2 + 1 Dimensional Gravity As An Exactly Soluble System PDF
2 + 1 Dimensional Gravity As An Exactly Soluble System PDF
2 + 1 Dimensional Gravity As An Exactly Soluble System PDF
North-Holland, Amsterdam
Edward WITTEN*
School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA
1. Introduction
* F o r the use of this principle to construct the canonical formalism in a manifestly covariant way, see
refs. [8, 9].
48 E. Witten / 2 + I dimensional gravity
the past fight cone, consisting of points of t < 0 and t2 _ x2 _ y 2 > 0. The 2 + 1
dimensional Lorentz group is SO(2,1); the 2 + 1 dimenisonal Poincar4 group is
ISO(2,1) (the ' T ' means that we are including the translations). A happy fact is that
SO(2, 1) and SL(2, R) are equivalent. Moreover, the hypersurface H ' in X +, defined
by
t 2 _ X 2 _ y 2 = 1, (1.1)
(and t > 0) is isomorphic with H. Thus, we can regard the group F ' as a subgroup of
SO(2,1) acting on H'; the quotient H'/F' is a Riemann surface of genus g. Now, to
get a flat space-time, we consider F ' to act not just on H ' but on the whole future
light cone X +. The quotient Y +-- X +/F' is flat, since X + is flat and F ' preserves
the metric of X. If we regard the hypersurfaces
as surfaces of "equal time", with r playing the role of "time", then the equal time
slices of this flat space-time are Riemann surfaces of genus g. Eq. (1.2) describes an
expanding universe, expanding from an initial singularity at r = 0. Likewise, simply
by considering X - / F ' instead of X +/F', we can obtain flat space-times with a final
singularity. These space-time models depend on the 6 g - 6 real moduli of a
Riemann surface of genus g (which enter in the choice of F').
Are these all of the flat space-times in which "space" is of genus g? Certainly not.
^
The problem can be analyzed as follows. Let M be a flat space-time and let M be its
simply connected universal cover. Being flat and simply connected, 1~I is automati-
cally isometric to Minkowski space, X, or perhaps to a subspace thereof. Let 3' be a
noncontractible loop in M; such a loop is a map of a circle into M such that
y ( o + 2rr) -- "/(o). If such a loop is lifted up to lVl c X, it does not close; it will only
close modulo an isometry, that is, an element of ISO(2,1). Let us denote the element
of ISO(2,1) associated in this way to a loop 3' in M as ~(3'). It is easy to see that the
m a p y ~ qffy) must be a homomorphism. Thus, flat structures on a given manifold
M give homomorphisms of %(M) into ISO(2,1). Conversely, given a homomor-
phism of ~rl(M ) into ISO(2,1), the image of %(M) is a subgroup F of ISO(2,1), and
from this data we can reconstruct a flat three manifold, namely X/F, X being, as
before, three-dimensional Minkowski space.
In our case, with M = 2; R 1, since R 1 is contractible, %(M) reduces to %(2;).
Thus, flat structures on 2; x R1 correspond more or less to homomorphisms of % ( ~ )
into ISO(2,1). I say " m o r e or less" because given a homomorphism, the space-time
that one would reconstruct from it may have very nasty singularities. We have
already given examples with initial and final singularities. Because of a rather
non-trivial theorem that will be discussed in subsect. 3.1, certain even worse
ailments, such as totally collapsed handles on the Riemann surface 2;, will not arise.
(An interesting example of an exotic type of singularity which one might expect to
E. Witten / 2 + i dimensional gravity 49
run into when the vierbein and spin connection are independent variables is that
discussed in ref. [10]. We will see later that this is one type of singularity that we will
definitely have to allow.) Much of the interest in trying to quantize 2 + 1 dimen-
sional gravity is precisely the question of what class of objects should be considered
in defining the "space of all classical solutions". In sect. 2, we will follow a simple
canonical analysis which will lead us to consider the moduli space of all homornor-
phisms of ~ri(~7) into ISO(2,1).
How many parameters are required to specify a homomorphism of rrl(S ) (with
a Riemann surface of genus > 1) into ISO(2,1), or more generally, into any Lie
group G? This question may easily be answered as follows. The fundamental group
7rl(~ ) is naturally described with 2g generators (the a and b cycles) and one
relation. A homomorphism ~ri(2~) ~ G can be described by giving 2g elements of G,
one for each generator, obeying one relation. In addition, we must identify two
homomorphisms if they differ by conjugation by an element of the group. This
enables us, as far as counting parameters is concerned, to remove another element
of G from the description of the hornomorphism. Thus, the dimension of the moduli
space is 2g - 2 times the dimension of G.
