91-25

October 1, 2018

Ward Identities of Liouville Gravity

†
coupled to Minimal Conformal Matter
arXiv:hep-th/9111024v1 12 Nov 1991

Ken-ji Hamada

Institute of Physics, University of Tokyo

Komaba, Meguro-ku, Tokyo, 153, Japan

ABSTRACT

The Ward identities of the Liouville gravity coupled to the minimal conformal
matter are investigated. We introduce the pseudo-null fields and the generalized
equations of motion, which are classified into series of the Liouville charges. These
series have something to do with the W and Virasoro constraints. The pseudo-null
fields have non-trivial contributions at the boundaries of the moduli space. We
explicitly evaluate the several boundary contributions. Then the structures similar
to the W and the Virasoro constraints appearing in the topological and the matrix
methods are realized. Although our Ward identities have some different features
from the other methods, the solutions of the identities are consistent to the matrix
model results.

† Talk given at YITP Workshop on Developments in Strings and Field Theories, Kyoto,
Japan, Sept. 9-12 1991.
 1. Introduction

The two dimensional quantum gravity has been studied as a toy model of the
four dimensional quantum gravity. It is important to discuss what are common
structures independent of the dimension. Recently there are remarcable develop-
ments in this direction [1, 2]. The problem of non linearity is the one of the most
important issues of gravity. In two dimension the quantum gravity becomes exactly
solvable [3, 4] and the non linear structures such as the string equations or the W
and the Virasoro constraints [5, 6, 7, 8] are realized in the approaches of matrix and
topological models. To make a comprehension deeper, we reexamine these struc-
tures in terms of the Liouville gravity [9, 10]. Really the non linear structures are
directly related to the nature of the Hilbert space [1], the factorization of ampli-
tudes [2, 1] and so on. Then it is important to discuss whether we have to give up
the superposition principle or not. As we will see later, at least in two dimension,
it appears that there is no need to abandon it.

In this talk we will discuss how the non linear structures appear in the Liouville
gravity coupled to the minimal conformal field theory (CET). The quantum Liou-
ville theory has the different features from the standard quantum field theory. The
theory has two kinds of states [1]: microscopic and macroscopic ones. Microscopic
states, which correspond to a branch of the local operators of Distler-Kawai [11],
are dominated at small area of surface and are non-normalizable, while macroscopic
states are normalizable and correspond to macroscopic loops in surface. The exis-
tence of two kinds of states is important when we discuss the non linear structures
of the Liouville gravity.

In Sect.2 we summarize the results of quantum Liouville theory. Here the

macroscopic states are introduced as the Hilbert space of the Liouville theory.
The intermediate states of amplitudes are expanded by the macroscopic states. It
is natural because these states include informations of fluctuating surface and are
normalizable. The factorization of matter part is given by BPZ theory [12] in which
the metric on the space of primary fields is diagonal. Then we have a question how

2
the non-trivial metric of scaling operators appearing in the topological and the
matrix approaches, or the structures of W and Virasoro constraints, are realized
from the diagonal metric of BPZ. This matter is discussed in Sect.4 and 5.

In Sect.3 we introduce the pseudo-null fields [9, 10], where “pseudo”means that
they become exact null fields for the free theory (the cosmological constant µ = 0
). The pseudo-null fields can be rewritten in the form of BRST commutator. But
it does not mean that they are trivial. We must take into account the measure
of moduli space. Then the BRST operator picks up the non-trivial contributions
from various singular boundaries of moduli space. Thus the pseudo-null fields are
essentially non zero and should satisfy the non-trivial relations, which we could
see just as a generalization of the equation of motion in quantum Liouville theory
coupled to the minimal conformal matter. The pseudo-null fields can be classified
in the m − 1 series of the Liouville charge, where the central charge of minimal
matter is cm = 1 − 6/m(m + 1).

In Sect.4 and 5 we explicitly evaluate the boundary contributions and derive

the various Ward identities of two dimensional gravity. Then the factorization
property discussed in Sect.2 and the fusion rule of CFT are used. The derived
equations have similar structures to some of the W and Virasoro constraints L0 ,
L1 and W−1 . For the Ising model we can discuss in detail and derive a closed set
of the Ward identities [10]. The solutions of the identities, which are summarized
in appendix, are consistent to the matrix model results.

Sect.6 is devoted to conclusions and discussions. We consider the similarities

and the differences between the Liouville gravity and the other methods. We also
discuss another BRST invariant fields found by Lian-Zuckerman [13]. Since these
fields have the non-standard ghost number, there are some difficulties when we
consider the correlation functions of these fields.

