Read slacpub4760.pdf text version
SLACPUB4760 October 1988 CT)
.
The Topology
of Moduli
Space and Quantum
Field Theory*
DAVID
MONTANO Stanford
AND JACOB SONNENSCHEIN+ Center 9.4309
Linear Accelerator Stanford,
Stanford
University,
California
ABSTRACT
We show how an SO(2,l) used to describe the topology the theory correspond  . . . is an extension investigate gauge theory of the moduli with a fermionic symmetry may be of
space of curves.
The observables
to the generators
of the cohomology field theory instanton
of moduli
space. This by Witten to
of the topological
quantum
introduced space. examine
the cohomology
of YangMills quantum
moduli
We explore a toy U(1)
the basic structure model,
of topological
field theories,
and then realize a full theory theory,
of moduli
space topology.
We also discuss
why a pure gravity
as attempted
in previous
work, could not succeed.
Submitted _. _
to Nucl.
Phys. I3
* Work supported by the Department of Energy, contract DEAC0376SF00515. + Work supported by Dr. Chaim Weizmann Postdoctoral Fellowship.
1. Introduction
.. 
There is a widespread belief that the Lagrangian is the fundamental study in physics. Lagrangian. The symmetries of nature are simply properties
object for
of the relevant physics
This philosophy
is one of the remaining
relics of classical Witten
where the Lagrangian a class of quantum logical quantum by a Hilbert
is indeed fundamental.
Recently,
has discovered These topocharacterized shown, they
field theories which have no classical analog!" are, as their name implies, As has been recently
field theories (TQFT)
space of topological by a BRST is zero modulo
invariants. gauge fixing a topological
can be constructed sical Lagrangian
[31 of a local symmetry. [2' The clasinvariant. Thus, we see that the The Lagrangian Topological moduli thespaces target by the 
heart of the matter
is in the symmetries determined
one chooses to study. by the symmetries. the instanton
is secondary, being completely
ories have so far been constructed of YangMills
for investigating
[llI41[51
theories in 4dimensions
and nonlinear
I61[71[Sl
sigma models with This was motivated
knanifolds having an almost complex structure. works of Atiyah, .. . . Donaldson and Gromov.
In this paper we follow up on previous work on two dimensional It is wellknown the classical moduli that pure gravity &s fiR, in two dimensions is a topological is just the Euler characteristic.
gravity!51' g1 theory, since The relevant
action,
space will be the familiar The topology and physics.
one of complex structures of this moduli String theory
of Riemann surfaces interest in
of genus g, M,. both mathematics a theory
space is of current
is believed to be fundamentally In particular it is known that on moduli space!"' of moduli space. ; .._
defined on the moduli partition function
space of curves.
the superstring
is locally a total derivative
Thus, the partition a _P_ ic _Mathematicians
function
will only depend on the topology in the topology
have also been interested
of moduli
space, since little
it is known to be highly nontriviaI!lll
Unfortunately,
there is still relatively
known about it. In this paper we will describe how elements of the cohomology of moduli space may be computed as observables of the theory. 2
We have found that * . ,cis to treat
the most fruitful
approach
to two dimensional This
gravity
it as a topological gravity
gauge theory with
of SO(2,l). cosmological
is the isometry We will of
group of 2dimensional
a positive
constant.
show that by an appropriate M, for g 2 2. is SO(1,2). topological When
gauge fixing
this theory
describes the topology
g = 1 the relevant differs
group is ISO(l,l), from
and for g = 0 it work15' [g' on
Th is approach gravity
dramatically
the previous
where the emphasis in a purely
was on the symmetry theory
of pure gravity. the topology whose
It was not possible of M,. moduli Roughly
gravitational
to investigate
speaking,
one has to study the topological
gauge theory
space is M,.
We will
discuss the reasons for this in section YangMills, except that
5. This is 
analogous with
to the case of topological group.
one must now deal
a noncompact
In this paper we will explicitly logical gauge theory of SO(2,l) field strength; this implies
construct
two dimensional
gravity
as the topoof the
for g 2 2. The gauge choice is the vanishing conditions that the zweibein
the standard
(a compo
nent of the gauge field) be covariantly constructed
_..._ . .
conserved and the curvature constant. Indeed,
of the metric it is essential fields which are It is
from
this zweibein
` a negative be be treated
that the zweibein related
and spin connection
as independent
only by the gauge fixing
condition
that we have a flat connection.
significant Riemann topology
to observe that the SO(2,l) surface of g 2 2 is equivalent
gauge theory on an arbitrary to two dimensional gravity
background with the same
. In the case of g = 1 the gauge group is ISO(l,l), implies that the curvature and the gauge fixing
and the vanishing for the sphere to be
of the field strength
is zero. Similarly, constrains
the gauge group is SO(1,2), a positive constant.
the curvature
In section 2, we will discuss the general relationship a _T_ for _and TQFT' s. constructing gravity.
between moduli concepts
spaces
;
Section 3 will be devoted to the fundamental TQFT' s emphasizing
necessary
the fields and symmetries
of topological
In section 4, we will work out completely
the toy example of a topological work on topological
U(1) theory.