Various choices of G are of interest. The moduli space of homomorphisms of
~rl(~ ) to G, for G = SL(2, R), is closely related to the moduli space of complex
structures that can be put on 2~*. For G = ISO(2,1) it is closely related to the space
of flat structures on ~ x R1. These are the two examples that we have discussed
above. If we include a cosmological constant in general relativity, then Minkowski
space is replaced by de Sitter space or anti-de Sitter space, and ISO(2,1) is replaced
by SO(3,1) or SO(2,2). The homomorphisms of ~rl(S) into one of these groups
correspond more or less to the solutions of Einstein's equations with a cosmological
constant of appropriate sign. If we replace the usual Einstein-Hilbert action of
three-dimensional gravity with a pure Chern-Simons action, which is conformally
invariant, then the symmetry ISO(2,1) of Minkowski space is enlarged to the
conformal group SO(3,2). Homomorphisms of ~ri(S ) into this group correspond
more or less to conformally flat structures on 2 x R 1.
In the case of ISO(2,1), since this group is six dimensional, the space of flat
structures on 2 X R 1 has dimension 1 2 g - 12, exactly double the dimension of the
family that we found in the discussion surrounding eq. (1.2). The discrepancy
obviously came from considering only the Lorentz transformations and not the
translation generators in ISO(2,1).
Even if there were no singularities to raise perplexing questions of principle about
what we mean by "the space of classical solutions", it would not be satisfying to
construct this space, formulate quantum mechanics on it, and dogmatically declare
* This is so because if we regard X as H/F', then a noncontractible loop on ,Y lifts on F' to a loop that
closes only modulo an SL(2, R) transformation. The moduli space of homomorphisms to SL(2, R)
will be further discussed in subsect. 3.1
50 E. Witten / 2 + 1 dimensional gravity
* For the foundations of the canonical formalism of general relativity, see refs. [11-13].
E. Witten / 2 + 1 dimensional gravity 51
sional q u a n t u m gravity. I also learned after writing this paper of new geometrical
results [19] about the locally homogeneous lorentzian space-times that we will be
studying.
or simply as R = da~ + oa/x o~. It can be regarded as a two form with values in Ae V.
( / k k V will denote the k t h antisymmetric tensor power (exterior power) of V.)
Let us consider, or instance, the case of d = 4, which is the physical case at least
macroscopically. The Einstein-Hilbert action can be written
[" ijkl
I=TJM~
1 [
Cabcd~eiej~kl
a br. c'd~
). (2.2)
This formula m a y be interpreted as follows. The expression e/x e/x R is a four form
on M with values in V V A2 V, which maps to A4 V. But since V, with structure
group SO(3,1), has a natural volume form, a section of A4V may be canonically
regarded as a function. Thus, there is an invariantly defined integral f e / x e/x R,
and this is what is written in eq. (2.2).
T o verify that eq. (2.2) is indeed the appropriate action for the Einstein theory of
gravity, one proceeds as follows. The metric 7/ab on V, together with the isomor-
phism e ia between T and V, give a metric gij = eia ej b 7lab o n T ; t h i s is the same as an
52 E. Witten / 2 + 1 dimensional gravity
ordinary metric on the manifold M. The connection o~, having structure group
S O ( d - 1,1), is metric compatible. Varying eq. (2.2) with respect to the connection
one learns that
D,e 7 - Dje~ = 0, (2.3)
where D~ is the covariant derivative with respect to the connection 0. Eq. (2.3)
precisely says that the metric compatible connection ~0 is also torsion free. These
conditions uniquely identify ~0 as the riemannian or Levi-Civita connection associ-
ated with the metric g~j on M. Finally, varying eq. (2.2) with respect to e we learn
that
(here eka i s the inverse matrix of e). This is equivalent to vanishing of the Ricci
tensor Rij = ejbek a Rikab . So eq. (2.4) is tantamount to the Einstein equations in
vacuum.
Actually, there is a key limitation in the above argument. We assumed that the
vierbein ei a is invertible, so that the inverse matrix exists. This is related to the fact
that in general relativity, the metric tensor gij = e~a ej b ~l,~b is supposed to be
non-degenerate. In fact, since the definition of the Riemann curvature tensor uses
the inverse of ggj, a configuration in which ei~ is not everywhere invertible must be
regarded as a singularity in classical general relativity. This is precisely the type of
singularity studied in ref. [10] (for the same reason - singularities of this type are
very natural when the vierbein and connection are regarded as independent vari-
ables). Permitting ei a t o not be invertible may seem like a minor change, since the
invertible e~~ 's are in any case dense in the space of all possible e~a. However, if one
attempted to project eq. (2.2) onto a subspace of invertible eia's, it would ruin the
following discussion at crucial stages. We will see at the end of sect. 3 that in a sense
the attempt to make such a projection is what leads to the alleged unrenormalizabil-
ity of 2 + 1 dimensional gravity. So the only definite statement we will make in this
paper about the role of singularities in quantum gravity is that from the point of
view that we will develop, the type of "singularity" related to a non-invertible
vierbein must be permitted to make sense of the quantum theory, at least in 2 + 1
dimensions.