3
 2. Quantum Liouville Theory

The two dimensional quantum gravity is defined through the functional inte-
grations over the metric tensor of two dimensional surface gab and the matter field
m. The partition function is

 
µ
X Z Z
−χ 2
p
Z= κ [dgab ][dm] exp − d z |gab | − Sm , (2.1)
χ
2π
Mχ

where Sm is a matter action. χ is the Euler number of surfaces: χ = 2 − 2g and κ

is the string coupling constant. We use the conformal gauge gab = eγφ ĝab (t̂), where
γ is a parameter given below and ĝab (t̂) is a background metric parametrized by
the moduli t̂. We choose the locally flat background metric. φ is well-known as
the Liouville field. After fixing the reparametrization invariance, one can rewrite
the two dimensioal quantum gravity as

X
<O>= < O >g
g
X Z Z (2.2)
−χ 2 −Sm −Sφ −Sgh
= κ d t̂ [dmdφdbdc]µ(b)Oe ,
g

where O is some operator. Sφ is the Liouville action

1 √
Z
Sφ = d2 z ĝ(ĝ ab ∂a φ∂b φ + QR̂φ + 4µeγφ ) (2.3)
8π

and Sgh is the ghost action. When the matter system is the minimal conformal
field theory with central charge cm = 1 − 6/m(m + 1), the parameters Q and γ are
defined as
4m + 2 2m
Q= p , γ=p . (2.4)
2m(m + 1) 2m(m + 1)

µ(b) is the measure of the moduli space. n is the number of operators.

4
 The Liouville action has the following scaling property

Q
Sφ → Sφ + χδ (2.5)
2

when we change the field φ and the cosmological constant µ as φ → φ + δ, µ →

µe−γδ . The constant shift of eq.(2.5) can be renormalized into the string coupling
κ.

2.1. Canonical Quantization of Liouville Theory

To discuss the Hilbert space of quantum Liouville theory we use the canonical
quantization. We first change the variable from the plane coordinate z to the
cylinder one w = τ +iσ; z = ew . After Wick rotating τ → it, we reach the Liouville
theory in Minkowski space. Then we can set up the equal-time commutation
relation

[φ(σ, t), Π(σ ′ , t)] = iδ(σ − σ ′ ) , (2.6)

1
where Π(σ, t) = 4π ∂t φ(σ, t) is the conjugate momentum. As a normal ordering, we
adopt the free field one as used by Curtright-Thorn [14].

The energy-momentum tensor of the Liouville theory Tφ±± ≡ 12 (Tφ00 ± Tφ11 ) is

given by
1 Q µ Q2
Tφ±± = (4πΠ ± φ′ )2 ∓ (4πΠ ± φ′ )′ + eγφ + . (2.7)
8 4 2 8

The Hamiltonian Hφ = L◦ + L◦ is

Z2π i Q2
1 dσ h
Hφ = (4πΠ(σ))2 + φ′ (σ)2 + 4µeγφ(σ) + . (2.8)
4 2π 4
0

The prime means the derivative with respect to σ. Tφ±± satisfy the Virasoro algebra
2
only if the parameters Q and γ satisfy the relation Q = γ + γ. This relation is
same as that derived by Distler-Kawai [11]. The central charge is found to be

5
cφ = 1 + 3Q2 . Also it can be seen that Tφ±± depends only on t ± σ. The conformal
weight of the operator eαφ is shown to be hα = 12 (αQ − α2 ). Thus in spite of the
interaction the values of cφ and hα are as if φ is a free field. The differece is, as
Q
we will see below, that we must take the branch α < 2 out of two solutions of
hα = 12 (αQ − α2 ).

2.2. Hilbert Space of Liouville Theory

To manage the interaction term, we simplify discussions by considering the

R 2π
mini-superspace approximation i.e. eγφ(σ) → eγφ◦ , where φ◦ = 0 dσφ(σ)/2π is
the mean value of φ. In this approximation, the Hamiltonian Hφ = Lφ◦ + L̄φ◦ is
simply
∂2 γφ◦ Q2
Hφ = − + µe + + N + N̄ (2.9)
∂φ2◦ 4

where N and N̄ are the left and right-moving oscillator levels. In the case of
conformal matter c ≤ 1, however, the oscillator modes are canceled out by the
oscillator modes of the ghost and the matter parts. In fact, when we consider
the partition functions on the torus, the Dedekint η-functions which come from
the determinants of oscillator modes are canceled out and only the zero mode
contributions survive [1, 15, 16]. The derived results are exactly same as those of
the matrix models. So in the following we do not attend to the oscillator modes.

The normalizable wave function for N = N̄ = 0 is given by using the modified

Bessel function as

Q2
 
2
Hφ Ψp (l) = p + Ψp (l) ,
4
 1 (2.10)
2 2π 2 √
Ψp (l) = psinh p K2ip/γ (2 µl/γ)
γ γ

1
for real p, where l = e 2 γφ◦ . Since Ψ−p = Ψp , one can take the region p > 0. To
obtain the wave function we use the boundary condition Ψp ∼ sinpφ◦ (p > 0) at

6
the limit l → 0, which comes from the fact that the incoming wave completely
reflect by the potential eγφ◦ . Note that there is no p = 0 ground state. Since the
ground state is not included in the Hilbert space, we cannot define the states by
acting the operators on the ground state as in the standard conformal field theory.

