B.
Morawitz, P.
Moretti, S.
Morozov, A.
Morris, T.R.
Moultaka, G.
Mourad, J.
Mueller, A.H.
Mukhopadhyay, S.
Mller, D.
Multamki, T.
Murakami, K.
Nachtmann, O.
Naculich, S.G.
Nagao, K.
Nakagawa, M.
Nakamura, A.
Nauta, B.J.
Nauta, B.J.
Navarro, I.
Nekrasov, N.A.
Nelson, B.D.
Nierste, U.
Nilles, H.P.
Nishino, A.
Nishino, T.
Noguchi, T.
Nolte, D.R.
B571 (2000) 26
B576 (2000) 313
B577 (2000) 279
B573 (2000) 377
B577 (2000) 263
B571 (2000) 151
B575 (2000) 383
B573 (2000) 652
B574 (2000) 263
B571 (2000) 3
B577 (2000) 88
B576 (2000) 399
B571 (2000) 632
B575 (2000) 504
B576 (2000) 501
B577 (2000) 240
Ohnuki, T.
Okunishi, K.
Oleari, C.
Oleari, C.
Ooguri, H.
Orland, P.
Osborn, H.
Oura, Y.
Ovrut, B.A.
Panagopoulos, H.
Parentani, R.
Parente, G.
Passarino, G.
Pelissetto, A.
Pea-Garay, C.
Peng, D.
Penin, A.A.
Pepe, M.
Perazzi, E.
Prez-Lorenzana, A.
Perkins, W.B.
Pernici, M.
Petkova, V.B.
Petrov, A.Yu.
Philipsen, O.
Pope, C.N.
Pope, C.N.
Pope, C.N.
Pospelov, M.
Pradisi, G.
Pradisi, G.
Pronin, P.I.
Provero, P.
Rabadn, R.
Raciti, M.
Rajagopal, K.
Ramgoolam, S.
Ramgoolam, S.
Rasmussen, J.
Rattazzi, R.
Ray, K.
Reina, C.
Reisz, T.
Rey, S.-J.
Rey, S.-J.
Ridolfi, G.
Ridout, D.
Rigolin, S.
Ritz, A.
Riva, F.
Rivelles, V.O.
Romanino, A.
Rmelsberger, C.
Rothe, H.J.
Rovelli, C.
Rubbia, A.
B572 (2000) 36
B577 (2000) 183
B574 (2000) 571
B572 (2000) 387
B574 (2000) 809
B576 (2000) 347
B575 (2000) 661
B575 (2000) 269
B575 (2000) 401
B574 (2000) 331
B574 (2000) 525
B576 (2000) 313
B571 (2000) 71
B572 (2000) 211
B574 (2000) 43
B571 (2000) 120
B576 (2000) 627
B573 (2000) 275
B573 (2000) 768
B573 (2000) 40
B576 (2000) 578
B572 (2000) 266
B571 (2000) 287
B573 (2000) 434
B573 (2000) 405
B572 (2000) 517
B576 (2000) 430
B575 (2000) 485
B571 (2000) 515
B571 (2000) 71
B571 (2000) 237
B573 (2000) 434
B577 (2000) 461
B575 (2000) 401
B572 (2000) 478
B576 (2000) 660
B577 (2000) 263
B577 (2000) 500
B572 (2000) 95
B573 (2000) 617
B573 (2000) 149
B575 (2000) 285
B575 (1999) 61
B576 (2000) 3
B573 (2000) 87
B574 (2000) 70
B576 (2000) 265
B577 (2000) 3
B573 (2000) 349
B573 (2000) 275
Takagi, T.
Takahashi, K.
Takaishi, T.
Tan, C.-I.
Taormina, A.
671
Tarasov, O.V.
Tatar, R.
Tateo, R.
Tavartkiladze, Z.
Taylor IV, W.
Terao, H.
Terashima, H.
Termonia, P.
Terras, V.
Teschner, J.
Tetradis, N.
Thomas, S.
Tomasiello, A.
Troyan, S.I.
Tsimpis, D.
Tsvelik, A.M.
Tuchin, K.
Uchida, Y.
Ujino, H.
UKQCD Collaborations
Umeda, T.
Uranga, A.M.
Uranga, A.M.
Urban, J.
Vafa, C.
Valent, G.
Valle, J.W.F.
Valle, J.W.F.
Van Raamsdonk, M.
Van Weert, Ch.G.
Veretin, O.L.
Vermaseren, J.A.M.
Vicari, E.
Vicari, E.
Vilja, I.
Vitev, I.
Wadati, M.
Wadia, S.R.
Wagner, C.E.M.
Watts, G.M.T.
Weisz, P.
Wells, J.D.
West, P.C.
Wiese, U.-J.
Wittig, H.
Wolff, U.
Yamada, A.
Yamada, Y.
Yang, J.-J.
Yang, S.-K.
Yogendran, K.P.
672
Yoneya, T.
Youm, D.
Youm, D.
Youm, D.
Youm, D.
Youm, D.
Yue, R.
Zaffaroni, A.
Zagermann, M.
Zampa, A.
Zoupanos, G.
Zraek, M.
Zucchini, R.
Zumino, B.
Zwirner, F.