In the last twenty years, many physicists have wished to combine together the
vierbein ei~ and the spin connection ~i~b into a gauge field of the group ISO(d - 1,1).
The idea is that the spin connection would be the gauge field for Lorentz transfor-
mations, and the vierbein would be the gauge field for translations. One then tries to
claim that "general relativity is a gauge theory of ISO(d - 1,1)'. However, there has
E. Witten / 2 + 1 dimensional grauity 53
always been something contrived about attempts to interpret general relativity as a
gauge theory in that narrow sense. One aspect to the problem is that in four
dimenisons, for instance, the Einstein action (2.2) is of the general form fe A e A
(d~0 + o~2). If we interpret e and ~ as gauge fields, we should compare this to a
gauge action fA A A A (dA + A2). But there is no such action in gauge theory. So
we cannot hope that four-dimensional gravity would be a gauge theory in that sense.
In three dimenisons, the situation is rather different. For a space-time manifold
M of dimension three, the Einstein-Hilbert action would be
(2.5)
If we interpret the e's and to's as gauge fields, this is of the general form
A d A + A 3, and might conceivably be interpreted as a Chern-Simons three form.
The study of such terms in three-dimensional gauge theory has a relatively long
history. Indeed, the Chern-Simons action in non-abelian 2 + 1 dimensional gauge
theory was studied in refs. [20, 21], where it was considered as an additional term
added to the unusual Yang-Mills action. In ref. [22], a quantization law associated
with the Chern-Simons term was discovered. Abelian gauge theory with only the
Chern-Simons term was studied by Schwarz [23] and in unpublished work by
Singer; those authors related this theory to certain topological invariants
(Ray-Singer analytic torsion). Non-abelian gauge theory with only the C h e r n -
Simons interaction has recently turned out to be exactly soluble [18]. It is also
interesting to note that string field theory can be formulated as a more abstract
version of a 2 + 1 dimensional gauge theory with only a Chern-Simons action [24].
We will claim that three-dimensional general relativity, without a cosmological
constant, is equivalent to a gauge theory with gauge group ISO(2,1) and a pure
Chern-Simons action.
Let us recall some facts about the Chern-Simons interaction. For a compact
gauge group G, this may be written
Here we are regarding the gauge field A as a Lie-algebra-valued one form, and " T r "
really represents a non-degenerate invariant bilinear form on the Lie algebra.
Thus, if we choose a basis of the lie algebra, and write A = A~T~, then the
quadratic part of eq. (2.6) becomes
Here dab ----- Tr(T~Tb) plays the role of a metric on the Lie algebra, and this should be
non-degenerate so that eqs. (2.6) or (2.7) contains a kinetic energy for all compo-
nents of the gauge field.
Thus, before we ask whether gravity in 2 + 1 dimenisons is equivalent to ISO(2,1)
gauge theory with a Chern-Simons interaction, we should ask whether such a
C h e r n - S i m o n s interaction exists, or in other words whether there exists an invariant
and non-degenerate metric on the Lie algebra of ISO(2,1).
Let us first consider the general case of ISO(d - 1,1). The Lorentz generators are
j~b, and the translations are pa, with a, b = 1 . . . . . d. A Lorentz-invariant bilinear
expression in the generators would have to be of the general form W = XJab j a b +
yPa Pa, with some constants x and y. However, in requiring that W should
c o m m u t e with the P b, we learn that we must set x = 0. At that point we are clearly
no longer constructing a non-degenerate bilinear form on the Lie algebra, so there
would be no reasonable Chern-Simons three form for I S O ( d - 1,1) for general d.
The magic of d = 3 is that in this case we can set W = CabcPaJ bc. This is easily
seen to be ISO(2,1) invariant as well as non-degenerate. Therefore, a reasonable
C h e r n - S i m o n s action for ISO(2,1) will exist. It remains to construct it and compare
it to 2 + 1 dimensional general relativity.
F o r d = 3 it is convenient to replace j a b with J ~ = 7c 1 ~b~,
abe. The invariant
quadratic form of interest is then
[L, Yb] = ~ . b c J c,
[Ja, Pbl = %bee c,
[Ca, ~b] = 0. (2.9)
(The fact that this is ISO(2,1) and not ISO(3) is hidden in the fact that it is the
Lorentz metric that is used in raising and lowering indices. This will not always be
indicated explicitly.)