Now we define the state/operator identification formally by using the path

integral method just like the Hartle-Hawking wave function
Z
Ψp (l) = [dφ]ψp (φ)e−S , (2.11)
D

where D is the disk with boundary |z| = 1 and the boundary value of φ is fixed.
The operator ψp (φ) is located at the centre of the disk z = 0, which has the
Q2
conformal weight h = h̄ = 12 p2 + 8 . Such a operator is given by

Q Q
ψp (φ) = µip/γ e(ip+ 2 )φ + µ−ip/γ e(−ip+ 2 )φ . (2.12)

In general the state corresponding to the operator eαφ is constructed by replac-

ing the operator ψp into eαφ in (2.11). Let us consider the case that α is real, which
corresponds to the operator of Distler-Kawai. In the mini-superspacce approxima-
Q
tion this state behaves like Ψα = e(α− 2 )φ◦ at φ◦ → −∞. If we adopt the branch
Q
α< 2 as a solution of h = 21 (αQ − α2 ), the state diverges at φ◦ → −∞. While for
Q
the branch α > 2, Ψα vanishes and gives no contributions. Therefore we should
Q
take the branch α < 2. The limit φ◦ → −∞ corresponds to the small area of the
Q
surface gab = eγφ ĝab . So the state with α < 2 is peaked on the small area region.
We call this type of state “microscopic state”. This state is non-normalizable.
On the other hand, the normalizable eigenstate of the Liouville Hamiltonian Ψp
Q
corresponds to α = ±ip + 2 and oscilates at the small area region. We call it
“macroscopic state”.

7
 2.3. Factorization of Amplitudes

Q
Let us consider how the correlation function < i Oαi >Σ on the Riemann
surface Σ with genus g factorizes into the two surfaces Σ1 with genus g1 and Σ2
with g2 . Here Oαi is the operator with the Liouville charge αi . The total Liouville
P P
charges of each part are i∈Σ1 αi and i∈Σ2 αi , respectively. If they satisfy the
normalizability conditions

X Q X Q
αi + (2g1 − 1) > 0 , αi + (2g2 − 1) > 0 , (2.13)
2 2
i∈Σ1 i∈Σ2

the intermediate states are expanded by the normalizable macroscopic states. We

normalize the macroscopic state as

¯ 2πC(p2 )
< c̄cψp Φ∆ (w̃ = 0)(∂c̄)(∂c)c̄cψq Φ∆′ (w = 0) >g=0 = δ(p − q)δ∆,∆′ , (2.14)
κ2

where the two frames w and w̃ are identified as w w̃ = 1. Φ∆ is the primary field
of minimal CFT. Then the factorization [2, 1] is

∞
X Z dp 1
< O >Σ = ¯
< O1 (∂c̄)(∂c)c̄cψp Φ∆ (w = 0) >Σ1
2π C(p2 ) (2.15)
∆ −∞

× < c̄cψp Φ∆ (w̃ = 0)O2 >Σ2 ,

where we neglect the oscillator modes which do not contribute to the boundary
terms.

8
 3. Pseudo-Null Fields and
Generalized Equations of Motion

Consider the Liouville system as CFT with central charge cφ = 25+6/m(m+1).

There are several null fields, for example

χφ1,1 = Lφ−1 · 1 ,

m + 1 φ2
 √ m √ m
φ φ − 2(m+1) φ (1,2)
χ1,2 = L−2 + L−1 · e ≡ D−2 · e− 2(m+1) φ , (3.1)
m

m
 √ m+1 √ m+1
φ φ φ2 (2,1)
χ2,1 = L−2 + L−1 · e− 2m φ ≡ D−2 · e− 2m φ .
m+1

In general there exist the null field χφp,q at the level pq of the primary field eβp,q φ
with conformal weight hp,q for 1 ≤ p ≤ m − 1, 1 ≤ q ≤ m

1 n 2 2
o
hp,q =− p(m + 1) + qm − (2m + 1) ,
4m(m + 1)
1   (3.2)
βp,q =p 2m + 1 − p(m + 1) − qm .
2m(m + 1)

(p,q) (p,q)
We write the null field as χφp,q = D−pq · eβp,q φ , where D−pq is the proper combina-
tion of Lφ−n (n > 1) with level pq. For example see eq.(3.1).

Now we construct the pseudo-null fields [9, 10]. The first non-trivial one is

φ
N1,1 = Lφ−1 L−1 · φ . (3.3)

The dot “·”denotes that the contour surrounds the operator located on the r.h.s. of
it. From the equation of motion N1,1 is proportional to the cosmological constant
operator
γ γφ
N1,1 = µe . (3.4)
2

In the limit µ → 0, the r.h.s. of eq.(3.4) vanishes. So we call it a pseudo-null field.