Then we will review, in section 5, the previous 3
gravity A. '
, its shortcomings
and its relation
to the current
work.
In section 6, we The
will discuss the degenerate case of g = 1 where the gauge group is ISO(l,l). explicit construction of the Lagrangian 8 we will of the SO(2,l)
gauge theory will be given to moduli space and its on the
in section 7. In section invariant observables.
discuss its relation
Finally,
we will summarize
the work and comment
open questions
remaining.
2. Moduli
In gauge theories, equations correspond treatment integration Euclidean modulo moduli
Space and TQFT
of the classical since they A proper requires an 
spaces are the spaces of solutions They partial with are finite differential
gauge equivalence. of nonlinear field theory moduli provide
dimensional, equations.
to solutions of a quantum
a local gauge symmetry theories
over the relevant space, instantons The moduli
. In YangMills an important
on 4dimensional nonpertur
tool for computing
bative effects.
spaces must then be understood
in order to complete
the final integration.
_..._ . .
Knowledge
about
the topology
of these spaces has recently topological realization
been dramatiYangMills the
cally increased by the work of Donaldson!71 Witten' s ories (TYM) work!"In morphisms figurations complex expected a _r_ _h ave provided theory a quantum field theoretic
of Donaldson' s diffeo
string
where the gauge symmetry
is that of worldsheet
one has an analogous structure. (i.e. the metric) simplifies
The integration
over the gauge conspace of
into an integration
over the moduli
structures
of Riemann
surfaces, M, . Indeed, string theory measures are space as advocated by Belavin
to be sections of line bundles on moduli
and Knizhnik!` *' Knowledge ing a better quantum about the topology of string of moduli theory. space is then important Witten for gain
understanding
has shown that topological moduli space.
field theories may provide a fruitful 4
new tool for studying
Mathematicians .?. principal Gbundles
investigate
the topology spaces.
of moduli Topological
space using the theory of quantum field theories we
and classifying
can also be used to study will clarify the relation
the topology
of moduli
space.
In this section,
between
these two approaches.
Readers who are only 3. In section 8, we will
interested
in the field theory
should proceed to section
show precisely how the observables of TQFT' s bundle. In the following
probe the topology
of the universal needed
we outline the relevant mathematical of TQFT' s. is a fiber bundle with fiber
terminology
to describe the structure A principal Gbundle
given by the structure g : P x G + P. P 
group G; it contains is called the total manifold
a mapping
K : P + B and a mapping
space; B is the base space, and x is the projection. The coset space B := P/G
P is a can also If P is
on which the group G acts freely. such that
be made into a manifold is contractible
the projection bundle
x : P + B is smooth.
, then the principal
is called universal, as BG
and B = P/G
called a classifying
space for G and often denoted
(base space of G). of classifying spaces
A good deal is known about the cohomology
and homotopy
which is closely related to that of' the moduli space. The moduli _..._ . . dimensional
space is the finite are allowed. space.
subspace where only the classical gauge configurations spaces are then useful in studying we can choose a principal Gbundle over some manifold Gbundle M}, the topology
The classifying In particular on a fixed the group concerned
of moduli
where P = A = {connections = A/$ with 9 being
then BG BG
of gauge transformations with in quantum
. Indeed,
is the space one is usually
field theory,
since the space of gauge fields in the the subspace of
path integral A/$ with
is defined to be A/S. field strengths
For the case of instantons is the moduli space.
selfdual
The observables of TQFT' s z _P_ _of instanton lore moduli
are the generators of H* (M) ,the cohomology
ring
; 
space. These are also elements of H*(A/$). which mathematicians of Gbundles; call characteristic
There is a whole classes. They of
on these observables
are used to classify the topology the classifying
they generate the cohomology connected,
space of a Lie group, G. In general, for a simply 5
finite
dimensional ,c. rings: __ 
Lie group G, we have the following
generators
for the cohomology
H* (BG) H*(G) whereR[...] is a polynomial
= $1~1 , . . . . yk], degree yi = di + 1, cz E[zl, . . . . zk], degree z; = di, (2.1)
algebra over the reals and E[...] is an exterior
algebra.