Let us use these formulas and construct gauge theory for the group ISO(2,1). The
gauge field is a Lie-algebra-valued one form
~A i = - O i u , (2.11)
E. Witten / 2 + 1 dimensional gravity 55
where by definition
Diu = Oiu + [ A i, u]. (2.12)
1 fy
' i j k l ( Oie j a --
? j e ia+ ~ a b c ( OOibejc + eib~Ojc))
d e
x ( ok ,. - + ). (2.15)
fM a a b c (2.16)
By this construction, eq. (2.16) is automatically invariant under the gauge transfor-
mations (2.13). In any case, this is easy to verify.
Now, a look back to eq. (2.5) reveals that the ISO(2,1) Chern-Simons action
(2.16) precisely coincides with the 2 + 1 dimensional Einstein action. Thus, we have
essentially succeeded in showing that 2 + 1 dimensional gravity may be interpreted
as Chern-Simons gauge theory. However, there is still an important point to clear
up. The transformation laws (2.13) do not coincide with the usual transformation
laws of 2 + 1 dimenisonal gravity. There is no problem with the local Lorentz
transformations whose generators have been called r a in eq. (2.13); the terms in eq.
(2.13) proportional to r a are the standard formulas for local Lorentz transforma-
56 E. Witten / 2 + 1 dimensional gravity
tions. The problem is with the generators O~ which hopefully should be related to
diffeomorphisms. From eq. (2.13) we see that under a transformation generated by
the p's, the transformation law is
At first sight, eq. (2.17) does not seem to have much in common with the standard
formulas for transformation under diffeomorphisms, but we want to show that they
are equivalent. Under a diffeomorphism generated by a vector field - d , the
standard transformation law would be
k
(ake ,a - O i e ka ) - O , ( o e
k
~eia=-v ka ),
If we let p -
_a __ oke k, then we find that the difference between eqs. (2.17) and (2.18) is
a
~ a a k a a
3 e i -- 3 e i = - - o ( D k e i -- D i e k ) + c abcvkcokbeic. (2.19)
and this term vanishes by the equations of motion. The remaining term on the
fight-hand side of eq. (2.19) is a local Lorentz transformation with infinitesimal
parameter
r" - Vkcok". (2.21)
The equations of motion now say not that space-time is flat but that space-time is
locally homogeneous, with curvature proportional to X. The simply connected
covering space of such a space-time is not a portion of Minkowski space, but a
portion of de Sitter or anti-de Sitter space, depending on the sign of ~. These latter
spaces have for their symmetries not ISO(2, 1) but SO(3,1) and SO(2,2), respec-
tively. Thus, it is reasonable to guess that if three-dimensional gravity without a
cosmological constant is related to gauge theory of ISO(2, 1), then three-dimensional
gravity with a cosmological constant will be related to gauge theory of these latter
groups. This proves to be the case.
To begin with, we generalize eq. (2.9) to
The formula (2.8) gives an invariant quadratic form on the generalized Lie algebra
(2.23). Using it, we find that the Chern-Simons three form comes out to be
precisely the Einstein lagrangian (2.22) with cosmological constant included! The
equations of motion derived from this lagrangian are precisely the vanishing of the
field strength (2.25). Vanishing of the coefficient of P~ in eq. (2.25) is the assertion
that ~o is the Levi-Civita connection; and vanishing of the coefficient of J~ is then
the Einstein equation with a cosmological constant.
2.3. A M O R E G E N E R A L L A G R A N G I A N
f a a 2 (~0 b c]
a a ~,, a b c~
+ X # ' ( 0 k e t - Ote k ) + ~a,,b,~oj eke, ). (2.27)
Therefore, eq. (2.27) is invariant under eq. (2.24) and it makes sense to add it, with
an arbitrary coefficient, to the original Einstein lagrangian (2.22). For generic values
of this coefficient, the classical equations are u n c h a n g e d - they still assert the
vanishing of the field strength (2.25). This is rather s t r a n g e - the more general
lagrangian is equivalent to eq. (2.22) classically, but this will not be so quantum
mechanically.
E. Witten / 2 + 1 dimensional gravity 59
(1t
J + = ~ J~ _+ ~ - P ~ . (2.28)
Of course, this step only makes sense for 2~=g 0. If ?~ is negative, the J~ are complex.
The Lie algebra (2.23) becomes simply
[ J + , J ; ] =- abcJ c +, [ J a , J b ] = Eabc J c - ,
[ J + , J b ] = 0. (2.29)
Obviously, for positive ),, eq. (2.29) is the Lie algebra of SO(2, 1)X SO(2,1), or
SL(2, R) SL(2, R).
The corresponding connections are
They are related as follows to the actions that we constructed earlier. The "stan-
dard" Einstein action (2.22) is 1 = ( I + - I )/47~-, and the "exotic" term (2.27) is
[= ~(~++ I-).