9
Note that N1,1 field is constructed by modifying the trivial null field χφ1,1 as

∂ φ
N1,1 = (Lφ L · eβφ )|β=0 . (3.5)
∂β −1 −1

In the same way we can construct the pseudo-null field corresponding to the null
field χφp,q as
∂ (p,q) (p,q)
Np,q = (D−pq D −pq · eβφ Φp,q )|β=βp,q
∂β (3.6)
(p,q) (p,q) βp,q φ
= D−pq D −pq · φe Φp,q ) .

Here, to make the physical operator, we combine the Liouville field and the matter
field. Φp,q is the primary field of matter system with conformal dimension

[p(m + 1) − qm]2 − 1
∆p,q = . (3.7)
4m(m + 1)

In the analogy of the equation of motion, it is expected that Np,q is proportional

to the dressed physical field of Φp,q

Np,q = Cp,q µxp,q eαp,q φ Φp,q , (3.8)

where
2m + 1 − |p(m + 1) − qm|
αp,q = p ,
2m(m + 1) (3.9)
1
xp,q = [p(m + 1) + qm − |p(m + 1) − qm|] .
2m

Cp,q is the proportional constant, which have to be determined later. The exponent
of µ is determined from the scaling property of the Liouville action (2.5).

Since there is the relation hp,q = hm+p,m+1−q + (m + p)(m + 1 − q), the null
state itself contains a null state. Therefore we can construct the family of null

10
physical states satisfying the physical state condition

1 = ∆p,q + hp,q + pq
= ∆p,q + hm+p,m+1−q + (m + p)(m + 1 − q) + pq
(3.10)
= ∆p,q + h2m+p,q + (2m + p)q + (m + p)(m + 1 − q) + pq
···

The pseudo-null field Np,q corresponds to the relation of the first line. From the
second relation we obtain

(p,q) (p,q) (m+p,m+1−q) (m+p,m+1−q)

Mp,q = D−pq D −pq D−(m+p)(m+1−q)D−(m+p)(m+1−q) · φeβm+p,m+1−q φ Φp,q , (3.11)

which is also proportional to the dressed physical field of Φp,q . In the following we
write the dressed physical field of Φp,q as Op,q = c̄ceαp,q φ Φp,q , where we combine
the ghost field. We also define Ñp,q = c̄cNp,q and M̃p,q = c̄cMp,q . Then these fields
are summarized in the table

n = 0 1 2 ...
βn1 : O1 − Ñ1,1 ... Ñ1,m − M̃1,m M̃1,m−1
βn2 : O2 O2,1 − Ñ2,1 ... Ñ2,m − M̃2,m
.. .. .. .. ..
. . . . .
βnm−1 : Om−1 ... ... Om−1,1 − Ñm−1,1 ... Ñm−1,m
(3.12)
Here Op = Op,p (p = 1, · · · , m − 1). The Liouville charges βnp (p = 1, · · · , m − 1)
are given by

(n + p − 3)m + p − 1
βnp = − p (n 6= p + 1 mod m + 1) (3.13)
2m(m + 1)

It is expected that the series βn1 have something to do with the Virasoro constraints
and βni (i = 2, · · · , m − 1) with W constraints. In the following section we dicuss
this correspondence.

11
 The pseudo-null field can be rewritten in the form of BRST commutator Ñp,q =
QB · Wp,q , for example W1,1 = Qs b−1 b̄−1 · c̄cφ, where QB = Qs + Q̄s is the BRST
charge. Thus the generalized equation of motion (3.8) can be rewritten as

Cp,q µxp,q Op,q = QB · Wp,q , (3.14)

In the following section we consider the Ward identity which is given by inserting
the identity (3.14) into the correlator of scaling operators.

4. Ward Identities of 2D Quantum Gravity

coupled to the Ising Model

Let us first discuss the case of the Ising model [10]. We derive various Ward
idetities obtained by inserting the pseudo-null field relations (3.14) into the corre-
lation functions: < O >g =< n1 OI n2 Oσ n3 Oε >g , where OI = O1,1 is the
Q Q Q

cosmological constant operator. Oσ = O1,2 and Oε = O2,1 are the dressed spin and
energy operators.

4.1. Equation of Motion

The pseudo-null field relation of N1,1 is nothing but the equation of motion. In
this subsection we treat this operator. As evaluating the boundary contributions,
there is a problem. The operator Wp,q of eq.(3.14) is in general not well-defined
on the moduli space, or it is not annihilated by the action of b◦ and b̄◦ . Therefore
we introduce the well-defined operator Xp,q satisfying the condition b◦ · Xp,q =
b̄◦ · Xp,q = 0. The operator X1,1 is defined by modifying W1,1 slightly as X1,1 =
Lφ−1 b̄−1 · c̄cφ. Then the identity (3.14) for the Ising case becomes

3
√ µOI = QB · XI + KI . (4.1)
2 6

where XI = X1,1 and KI is called “ghost pieces”: KI = − √76 c−1 b̄−1 · c̄c, which has
the non-standard asymmetric ghost number.