The generators,
zi, yi, are forms, and by degree we mean the degree of the form. N H*(A/G). Th us, the observables yi. of TYM will be a poly
Recall that H*(BG) nomial
algebra over the generators, Thus, TYM theories
These are referred quantum
to as the Donaldson techniques
polynomials. for the explicit ref.[l].
provide
field theoretic The details
construction
of these polynomials.
are discussed in
For the moduli cation is, however,
space of curves there is an analogous structure. much more difficult with
The classifi
because the gauge group is noncompact. Miller proved that there is an to the coof genus g
` We must then be content injective homology _..._ . . mapping from
a weaker result.
a polynomial
algebra, Q, over the rationals diffeomorphisms
ring of the moduli
sp.ace of the oriented
surfaces, A,: ` 131
Q[n1, ", Q21 ` + H* @A, $2) (2.2)
where /ci E H2i(BAg). Recall that diffeomorphisms. related!"' M, is the space of metrics As in YangMills theory, for genus g Riemann H, (BA,) and H, (M,) surfaces modulo are intimately
Indeed, their rational
cohomologies
are isomorphic:
fL(BA,$l) __T_ ix
=
H&&&l).
_
(2.3)
_
Mumford Mumford
gave a construction invariants.
for. the classes, Ki!141 We will
refer to them of
M,.
as
They are elements of the stable cohomology
Stability
is the statement
that for g >> i, Ki is independent 6
of g. In fact, for i 2 g  1, Ki
is a polynomial ,z..
in ~1, . . . . +2.
We will have more to say about these classes in of the TQFT will be discussed. in this paper, we expect that the Mumford the entire classes. Since
section .8 where the observables For the quantum our topologically the Mumford might
field theory we will construct observables will include
invariant
classes do not necessarily observables. Indeed,
generate
H*(M,),
we also field theo
find other
we hope that will
our quantum
retic construction of H*(M,).
of the generators
of H* (M,)
increase our understanding
We will have more to say about the invariant
observables
in section 8. In the in particular 
next section, we will outline basic issues for the assembly of TQFT' s, those related to gravity and the groups ISO(l,l) and SO(2,l).
3. Symmetries,
Our objective Therefore, (TG).  . . .
Fields and Actions
in Topological
Gravity
space of curves. gravity un
is to obtain a topological candidate quantum
theory for the moduli is obviously
a natural
for such a TQFT
topological
Topological
field theories are characterized
by an invariance
der a local symmetry be metric independent.
which guarantees that all the observables of the theory will In TYM, for example, the symmetry is 6A, = 0,(z) and
the observables
are the so called Donaldson and fields relevant
polynomials. for describing
In the following topological
we will of
discuss the symmetries gravity. The fields usually g,p, zweibein
theories
used to describe two dimensional The zweibein vector
gravity
are the metric
and spinconnection. a world
ecra, defined by gap = ezepa space vector V, via ;
, is used to transform a _r_ _V, = e,,V". derivative
Va into a tangent
The action by D,va
of the spin connection
w, is defined in the covariant
= d,V"
+ W,cabT/b. It can be expressed in terms of eora by namely Dpe,, = 0. The symmetries of TG
using the requirement should include
of no torsion,
diffeomorphisms
(60 (v)), 7
Weyl resealing
(6~ (p)), local Lorentz
transformations ,z..
(6~ (A)),
and the additional
"topological
symmetry"
(&G(O))
which allows the metric the fields g,p,
to be gauged away locally. as follows:
~Lgczp
Under these transformations,
eaa and wa transform
GDg,p = &up + Dpv,,
= 0,
6Dez = epaD,vp,
6Lei = babe&, bLWa = &A,
c?DW~= EabDaDcyVb,
(3.1)
~wgap = PSaP,
6Thp = kp + opa,
&et
= ipet,
&fez = e$ t&d, = Eabdat&,. was presented the EinsteinIn

6wwa = +,pdPp, A different by Witten Hilbert approach to the symmetries
of the theory of gravity He showed that
for the case of three dimensions!151 was equal to a ChernSimons
action
action for the group ISO(2,l). First,
analogy, we want to analyse the ISO(l,l) gauge transformations _..._ . . given in (3.1).
theory.
we want to compare the to those
of ecva and We , the gauge fields of the ISO( l,l), gauge fields by +
Wa
We denote the ISO(l,l) A, = e:P,
ff abJab of the two translations algebra: = EabPb. = D,u
(3.4 and the
where Pa and J = $tabJab are the generators Lorentz transformation.
[pa,pb]
They obey the following
=
[J, J] = 0,
[J,P,]
(3.3) where u = vaPa +
Using the usual transformation I __T_  _AJ, one gets :
for gauge fields, 6A,
6(1,1)ez = daVa
q1,1pJa = aan.
+ kabeab
 EabWaVb (3.4
The A transformation
is identical
to the Lorentz transformation
8
given in (3.1).
If one uses Dpez = 0 when A = 0, then the difference ,*.. only a Lorentz use the ISO(l,l) Lorentz usual D, transformation with a parameter
between
(3.4) and (3.1) is
A = V"w,.