2.5. C H E R N - S I M O N S G R A V I T Y
We will conclude this section with a brief discussion of what would usually be
called 2 + 1 dimensional gravity with a Chern-Simons action [22]. (The terminology
is of course somewhat misleading since we are claiming that ordinary 2 + 1
dimensional gravity has a Chern-Simons interpretation.) Chern-Simons gravity
means the following. The fundamental variable is a vierbein e 7. The spin connection
is defined as a functional of ei~ by requiring it to obey
I ' is not varied with respect to o~ regarded as an independent variable. Rather, one
regards o~ as a functional of e via eq. (2.33) and varies eq. (2.34) with respect to e.
The field equation obtained in this way is
with RU = R u - gu R"
The lagrangian (2.34) - with ~o defined in terms of e via eq. (2.33) - is invariant
under local Weyl transformations e i a ( x , y , t ) ~ e q'(x' y' t) . eia, even though this is not
manifest in the way that eq. (2.34) is written. Consequently, eq. (2.35) is a
conformally invariant equation. Indeed, the left-hand side of eq. (2.35) is the
three-dimensional analogue of the Weyl tensor, and vanishing of eq. (2.35) is the
E. Witten / 2 + 1 dimensional gravity 61
3. Quantization
We now turn to constructing a canonical formalism, with a view toward quantiza-
tion. Thus, we consider the lagrangian (2.22) on a three manifold M = Z R 1, with
Z being a Riemann surface that plays the role of an "initial-value surface". Some
subtleties arise in the canonical formulation because of the gauge invariance. A
convenient reference on the general procedure is ref. [25]. The first step in construct-
ing a canonical formalism is to introduce new variables, if necessary, to get a
lagrangian that is linear in time derivatives. We can skip this step, since eq. (2.22) is
already linear in time derivatives. If possible, one then separates out the variables
into variables whose time derivatives are present in the lagrangian and variables
whose time derivatives do not appear* In our case, this is easily done. The variables
whose time derivatives appear in eq. (2.22) are the "spatial" components of the
vierbein and connection, namely eia and ~0ia, for i = 1, 2. The variables whose time
derivatives are absent in eq. (2.22) are the " t i m e " components e0a and ~0a. This
convenient, global separation between variables whose time derivatives appear in
the lagrangian and variables whose time derivatives do not appear, and the fact that
the lagrangian is linear in the latter, make the construction of a canonical formalism
relatively straightforward.
Eq. (2.22) may be rewritten as
A
I= -2[dt f UJei~ ~-- a
J J~ dt %
In discussing a closely related problem in ref. [18], I have adopted the possibly more familiar
language of "picking the gauge .40 = 0". In the gravitational problem that we are considering here, a
different and perhaps more careful and canonical language seems appropriate.
62 E. Witten / 2 + 1 dimensional gravity
The Poisson brackets can be read off from the terms in eq. (3.1) that contain time
derivatives. They are
{ %~(x),ej~(Y)} = ,iffl~b82(x - y ) ,
In addition, we must impose the constraint equations. They are simply the equations
8I/6eo a = 8I/&oo ~ = 0, or
* The problem of imposing these particular classical constraints to reduce this particular phase space
was one element in the work of Atiyah and Bott on equivariant Morse theory, two-dimensional gauge
fields, and the moduli space of holomorphic vector bundles [26]. Of course, 2 + 1 dimensional gravity
gives a new context for this problem.
E. Witten / 2 + 1 dimensional gravity 63
This should come as no surprise; in gauge theories, 6~q~/SAo is always the generator
of gauge transformations. Thus, to construct the classical phase space which should
be quantized, one simply takes the space of solutions of the constraints - namely
the space of flat c o n n e c t i o n s - and divides by the group generated by the con-
s t r a i n t s - namely, the group of gauge transformations. Consequently, the phase
space ,//t' of 2 + 1 dimensional gravity is the same as the moduli space of flat G
connections modulo gauge transformations.
The Poisson brackets on ~ ' are just the original Poisson brackets on Y~,
restricted to gauge-invariant functions. In other words, a physical observable is a
function on ~t'. Modulo the constraints, functions on J// are the same as gauge-
invariant functions on ;of', and the Poisson brackets of gauge-invariant functions on
Y/" are computed using the Poisson brackets (3.2).
In the introduction we have explained heuristically why the physical phase space
of 2 + 1 dimensional gravity is related to the moduli space of flat G bundles. Now
we have derived the result from a more conventional field-theoretic analysis. It
remains to quantize the system.
64 E. Witten / 2 + 1 dimensional gravity
* The idea that it would be best to view the connection data as the coordinates was envisaged by
Ashtekar [16] in 3 + 1 dimensions.