12
 The Ward identity we discuss is

3
√ µ < OI O >g =< QB · XI O >g + < KI O >g . (4.2)
2 6

The correlation function with the ghost piece KI vanishes because it has the non-
standard ghost number. The first term on r.h.s. is evaluated as follows. Taking
into account the moduli and the measure for the position z = z1 of QB · XI , we
obtain

1
Z I
2 B
d z1 b−1 b̄−1 Q · XI (z1 )Oα (zi ) = − dz1 ∂φ(z1 )Oα (zi ) . (4.3)
2i
|z1 −zi |=ǫ

Here the BRST algebra {QB , b−1 } = L−1 = ∂ is used. The r.h.s. of eq.(4.3)
becomes a total derivative with respect to the moduli so that finite contributions
will come from the boundary of moduli space. In this case the relevant boundary
is where QB · XI (z1 ) approaches other operators Oα (zi ). To evaluate the bound-
ary term the small cut-off ǫ is introduced and, after the calculation, we take the
limit ǫ → 0. The contributions come from the singularity of operator product
∂φ(z)Oα (zi ). The leading singularity comes from the free field (µ = 0) OPE. As
the next leading singularity a µ-dependent term appear, but, in this case, does not
contribute because the power of singularity is too small to give the finite value at
the limit ǫ → 0.

We also have to analyze curvatures carefully. One can choose a metric which
is almost flat except for delta function singularities at the positions of scaling
√
operators; gR = 4π i νi δ 2 (z − zi ), so that i νi = χ, where χ is the Euler
P P

number of two dimensional surface with genus g: χ = 2 − 2g. We do not assign

the curvature to QB · XI . To use free field operator products it is necessary to
smooth out the curvature singularity in the neighborhood of the position of the
operator. This is done by the coordinate transformation: z − zi = (z ′ − zi′ )1−νi , or
dz ′ dz̄ ′
dzdz̄ ∼ |z ′ −zi′ |2νi . In the smooth z ′ -frame we can freely use the operator products.

13
 After evaluating OPE, we finally obtain the expression

n1 n2 n3 nY
1 +1 n2 n3
∂ Y Y Y µ Y Y
µ < OI Oσ O ε >g = − < OI Oσ O ε >g
∂µ 2π
  Y n1 n2 n3
(4.4)
5 1 7 Y Y
= − n1 + n2 + n3 − χ < OI Oσ O ε >g .
6 3 6

†
Thus the µ-dependence of the correlation functions is

n1 n2 n3
Y Y Y 7 5 1
< OI Oσ Oε >g = Zng1 ,n2 ,n3 µ 6 χ−n1 − 6 n2 − 3 n3 . (4.5)

Note that, since the path integral of Liouville field diverges, the derivation can not
Pn1 +n2 +n3
apply for the case i=1 αi − Q2 χ < 0, where αi is the charge of the exponential
operator. Therefore in this case we have to define the correlation function by using
d
the differential equation such as, for example, −2π dµ < Oσ Oσ >◦ =< Oσ Oσ OI >◦ .

4.2. Ward Identities corresponding to Virasoro Constraints

In this section we consider the Ward identities given by inserting the pseudo-
null field N1,2 = Nσ . In the following we consider the Ward identity

Y Y
Ca µ4/3 < Oσ Oα >g =< QB · Xσ Oα >g +ghost term, (4.6)
α α

where Oα ’s are only the “gravitational” primary fields OI and Oσ . Xσ is the well-
σ B σ 3
− 2√ φ
defined operator on moduli space: Xσ = D−2 −2 · c̄cφe
6 σ. To evaluate the
r.h.s. of eq.(4.6) we must take into acount the 3 types of boundaries.

† Zng=0
1 ,n2 ,n3
can be directly calculated by using CFT methods [17]

14
 As discussed in Sect.4.1 the first boundary arises from that Xσ approaches
other operators Oα . The relevant operator product is
Z
d2 z1 b−1 b̄−1 QB · Xσ (z1 )Oα(zi )
|z1 −zi |≥ǫ
(4.7)
1
I
=− (dz1 b−1 + dz̄1 b̄−1 ) · Xσ (z1 )Oα (zi ) .
2i
|z1 −zi |=ǫ

One can see that from the power of OPE singularities between b̄−1 · Xσ and Oα the
integral of dz̄ vanishes. The integral of dz gives the finite contributions for α = σ.
Then we get
1 5π
I
− dzb−1 · Xσ (z1 )Oσ (zi ) = √ Oε (zi ) . (4.8)
2i 9 6

In the interacting theory there will be the next leading OPE singularities which
depend on the cosmological constant µ. If one uses the free field OPE, one should
add the µ-dependent term, or the operators Oα (α = I, σ) behave in the neighbor-
hood of boundaries as follows
3 √4 φ
φ
+ ηI µ1/3 e
√
OI → c̄c(e 6 6 ),
5 9
(4.9)
√ φ 2/3 √ φ
Oσ → c̄c(e 2 6 + ησ µ e 2 6 ).