It is thus possible to and local is as
gauge transformations
instead of the diffeomorphism
transformations = d, + [A,,
when applied to ecxa, wa. The covariant ] and the field strength is
derivative
~~~ = [Do, Dp] = D[ae$Pa
+ d[awpl J
(3.5)
given in
where 1 1 denote antisymmetrization. (3.4) F,p transforms as follows:
Under the gauge transformation

b(,,,)F,p
=
(kabDlaepjb

Eabd[,Wp]Vb)pa
(3.6)
Note, however, that the transformations The ISO(l,l) group which
(3.4) d o not include the Weyl resealing. group of a flat Minkowski spaces with positive spaceand negThis is
is the isometry
time can be generalized ative _..._ . . curvatures,
to maximally
symmetric
the deSitter
and antideSitter constant,
spaces, respectively. X. For the deSitter
achieved
by introducing
a cosmological
space,
X > 0, the isometry
group is SO(2,l)
and for X < 0 it is SO(1,2).
The gauge field
is still given by (3.2), but the algebra is now:
[pa,
pb]
=
XEab J
[J, J] = 0,
[J, Pa] = EabPb.
(3.7)
The invariant operator __
quadratic
form which is consistent
with
a nondegenerate
Casimir i
is < J, J >= I, L. 3 < Pa, Pb >= and just x6ab. (3.8) X
For positive

X we can rescale Pa + 4x1
take X = 1. For negative
one can rescale with
and set X = 1.
9
As will be clarified
later, we will be
interested
only in X = 1 ; so from here on we will laws of the gauge fields are
dava + iiEabe,*b
discuss only this case. The
transformation
I
Eabf&Vb,
42,l)C q2,l)wa
= =
d,A
 eabeaavb.
(3g)
The fieldstrength,
which is now
Fcyp = [Da, Dp] = D[,$l
Pa + ($+~pl
t Eabhaepb)J,
(3.10)
transforms
under SO(2,l)
as follows:

6(2,1)Fap = (bcabDiaepjb
 Eabd[,wpjvb)pa
I EabDc*epavbJ.
(3.11)
The "topological" the "topological" simply _..._ . . The difference approach that
ISO(l,l)
or SO(2,l)
transformations
are now different The former
from are
gravitational
transformations
given in (3.1).
&A,
= 0, ===+
be: = Oz,
6w, = 6,. in the ISO(l,l) gauge fields.
(3.12) and SO(2,l) We expect In the
is obviously treat
due to the fact that
we, apriori,
eaa and wa! as independent emerge from
the relation geometric
wa(eora) will picture
the equations
of motion.
standard
of spacetime
the condition
for having
no torsion
relates wa and eora. The next stage in defining is choosing a _T_ ic _invariants the action. TG after specifying the fields and the symmetries, L: = 0 module topological
We advocate
the Lagrangian
and the elimination invariant
of auxiliary
fields. However, we believe that unless a nontrivial quantity left
there is a topological TQFT invariant cannot
expressed in terms of the giverrfields There must be some topological
be constructed.
which constrains
the global properties
10
of the ghost fields, for otherwise
the gauge field could be completely i . ,cthe instanton Euler number invariant invariant number
transformed
away.
For example,
in TYM the
must be left invariant,
andin two dimensional
gravity
must be left invariant. Lagrangian.
We, thus, prefer to construct In two dimensional gravity
a topological the natural
as our original
is the EinsteinHilbert
action: IQ = d2x&jR. (3.13)
s
This can be reexpressed f!o = &R
in terms of eaa and wa as follows: = det(e)eaaefRa; depce7d, (3.14) = Ppd,wp, for the Riemann two form assuming D,ez = 0. it ; 
= det(e)eaaeiRzqp
= det(e)eaae~dcadd~,wrl, = ~det(e)det(e` )Ppi3~awpl where we used R$ = cabdl,wpl
The last expression
raises two issues: (i) Since we get a total manifolds without
derivative, boundaries
looks as if the action is zero for two dimensional whereas, _..._ . . on the other hand, it is well known
that the Einstein
action
is in fact
the Euler number by noting that fiR
87r( 1  g) w h ere g is the genus of the manifold. is only locally a total derivative.
This is resolved
In fact it is an element to
of the second cohomology construct globally
group of the manifold
and measures our inability (3.13) is just
flat coordinates.
(ii) The action
a topological
abelian action
(i.e. the first Chern number): IO = J d2xPP&wp = f s to the topological d2xPpFmp = ; J Maxwell theory. F. (3.15) This 4
So it may look as if TG is equivalent statement I __e_  relation equivalent is incorrect, D,e; since the action
(3.15) is independent of motion. Nevertheless,
of eora; thus, the Hence, TG is not we now want to
= 0 cannot
emerge as an equation Maxwell theory.
to the topological
analyze the topological TG.
U(1) theory,
and later in section 5 we will come back to
11
4. Topological
.. _
Abelian
Gauge Theory
Leaving aside momentarily
TG theory, we now proceed to analyze the action
given in (3.15). The field w, is now an abelian gauge field which is not related to the twodimensionalmetric. formalism of topological This will be a toy model useful for understanding gauge theories though there is no interesting as we have seen it is closely related to gravity, group is just U(1). 0 ur initial topological the
topology since the is
for this case. However, twodimensinal Lorentz
invariant
the first Chern number cl = & s F. Considering (as is often done for instanton applications)
a noncompact
Euclidean
space
one has,

J J
F
=
d2xPPF aP =
J WC2 121=00 f 6C
=
w,dx",
(44
where the boundary QED instantons) breaking!16' .. . .