E. Witten / 2 + 1 dimensional gravity 65
The second equation in (3.3) says that the curvature of the connection to should
vanish, and the second equation in (3.5) instructs us to identify two connections that
differ by an SL(2, R) gauge transformation. Taken together, these conditions say
that we should introduce the moduli space sV" of flat SL(2, R) connections modulo
gauge transformations. The quantum Hilbert space, incorporating the constraints, is
not the space of square integrable functionals on Y but the space of square
integrable functionals on sV.
Geometrically, the situation may be described as follows. For zero cosmological
constant, the relevant gauge group is ISO(2,1). The ISO(2,1) group manifold is the
total space of the cotangent bundle of the SO(2, 1) manifold. Correspondingly, the
moduli space Jr' of flat ISO(2,1) connections is the total space of the cotangent
bundle of the moduli space .A/"of flat SO(2,1) connections.
Also, the Poisson brackets (3.2) induce on ~t' its natural symplectic structure as
the cotangent bundle of .AP. Therefore, quantum mechanics on J g is very
simple - the quantum Hilbert space is the space of L 2 functions on ~'.
For clarity, let me make this candidate for the "physical Hilbert space" of
quantum gravity with zero cosmological constant completely explicit. The funda-
mental group of a Riemann surface X of genus g > 1 can be defined via 2g
generators, which we denote
A point in .Ar is a homomorphism of the group with generators (3.6) and relation
(3.7) into SO(2,1). Such a homomorphism is described by representing the a / a n d bj
by elements of SO(2,1) which we will denote as U~ and Vj. These must obey
along with a topological condition that will be described presently. Two representa-
tions are equivalent if they differ by a global gauge transformation
for some fixed element E of SO(2,1). The quantum Hilbert space Jf~ is the space of
functions '/'(U, Vj), such that: (i) g' is defined on the hypersurface that is defined
by eq. (3.8) together with a certain topological condition that we will describe next;
(ii) qs is invariant under the transformation (3.9).
66 E. Witten / 2 + 1 dimensional grauity
By the way, the Hilbert space structure on these functions is defined by the norm
I ~ l 2 = f , / , . g,. Evidently, for this to be invariant, ~ must be a half-density on
rather than a "function". It is generally true that quantization leads naturally to
spaces of half-densities rather than spaces of "functions". This subtlety is usually
slurred over in quantum mechanics texts, something which is possible because in
most physical problems there is an evident measure on coordinate space which can
be used to give a canonical isomorphism between the space of half-densities and the
space of functions. Even in the problem at hand, there are a variety of more or less
natural measures on Jg" which could be used to identify functions with half-densi-
ties, but since none of these has compellingly appeared in the above construction, it
is most natural to think of the quantum Hilbert space X as the space of
half-densities on JV.
3.1.1. A topological discursion. We must now discuss a certain subtle but impor-
tant topological point, which we have suppressed until this point. Actually, the
moduli space ~V" of flat SO(2,1) connections on a Riemann surface Z is not
connected, but contains several components. These arise as follows. A flat SO(2,1)
connection is of course the same as a flat SL(2, R) connection. Since SL(2, R)
naturally acts on a real two-dimensional vector space, a flat SL(2, R) connection
defines a real two-plane bundle o~ over Z. Such bundles are classified topologically
by an integer, the Euler class. In general, the Euler class of a real two-plane bundle
o~ on a Riemann surface may have any integer value, but for a flat two-plane
bundle, there is an upper bound - if ~ admits a flat SL(2, R) connection, its Euler
class can be no bigger in absolute value than 2g - 2, which is the Euler class of the
tangent bundle of a Riemann surface Z of genus g.
Now, general relativity is supposed to be a theory of the dynamics of geometry.
As we indicated in sect. 1, the relation of homomorphisms ep: ~rl(X ) ~ SL(2, R) to
geometry is as follows. Suppose that the genus g of Z is greater than one. (For
genus zero there are no non-trivial homomorphisms to discuss; for genus one the
situation is more complicated than the simple situation that I will now summarize
for g >~ 2, and I will not attempt to discuss this case.) Let H denote the complex
upper half plane. If dp embeds vrl(Z ) as a subgroup P of SL(2, R), and if moreover P
is a discrete subgroup of SL(2, R), then H / F is a complex Riemann surface with a
complex structure determined by the homomorphism ft. All complex structures arise
in this way, and the moduli space JV" of homomorphisms (3.8) that give discrete
embeddings of the fundamental group of Z in SL(2, R) can be identified (if g >/2)
with Teichmuller space. Thus, the quantum wave function ~/'(Ui, ~ ) described
earlier can - if restricted to homomorphisms that give discrete embeddings of the
fundamental group - be regarded as a function on Teichmuller space, or in other
words as a function on conformal geometries.