The charge of the Liouville mode and the power of µ are determined from the
ristriction of conformal dimension and the scaling symmetry of the Liouville action.
The phases ηI and ησ are determined by the consistency.

Next we discuss the boundary-2 where the field Xσ approaches the pinched
point which divides the surface into two pieces. Then the factorization discussed
in Sect.2 is important.
∞ Z 2
X Z dp 1 d z1 d2 q
Z
B B
< Q · Xσ O > Σ = < O1 b◦ b̄◦ Q · Xσ (z1 ) b◦ b̄◦
2π C(p2 ) z1 z̄1 q q̄
∆ −∞ |q|≥ǫ
¯
× q L◦ q̄ L̄◦ (∂c̄)(∂c)c̄cψp Φ∆ (w = 0) >Σ1 < c̄cψ−p Φ∆ (w̃ = 0)O2 >Σ2 ,
(4.10)
where Φ∆ = I, σ, ε. The coordinates w and w̃ are defined in the neighborhood

15
of the nodes of Σ1 and Σ2 . These are identified as w w̃ = q, where w = z and q
is the moduli that determines the shape of the pinch. By this identification the
normalization (2.14) changes so that the operator q L◦ q̄ L̄◦ is inserted. We explicitly
introduce the measure of moduli for z1 and q. The operator b◦ is defined by the
contour integral around w = 0. Using the BRST algebra one can rewrite the
expression into the derivatives with respect to the moduli q.

The boundary contributions come from the limit that z1 and q approach zero
simultaneously. In this limit the integrand is highly peaked and we can evaluate
the integral by the saddle point method. The final result becomes

Y X n Y Y
B
< Q · X1,2 Oj >b♯2
g ≃ < Oσ O α >g 1 < O I O j >g 2
j∈S S=X∪Y α∈X α∈Y
g=g1 +g2
Y Y o
+ < OI O α >g 1 < O σ O j >g 2 ,
α∈X α∈Y
(4.11)
where S = X ∪ Y means that the sum is over the posible factorizations satisfying
the conditions (2.13). To derive these structures we use the fusion rule of the
minimal CFT. Here we neglect the curvature contributions. Note that the metric
we first introduceed (4.10) is the diagonal one, but after evaluating the boundary
the metric structure changes to the asymmetric form and Oε disappears such as
the topological and the matrix models. This structure really corresponds to the
L1 Virasoro constraints.

The boundary-3 is a kind of boundary-2, where a handle is pinched. In this

case the surface is not divided by the pinching. Thus we obtain

Y Y
< QB · X1,2 Oj >b♯3
g ≃< OI Oσ Oα >g−1 . (4.12)
j∈S α

Q Qn
Let us consider the Ward identity with operator insertion Oα = Oσ OI .

16
In general genus we get (n ≥ 3 for g = 0)
n
5 Y
0 = √ π < Oε O I >g
9 6
g X n  
1 2
 
X n−2  5 n−2  5 1 2
− 2λ χ1 − + χ2 −
k 12 3 k − 2 12 2
g1 =0 k=0
    
n−2  5 1  5 1 25 n − 2 X 2 (4.13)
−2 χ1 − χ2 − + νi
k − 1 12 3 12 2 144 k − 1
i
k
Y n−k+1
Y
× < Oσ Oσ O I >g 1 < O I >g 2
n+1
λ Y
− < Oσ Oσ OI >g−1 +ghost term.
72
Here the curvature singularities are considered, which are assigned to n OI op-
† 8π 2
erators . We determine the unknown constants except λ = C(p2 =−1/6) from the
consistency of the Ward identities on the sphere. In the end the next leading
terms of the boundary-1 and Ca -term are absorbed in the factorization form. In
the above expression it appears as if there were no restrictions like the inequalities
(2.13).

In general genus we need the contributions of the ghost term. Naively it does
not contribute because of the asymmetry of the ghost number. However, for g ≥ 1,
there will be the non-zero contribution when we evaluate the curvature singular-
ities. To calculate the curvature contributions we used the transformation from
the singular frame to the non-singular frame. Then the mapping analytic in the
moduli was used. This is correct on the sphere, but for g ≥ 1 one can not take such
a mapping globally to remove the curvature singularities. So there are posibilities
that the measure makes up for the asymmetry and the ghost term contributes.
Really eq.(4.13) is inconsistent if there are no contributions of the ghost term. Ex-
ceptional case is g = 1, then we can choose the flat metric where all νi ’s are zero.
In this case the ghost term will vanish.