6C is a circle at infinity.
Nonvanishing
results for cl (i.e.
are known to exist for scalar QED with spontaneous symmetry QED instantons. For _
There are, however, no pure 2dimensional
our case, even though a priori clarified below, the Maxwell symmetry
there is no equation of motion
for wa, as will be
equation will emerge as the equation of motion once is gaugefixed. Thus, these instanton configurations Riemann
the topological are not relevant surfaces without cohomology,
to us. Moreover, boundaries.
we are interested
only in compact cl #
For these manifolds i.e.
0 only if the second equation) but with
H2 (C) , is nontrivial;
dF = 0 (Maxwell' s
F # dw globally. over C.
The relevant configurations coordinates
are thus nontrivial
vector bundle vector fields. global
In holomorphic
these are the meromorphic
The cl which is also the Euler number measures our inability to construct z _T_ vector fields and is given by the number of the poles.  holomorphic
_
We now gauge fix the topological the ordinary U(1) s y mmetry.
symmetry,
6w,
= eoI, while maintaining
This is done following 12
the procedure we introduced
for other topological
quantum
field theories!"'
.
f$) = f$$,+FP) =&` ,[i~(~ap&wp  c + is)],
ffpa,wp where (GF + FP) stands for gauge fixing by 6~1 = ie&r constant  C)  iiPP&$p and FaddeevPopov, with E a constant + 9,
(4.2)
and the BRST anticommuting
transformation p.arameter.
is denoted Th e commuting
C has the same sign as the Chern number.
Under this transformation 2, and the auxiliary

the gauge field, wa, the ghost field, qp, the antighost, as follows:

field 5 transform
iTI", ` &+
= =
$, B,
i,,$ A
= 
0,
STUB =
0.
(4.3)
we expect an additional under the of a gauge
Since we have gaugefixed Iocal symmetry. ghost symmetry: field. _..._ . .
only one degree of freedom, to check that
It is straightforward $TrG, = ;aa$
(4.2) is invariant
which
is the U(1) transformation by:
We fix this additional
symmetry
c(2) = &,,
The BRST transformations
= 8Tl[i&qp]
2 of the ghosts i,
= l&j,&
2 71" and 4 are:
 ;+&p.
(4.4
&1X = 2ij
&+j
= 0
i&i
= 0.
(4.5)
Note that while L(r) was expressed in terms of forms, in fZc2) we had to introduce I __P_ _a metric. Therefore, each term of the Lagrangian @j, and the. derivatives diffeomorphisms. namely, @,w, 13 which does not include with cap is respect
4
iC in fact multiplied' by to the two dimensional
have to be covariant algebra
The BRST = a,J,
is closed up to a
U(1) gauge transformation;
and &,
= 0 on the rest of the
fields since they are neutral. G . L: = Lo + L(l) ;are left .with the following
Altogether,
after eliminating
the auxiliary
fields, we
U(1) gauge invariant
Lagrangian:
+ Lt2) =;c"pFap
+ f(6%,wp
 C)"  i~~ap&~p (4.6)
The anticommuting ten as :
part of the Lagrangian
(on a flat background
) can be rewrit
jypa
,Gp
+ I?ac#

gT
$4
P7)
where we have denoted the vector The operator 3 is given by,
(GO, $1) by 4 and the two scalars (G, i)
by 2.

where the gamma matrices
are:
7' = 01 and r"=
 .. _ .
03,
with
{r",rP}
= 26' Ip
(4.8) Writing eq.
We then see that (4.7) in holomorphic
b is just the usual 2dimensional coordinates we get,
Dirac operator.
gT $J = g+a&+ + ga&L
where d, = ai + idc, $J* = $1 f ;+e, gh = 21 f ;gc.
(4.9)
The anticommuting for ??;and the two the
zero modes will then be the 2g (anti)h o 1omorphic constant Index(a) _
differentials
zero modes for the scalars, = 2  2g which is just
2.
Interestingly,
we then have that
the Euler number. theory.
We will come back to this
later
when considering For the calculation
the SO(2,f)
of the invariants gauge symmetry. 14
of the theory, To maintain
we obviously
have to
gaugefix
the abelian
the closure of the BRST
algebra i . ,z.
, &,
= 0, o n each of the fields we have to use a covariant the familiar c, E, b ghosts. After eliminating
gauge fixing fields,
by introducing the Lagrangian
the auxiliary
(4.6) takes the form:
L = L(O) + L(l) + J!?2)+ l?d,d"c + pawy2,
and we have to modify and add new transformations
(4.10)
to (4.3) and (4.5) as follows:
&lWa = 4, + a,c &lc = b = &` da.