However, it is far from being true that all homomorphisms ~: ~rl(Z) ~ SL(2, R)
give discrete embeddings. The opposite of an embedding would be the trivial
h o m o m o r p h i s m U/= Vj = 1 in eq. (3.8). If we consider " b a d " homomorphisms of
E. Witten / 2 + 1 dimensional gravity 67
(in this case, any desired classical three geometry). It would be interesting to study
this more carefully and see if there really are wave functions that correspond, with
very high probability, to space-times with closed timelike curves.
3.1.3. Non-zero cosmological constant. We would now like to generalize this
rather explicit description of the Hilbert space for X = 0 to the case in which the
cosmological constant is not zero. If the cosmological constant is positive, the
relevant classical phase space J g is the moduli space of homomorphisms of rq(E) to
$0(3,1). It can be shown [27] that this space is the total space of the cotangent
bundle of the moduli space of homomorphisms of ~h(Z) to SO(3). Therefore, a real
polarization is available, and the quantization can be carried out somewhat along
the likeness of the above. However, the arguments are less elementary and will not
be described here. As far as I know, it is not possible to find a real polarization
when the cosmological constant is negative. There is another point of view about
quantization of Jg, which works for any value of the cosmological constant. This is
what we will discuss next.
3.2. K A H L E R POLARIZATION
i a i a
D(o) e~ = D(o) o~b = O, (3.11)
with D(o)' denoting the covariant divergence with respect to g(0). The gauge condi-
tion (3.11) may be implemented by introducing Lagrange multipliers fa, )~ab, and
adding to eq. (3.10) the gauge fixing term
In the usual fashion, this must be supplemented with Faddeev-Popov ghosts. The
point of this gauge choice is that eq. (3.12) (and the ghost action) preserves the
power-counting renormalizability of eq. (3.10).
70 E. Witten / 2 + 1 dimensional gravity
* This is related to the fact that usually, in discussions of the quantization of gravity, g is considered to
be dimensionless, while in our treatment one must consider g to have dimension two, since e has
dimension one.
E. Witten / 2 + 1 dimensional gravity 71
have dimension zero.) As e and w have positive dimension, the short-distance limit
must have e = ~o = 0. The problem is now that as eq. (3.13) has no quadratic term in
an expansion around e = 0 = 0, one cannot make sense of the " u n b r o k e n phase"
that should govern the short-distance behavior; that is the essence of the unrenor-
malizability of quantum gravity in four dimensions.
The discussion of 2 + 1 dimensional gravity that we have given above raises many
questions. In this concluding section I would like to briefly draw attention to a few
of these questions, without claiming to solve any of them.
4.2. E U C L I D E A N C O N T I N U A T I O N
natural to believe that it should hold even after any euclidean continuation that may
be valid. But this raises puzzling issues.
Usually, the Einstein action is real whether one is in Minkowski or euclidean
space. But the Yang-Mills Chern-Simons action is always imaginary in euclidean
space. The question has to do with what is being continued when one goes from
Minkowski to euclidean space. Usually, it seems obvious that in going from
Minkowski to euclidean space the tangent space group of general relativity goes
from SO(2,1) to SO(3). However, in Yang-Mills theory one does not make a Wick
rotation on the gauge group when one rotates from real to imaginary time.
Possibly, we should not think of Z x R 1, the manifold on which we have worked
in this paper, as space-time, but rather as an auxiliary space analogous to the world
sheet in string theory. The idea would be that space-time is reconstructed from data
on Z x R1 just as in string theory space-time is reconstructed from a world-sheet
theory. The idea of this reconstruction is that a flat ISO(2,1) connection on ~ x R1
is equivalent to a homomorphism of the fundamental group of Z into ISO(2,1). The
image of the fundamental group under this homomorphism is a subgroup F of
ISO(2,1), and we try to identify spacetime with X/F, with X being Minkowski
space. If we think of Z X R1 as a "world sheet", and the dynamical variables el, o~j
as tools in reconstructing space-time, then as there is no metric on the world surface
Z R x, there is no natural notion of whether this space has "Minkowski or
euclidean signature". It does make sense to ask whether space-time has Minkowski
or euclidean signature. The minkowskian case is the case, considered in this paper,
in which ei, 0j are a gauge field of ISO(2,1) or one of its generalizations SO(3,1) or
SO(2, 2). If we actually want to do euclidean gravity, meaning gravity with euclidean
space-time, then those groups would be replaced with their analytic continuations
ISO(3), SO(4), or SO(3,1), depending on the sign of the cosmological constant.