† Although the expression changes by how to assign the curvatures, the final results are
independent of the assignments. Furthermore as a consistency check we can see that the
expression (4.13) is indeed independent of the value of i νi2 for n ≥ 3, g = 0.
P

17
 We also cosider the Ward identity with n Oσ operators. The curvature singu-
larities are assigned to n Oσ operators. Then we obtain

n−1
5  5 X 2 Y
0= √ π n−χ− νi < Oε O σ >g
9 6 4
i
g n
X X n − 2 5

1 2
 
n−2  5 1 2
− 2λ χ1 − + χ2 −
k 12 3 k − 2 12 2
g1 =0 k=0
    
n−2 5 1  5 1 25 n − 2 X 2 (4.14)
−2 χ1 − χ2 − + νi
k − 1 12 3 12 2 144 k − 1
i
k+1
Y n−k
Y
×< O σ >g 1 < O I O σ >g 2
n+1
λ Y
− < OI Oσ >g−1 +ghost term.
72

4.3. Ward Identities corresponding to W constraints

In this section we consider the case of the pseudo-null field N2,1 = Nε :

Y Y
C1 µ < O ε Oα >g =< QB · Xε Oα >g +ghost term, (4.15)
α α

where α = I, σ. The operator Xε = X2,1 is defined as in the previous section by

ε B ε − √26 φ
Xε = D−2 −2 · c̄cφe ε. In this case we have to take into account the 4 types
of boundaries.

We do not repeat the calculations in detail. The second and third boundary
contributions have the following form

X Y Y Y
b♯2, 3 ≃ < Oσ O α >g 1 < O σ O j >g 2 + < O σ O σ Oα >g−1 .
S=X∪Y α∈X α∈Y α∈S
g=g1 +g2

(4.16)
The operator Oε does not appear on the nodes. The metric structure really corre-
sponds to the W−1 constraint.

18
 Furthermore we must take into account the boundary-4 that Xε and two Oσ ’s
approach at a point simultaneously. In fact one can easily see that, if there is no
boundary contribution of this type, the Ward identity becomes inconsistent. We do
not know how to evaluate this boundary directly. Instead we assume the following
form

QB · Xε Oσ Oσ → OI . (4.17)

Q Qn
Let us consider the Ward identity of the type: α Oα = Oσ Oσ OI . If the
curvatures are assigned only to n cosmological constant operators, we obtain the
following Ward identity

n−1
2π  ∂ 11 55 X 2  Y
0 =√ µ +n+ χ− νi < Oε Oσ Oσ O I >g
6 ∂µ 18 18
i
g n    
1 XX n h n−2
+ λ − 55 (1 − χ1 )2
36 k k
g1 =0 k=0
      
n−2 n−2 2 n − 2 X 2i
−2 (1 − χ1 )(1 − χ2 ) + (1 − χ2 ) + νi
k−1 k−2 k−1
i
k
Y n−k
Y
× < Oσ Oσ O I >g 1 < O σ O σ O I >g 2
4 n n+1
1 Y Y 32 2 Y
+ λ< Oσ OI >g−1 + π C4 < OI >g +ghost terms
72 3
(4.18)
Q Qn
For the case with the operator insertions α Oα = Oσ , if the curvatures are
assigned only to n Oσ , we get

19
 n
1 Y
0 = √ µ < Oε O σ >g
6
g n   hn − 2
1 XX n
− λ − 55 (1 − χ1 )2
72 k k
g1 =0 k=0
      
n−2 n−2 2 n − 2 X 2i
−2 (1 − χ1 )(1 − χ2 ) + (1 − χ2 ) + νi
k−1 k−2 k−1
i
k+1
Y n−k+1
Y
×< O σ >g 1 < O σ >g 2
n+2 
1 Y 32 2 n(n − 1) 77 11
− λ< Oσ >g−1 − π C4 − (n − 1)χ + χ2
72 3 2 60 24
X  n−2
11 2
Y
+ (n − 3) νi < OI Oσ >g +ghost terms.
48
i
(4.19)

Unfortunately we cannot determine the constants λ and C4 by the consistency.

The determination of these values and the ghost terms remaines as future problems.
We could determine these values by using the results of ref.18, which are given by
λ = √2 π and C4 = 5 1
√ . Then we can derive several correlation functions on
6 6 6π
the sphere and the torus, which are consistent to the results of the two matrix
model [19].

5. Ward Identities for Minimal CFT

In the previous section we obtain a closed set of Ward identities for the case of
the Ising model. For the general minimal series it is difficult to derive a closed set
of Ward identities because more complicated boundaries contribute and also the
number of primary fields increases. So we only concentrate on the pseudo-null field
N1,2 . Then it is expected that the structures like L1 equation, or metric on the
space of scaling operators appearing in the matrix and the topological methods,
are realized.