We will now discuss the conserved term fZ(O). The BRST
&` = 4 lC
(4.11)
currents Noether
for the action current which
in (4.6) without follows from the

the topological latter
action is given by:
Ja

;iaa$

e,pfaP$.
(4.12)
The energymomentum
tensor Tap now takes the form,
.. . .
Tap =;[(F&$  (&&pi + qvil?;p  ig,p(F,aF76 + a,iapJ + +cfL + 2C2)]  g,pd,id7$)  gcYp~,+P) (4.13)
where
I__T_
_
x ap
=i,&[(Fac + &api
+ ~~acZ)~;; L.
 ;gnp(F,se76
+ [email protected])]%
+ 7?lpa,i  gapq7d7i). It is straightforward 15 to check that D,T"p = 0.
Thus Tap is a BRST commutator.
To check for scale transformation
we find that
.
T," = i(FapFap
 C")
(4.14)
Therefore, dimensional This
the action is not invariant topological
under local scale transformations.
The four
theories are invariant
under global scale transformations. as a total derivative,
is not the case here, since T," can not be written Thus the action is not invariant U(1) ghost number symmetry of the invariants
T," # D,R".
under global scaling.
Nevertheless, role
there is a further in the construction (w,,$,, X,6,$,
which will play an important Under this symmetry
of the theory.
the fields
i) carry the charges (O,l,l,1,2,2). the observables we follow the procedure in the TYM is not a BRST 0 # &+` . theory!"We commutator outlined for the Doninvariant
In constructing aldson polynomials operator &ro
W(2")
0
first search for a BRST of another must operator
0 which
0' namely: ; independent. cycle
= 0, but
This operator
also be metric
Obviously,
= &Z
6 fulfills
this condition.
Thus, we take for the zero homology of W indicates the ghost number
w h ere the superscript is IO =< as follows:
. The as
_.. . .
sociated homology
observable invariants
l~Vo(~~)>.
We can now create the chain of higher
0 = i{Q, W,$2n)}, dW;2n) = i{Q,W,
(2n 1' )
(2nd29' ,
[email protected]")
wpz94
= @l
= &n1$ = &nl ,
dVV(2n1) =~ ;{Q,W, dw:2n2)
2
wl(2"2)
2
F + n(n  l)Jnb2&
A 4,
= o. (4.15) independent observables then take the form lizn) homology cycle. For the simplest 27rci. = srlr VVi2n) where
The metric I __zz_ _
rk is a kdimensional I;" =< 5 >,
case (n = 1) we get to TYM , in the
I1(I) = f 4 and ii
= Jx F 
Contrary
abelian theory zero modes.
the (4, i) system does not have a potential We, therefore, believe that 16 all the invariants,
and thus has constant < VVi2n) >, vanish.
By noting i . ;different
that the (E, c) system is the same as the (6, i) system apart from the statistics, will we see that cancel. in the partition function the zero modes of the lead
two systems to a vanishing observables
However,
for < Wc > the (c, c) zero modes will invariants,
result.
Just as in the case of the Donaldson values of operators
nonzero
are those expectation
which can absorb the zero obeying this condition surface. This operator explicitly.
modes of the anticommuting is lYI:&f&
ghost, 6. The only operator
* 72;) w h ere g is the genus of the Riemann
cancels the 2g zero modes of 4. Its expectation It is easily seen to be independent of the metric:
value can be computed
(fi / 4 A 4) = det[/ i=l c
$i"' A qj"` de$Jfa2, ]
(4.16)
where the 4:"' are the zero modes of 4. We choose the basis G(O) Pai( where the wi(z) are abelian differentials the differentials
= c,[BiWi(Z) +
HI(C)
and the 6i are anticommuting the canonical
parameters.
We normalize
by choosing
basis, ai, bi, such that
Wj = Sij.
(4.17)
_..._ . . Then, the period matrix
a; of the Riemann
Wj =
f
surface, C is given by:
Tii.
(4.18)
f
bi
Using the Riemann
bilinear
identity
for closed lforms,
pl,p2,
/P1~P2=~~~P1fP2fP1fP21, c
(4.19) bi bi ai _
i=l a, I L.

,it
is simple to show that,
_
det[ $t"' A $!"` = det[ wi A Qj] 3I s J a c c
17
(4.20)
where, i
.
,c
..
c
Wi A Uj
=
.I?l%(Tij).
(4.21)
J
Hence, we have that,
(fi
J 4 A 4) = det"/2(lmTij) l c
defF:;fa2.
(4.22)
Det' jJ
is just the determinant Since the theory structure,
of the Dirac operator is topological
with
all periodic
boundary on the 
conditions. complex
and thus does not depend
we have that, det' f?) det' 1/[email protected]
 [det(lm
~)]l/~.