A closely related question is whether the parameters in the gravitational la-
grangian should be quantized, by analogy with the corresponding phenomenon in
gauge theories [22]. This is inevitably related to the question of whether the
lagrangian is to be real or imaginary, since it is only imaginary terms in the action
that might be sensibly quantized. The standard Einstein action I of eq. (2.22) is
ordinarily real in "euclidean space", and we would like to preserve this. Thus, it
should not be quantized. The more exotic term I' of eq. (2.27) is less familiar and
we are willing to believe that it should be quantized and perhaps should appear in
the lagrangian with an imaginary coefficient. To investigate this, we note that
quantization depends on ~r3(G), where G is ISO(2,1) or one of its generalizations;
and here there seems to be a big difference between Minkowski space-time and
euclidean space-time, since for instance ~r3(ISO(2,1)) is zero, but %(ISO(3)) - Z. In
fact, for all of the minkowskian groups and all of the euclidean groups except SO(4),
there is no topological reason to quantize the ordinary Einstein action (2.22), and it
can be given the usual real coefficient if we are considering the euclidean case.
(SO(4) is from this point of view a mysterious exception that we will not try to
E. Witten / 2 + 1 dimensional gravity 73
elucidate.) However, for the minkowskian group SO(3, 1) and all of the euclidean
groups, the exotic interaction I ' is quantized and must appear in the lagrangian with
an imaginary coefficient.
4.2.1. Global Anomalies and Classical Singularities. This whole discussion of
global anomalies and quantization of couplings in 2 + 1 dimensional gravity may
seem rather bizarre. When 2 + 1 dimensional gravity is written in terms of the
metric (rather than vierbein and connection), it is manifest that there are no such
anomalies. So what is going on? Actually, the crucial point is that (as discussed at
the end of the introduction to sect. 2), in formulating 2 + 1 dimensional gravity in
terms of the vierbein and spin connection, we have dropped the requirement that
the vierbein should be invertible. Though the non-invertible vierbeins are of "mea-
sure zero", adding them changes the topology of field space (and of the space of
gauge transformations) and permits the occurrence of global anomalies that other-
wise would have been absent. In fact, the four-dimensional "instanton" studied in
ref. [10], which has a classical singularity (a degenerate vierbein) at its core, is
precisely the configuration which is manifested in three space-time dimensions in
terms of global anomalies.
4.3. UNITARITY
Some of the fundamental puzzles in the canonical formalism of quantum gravity
have to do with the physical interpretation of the Wheeler-de Witt wave function.
The following may illustrate some of the questions. Let PBA be the probability
amplitude for observing a final state B after having observed an initial state A (the
initial and final observations being on some specified spacelike hypersurfaces). It is
a fundamental fact of life in ordinary field theory that in a sequence of observations
(fig. 1) one has
Fig. 1. A two step transition from an initial state A to a final state C via an intermediatestate B.
74 E. Witten / 2 + 1 dimensional gravity
4.4. O B S E R V A B L E S
/ 44)
E. Witten / 2 + 1 dimensionalgravity" 75
should formally depend on the topological classes of the links C~. Any two
unknotted circles on the three sphere are equivalent, for example. One might
wonder whether such observables can actually be expected to make sense. A priori
one might well have been inclined toward a negative answer, but some recent
developments relating Yang-Mills theory to knot theory [18] strongly suggest a
positive answer. If the equivalence that we have proposed between quantum gravity
in 2 + 1 dimenisons and Chern-Simons gauge theory with gauge group ISO(2,1),
SO(3,1), or SO(2, 2) is really correct, then the expectation values (4.4) should be the
analogues for these groups of the Jones polynomials [29] of knot theory. This would
be a 2 + 1 dimensional version of the possible relation between quantum gravity and
knot theory conjectured in ref. [17].
where I is the standard Einstein action (2.22) with cosmological constant, and I ' is
the exotic action (2.27). In keeping with the above discussion of quantization of
couplings, the standard action 1 appears with a real coefficient, which we have
written as I/h; here h is Planck's constant. But - again in view of the discussion
E. Witten / 2 + 1 dimensional gravity 77
a b o v e - I ' has a quantized coefficient, with k an integer. Once h is explicitly
introduced in this way, one may as well set X = 1 in eq. (2.22). Now one wishes to
study the F e y n m a n integral over all choices of field variables on an arbitrary three
manifold M, to get the "partition function" defined by
I would like to thank M.F. Atiyah and W. Thurston for assistance on some
topological matters. I would also like to thank J. Horne for a critical reading of an
earlier draft of the paper, and T. Banks for some stimulating questions.
References
[1] S. Deser, R. Jackiw and G. 't Hooft, Ann. Phys. NY 152 (1984) 220
[2] S. Deser and R. Jackiw, Ann. Phys. NY 153 (1984) 405
[3] J.R. Gott and M. Alpert, Gen. Rel. Gray. 16 (1984) 243
[4] J. Abbott, S. Giddings and K. Kuchar, Gen. Rel. Grav. 16 (1984) 243
[5] T. Banks, W. Fischler and L. Susskind, Nucl. Phys. B262 (1985) 159
[6] E. Martinec, Phys. Rev. D30 (1984) 1198
78 E. Witten / 2 + 1 dimensional gravity