The Ward identity is given by substituting the relation (3.14) with (p, q) =
(1, 2) into correlation functions. For simplicity we consider the correlation func-

20
tion with the operators Oj ≡ Oj,j (j = 1, · · · , m − 1), which corresponds to the
gravitational primary fields. Then
m+1 Y Y
C1,2 µ m < O1,2 Oj >g =< QB · W1,2 O j >g . (5.1)
j∈S j∈S

The operator O1,2 = Om−1,m−1 corresponds to the first gravitational primary

Om−1. We evaluate the boundary contributions of the r.h.s. of eq.(5.1). The
contributions of the first boundaries are given by
Y X Y
< QB · W1,2 Oj >b♯1
g ≃ < Ok,k−1 Oj >g + next leading terms, (5.2)
j∈S k∈S j(6=k)
(k6=1)

where Ok,k−1 corresponds to the gravitational descendant σ1 (Ok ) (k = 2, · · · , m −

1). Here we neglect the curvature contributions and the normalization of scaling
operators. The contributions from the boundaries-2 and -3 are given by

Y m−1
X Y
B
< Q · W1,2 Oj >b♯2,3
g ≃ < Ok Om−k Oj >g−1
j∈S k=1 j∈S
X Y Y 
+ < Ok Oj >g1 < Om−k O j >g 2 ,
S=X∪Y j∈X j∈Y
g=g1 +g2

(5.3)
where S = X ∪ Y means that the sum is over the posible factorizations satisfying
the conditions (2.13). To derive these structures we use the fusion rule of the
minimal CFT. As discussed in the case of the Ising model, the l.h.s. of eq.(5.1)
and the next leading terms of eq.(5.2) are used to complete the factorization form
of eq.(5.3). Then we finally obtain the following structure
X Y
0≃ < Ok,k−1 O j >g
k∈S j(6=k)
(k6=1)

m−1
X Y
+ < Ok Om−k Oj >g−1 (5.4)
k=1 j∈S
X Y Y 
+ < Ok Oj >g1 < Om−k O j >g 2 ,
S=X∪Y j∈X j∈Y
g=g1 +g2

21
The equation really has the similar structures to the L1 Virasoro constraint. The
metric on the space of scaling operators appearing in the other methods are re-
alized explicitly. It is essentially determined by the fusion rule of CFT and the
conservation of the total Liouville charge.

It probably needs to discuss all Np,q fields to determine all correlation functions
of Op,q . It appears, however, that Mp,q and the others do not give the essentially
new informations as far as one considers only the correlation functions of the op-
erators in the Kac table Op,q .

The difference from the other methods is that the l.h.s. of eq.(5.4) vanishes
and the identity is closed only by the operators Op,q in the Kac table, while
L1 equation appearing in the other methods has the structure that the l.h.s.
of eq.(5.4) is the correlation function with the gravitational descendant σ3 (O1 ):
Q
< σ3 (O1 ) j∈S Oj >g .

6. Conclusions and Discussions

We have discussed the Ward identities of the Liouville gravity coupled to the
minimal CFT. We found the series of the pseudo-null fields and the generalized
equations of motion. The various Ward identities given by inserting these equations
into correlation functions are derived. Especially for the Ising model we give a
closed set of the identities. Then the several interesting structures similar to the
matrix and the topological methods appeared. The identities we discussed have the
similar structures to the W and Virasoro constraints; L◦ , L1 and W−1 for the Ising
model and L1 for the general case. Really the boundary-2 structure has the same
metric on the space of scaling operators as that in the other two methods. Also
the Ward identities corresponding to W−1 constraints require the new boundary
different from the Virasoro constraints as in the other methods. It should be
stressed that these non-linear structures are derived from the factorization (2.15),
where the intermediate states are expanded by the normalizable Hilbert states of
the Liouville and CFT. Since the states are defined by using the path integral just

22
like the Hartle-Hawking states, it might be expected that we also need not abandon
the superposition principle in higher dimensional quantum gravity.

The differences from the other methods are related to the problem of the grav-
itational descendants in the Liouville gravity. Our Ward identities are closed only
by the dressed operators corresponding to the Kac table of CFT and just have the
same form as the W and Virasoro constraints given by setting the gravitational
descendants outside of the Kac table to zero. The pseudo-null fields have indeed
the similar properties to the gravitational descendants outside of the Kac table.
These properties are ruled by the Liouville charges and the fusion rule of CFT.
Therefore the gravitational descendants should have the same Liouville charge and
matter field as that listed in table (3.12).

Lian and Zuckerman [13] found a series of BRST invariant states with these
properties as a candidate of gravitational descendants. For example one can easily
construct the states with the same Liouville charge and matter field as that of N1,2
and N2,1
 
m+1 φ
R1,2 = b−2 c1 + (L−1 − L−1 ) · eβ1,2 φ Φ1,2 ,
m
m
  (6.1)
m φ
R2,1 = b−2 c1 + (L − L−1 ) · eβ2,1 φ Φ2,1 .
m
m + 1 −1

The BRST invariance of these states are proved by using the null states of the
Liouville and the matter sectors. Note that these states have zero ghost number.
If the measure of moduli space is taken into account, the correlation function with
these fields vanishes by the ghost number conservation. This situation might have
something to do with that the Ward identities of the Liouville gravity have the
form mentioned above. If one wants the non-vanishing correlation functions, it is
necessary to change the measure of moduli.

The author would like to thank T. Yoneya for careful reading of the
manuscript. This work is supported in part by Soryuushi Shogakukai.

23
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24