(4.23)
For g = 1, we know that the above determinants
are given by:
det' $
_..._ . 
=(~)1/2~29~` ~~)T) =72 [q(r) 12,
1 = fi2jq(~)12,
(4.24)
det' lj2a2 which verifies
the general relation
given by eq. factor.
(4.23).
Indeed,
(detImT)1/2 our Gi(` ) in (4.23) in this
is just the zero mode normalization so the their determinant
We could have normalized
would be one; then the ratio of determinants
would also be unity. theory.
We thus see that there are no interesting are the identity
observables
In fact, the only observables
and the Chern number.
c a _T. L.
18
5. The Problems
i ,z.
of a Pure Gravity
Theory
expressions the
.
Since the topological for the desired invariants difficulties
U(1) cannot provide quantum field theoretic
on moduli space, we also want to briefly summarize gravity in two dimensions. action as in the U(1) theory.
with topological
We begin with
the same topological
However,
the gauge field is now identified forced to add to the Lagrangian tion of the 4 and the Christoffel
with wol, the spin connection.
We are, therefore,
(4.6) terms which emerge from the BRST variasymbols. For example, in the first stage of gauge where

fixing we have to add [&rti]fE, I' :, is the Christoffel connection.
and in the second stage fii[$Tlrzp]qp
The problem is that we cannot invert the rela&r&a in terms of 4. A natural way transformation of wa in terms
tion wa(eara) and express the transformation to resolve this difficulty of &rez
is to express the topological &SW,
= ?+!I:. As was given in (Xl), to a flat connection,
= EabDb?,baa= det(eW1)ep7D7tiap. = 0, we take,
Gaugefixing
Ppd,wp
c1 = f!&[~~"Pd,wp ]
_..._ .
= kCpd,wp
 det(e` )~Ppc7' D,D~~p7.
(5.1)
.
Using [email protected] = (detg)(g"rgp&
 ga6gp7), we get,  D,D*$J$), (5.2)
p = &@a awp  det(e)i(D,Dp$aP
= JijI(;iiR which is exactly  g(D,Dptiap
 [email protected])],
the same gauge fixed action for pure gravity gauge fixing of the additional
that we derived will also be
in the past!"The the same.
ghost symmetry
The BRST provided
algebra is closed up to a diffeomorphism the "ghost for ghost" 4" (a,,?& as was explained topological
and Lorentz is
transformation,
not a
is
a_the
BRST
scalar, 8~~4~ = er$E4 b. This,
in the introduction, invariants. Different
source of the inability gaugefixing
to derive nontrivial
~T,[E".P~Pw~]
procedures such as
or those presented in reference [9], ,
cannot escape f$~~rJ~ # 0. 19
At this stage it appears that we have reconfirmed ,c. that TG does not contain a quantum field theoretic invariant
the statements realization
made in [9]
of the Mumford This theory.
classes and that
the only topological
operator
is the identity. quantum
means that TG ( in both two and four dimensions) What is the reason for this situation the zero homology ? Technically,
is an empty
it is due to the fact that This whose
while in TQFT is obvious parameter
invariant
is a scalar, here it is a vector.
because the BRST is a vector.
algebra
is closed up to a diffeomorphism
Since both the topological ants on Riemann moduli
U(1) and the TG theories cannot
provide
invari
space, we must look elsewhere. between the ISO(l,l) (or SO(2,l)
We showed earlier that for g > 2) gauge transof w, and ez. It and SO(2,l) ISO(1, non1) will
there is an equivalence formations is, therefore,
and the diffeomorphism natural to ask whether
and Lorentz transformations the topological ISO(l,l)
abelian theories can save the day. The analysis of the topological be the subject of next section.
6. Topological
This provide ISO(l,l) know section will
ISO(l,l)
and g=l
Riemann
ISO(l,l)
Surfaces
theory. space
(Ml),
be devoted
to the topological
This will since We
a description
of the topology
of the genus one moduli
is the isometry from mathematicians
group of flat twodimensional that the only homology field theory
Minkowski groups of Ml
space.
are torsion (unless
groups!"` * it is through
We do not expect global anomalies).
a quantum
to probe torsion
Thus, interesting
observables
are not expected
for the ISO(l,l) theory. Nevertheless, let us see what we get from the quantum __T_ I field theory approach; this will be useful for comparing with the richer theory of  ic SO(2,l) for higher genus. . _* A torsion group is an abelian group in which every element has a finite order; e.g. ZN. 20
We begin with i . ,cdiscussed in section
the ISO(l,l) 3. Recall, that
gauge field,
A,
= e:P, Chern
+ waJ, number,
which
was
fZ(O), the first
is invariant
under the topological
transformation
of the gauge field, A,,
A
b(l,l)Aor
= *a
==+
h(1,&
= ti$
A b(l,l)Wor
= &Y
(6.1)
The strategy
for eliminating theories,
the redundent
(local)
degrees of freedom
is the
usual one for topological while maintaining %
namely gauge fixing the topological 1ocal gauge symmetry. having
symmetry
the ISO(l,l)
Since we are to describe Fcyp = 0. This 
we gauge fix to configurations that [email protected],wp
a flat connection:
implies
= 0, and [email protected],es
= 0. This configuration conserved metric
of gauge fields
can be simply
used to define a covariantly compact
with zero curvature. (g = 1). Let
Since we are dealing with us now write the topological
surfaces, this describes a torus Lagrangian,
gaugefixed
=i[
(iie
` @d,wp
+
[email protected],D,e~)
(6.2)
... . .
A
 i&%3
a' + ~aDcr$F i EabXaGaepb) &/3 variation. From here on exThe antighost x =
where
b(l,l)
=
~~~T(l,l)
denotes the topological
pressions written xaPa + jiJ
in group multiplet as follows:
will always be traced.
transforms
A
&r(i,i)Xa &(1,1)X = B ==+{
= =
Ba, B*
&(l,l)i
(6.3)
1, < Pa,Pb We know >= that babe the 
Note that
i2
we took
the quadratic "Casimir" of ISO(l,l)
form to be < J, J >= with Pa.
The corresponding _quadratic Casimir
does not commute is degenerate.
It is just form.
Papa.
At any rate, we will still
have no choice but to use the above quadratic 21
The Lagrangian
have an ISO(l,l) ,; . transformation.
gauge symmetry,
but x will transform
by a nonstandard
gauge
The gauge transformations
of the fields are given by eq. (6.9). theory which
These complications
will not appear in the higher genus SO(2,l) quadratic symmetry Casimir. was only partially Because the BRST we anticipate
does have a nondegenerate Since the topological an additional ISO(l,l) ghostly
fixed, we expect to have algebra will close up to an of Q under
symmetry.
gauge transformation
that the transformation
this symmetry
will be like that of a gauge field under ISO( l,l),
namely,
&,I)
By transforming it is invariant,
Xl?, = iD,@
===+
&y1,1)& &ii =
&y,,l,~: = i(DaqP I cab&ab)a
(64
the Lagrangian provided
(6.2) by (6.4), i t is straightfrward transform B as follows:
to verify that
we simultaneously
6 = cab&&,
This again could have been anticipated _..._ . . So far we have not introduced action was constructed
&` (@" = Eab&,.
by the closure of the BRST metric, algebra.
(64
a worldsheet
since our gaugefixing
only from forms.
In the second stage of gauge fixing, We have to decide whether metric, our
the introduction
of a metric
is unavoidable.
gauge fields, w, and ez, will be identified with ISO(l,l) Following g au g e fi e Id s ( completely
with the worldsheet
or simply metric). can
unrelated
to the twodimensional
the first alternative
will mean that the ISO(l,l) functionals
gauge symmetry $j
not be maintained, Christoffel a _=_ _. _
since the additional
of the metric, covariant
and the
symbols,, cannot be expressed in an ISO(l,l) we would like the theory
way. From the
bundle structure and
to possess (as discussed in section 2) are
the above considerations, related metric
we emphasize that the gauge fields of ISO(l,l) metric. As in TYM, we simply
not directly worldsheet
to the worldsheet as a background
treat the
field. 22
The second stage of gauge fixing thus
takes the form:
L:(2)` = $~+,, = &J&)[;
= ixD,[email protected]
` XD,9"  iqD,!lP
+ xB] + X[\k, , Qa] + B2 + ~&&3] (6.6)
In terms of the "component" ~(~1 =;&3aP$
+ &[ifja,@
fields the Lagrangian + X,D,D"4a
I iqa(Da$z 
is:
hbf$i+]
(6.7)
+ fi[icabXa$ab&=
6 h
+ 5," + B,B" i cST(~,~) = +,
+ iEab$axbi], 8T(1,1~Xa = va. Following instead we =
where we used &(r,r~X our work on TYM, of XD a KP[*` * In
= 2q, =+
we can insert in equation
(6.6) a term X(D,` P
 fa)
this case in order to conserve the U(1) global ghost number = i[d, A] ==+ &(r,r)rj = 0 &(r,r)va
have to take 0 = [@,q] and &(r,r)~ &&(&ib  kjb).
This will lead to an additional
term in the Lagrangian,
..._ . _
Let us now verify transformation can be realized
EabxdJqb

ij$b)]
=
;(&.k$b

cab+q,4b)s
(6.8)
that the BRST
algebra is closed up to an ISO(l,l) parameter Q = $J + @Pa. Explicitly,
gauge this
with a transformation directly
from the transformations: ===+8~11,1)e~ = Da+"  cabeab&
==+~;,,J)~: &(I,I)% = ~ab(&hab = EabXa4b, d' b&),
@G(,,~~A, = [email protected]
@(,~)*a = [@a,*]
B;(~Jya
&,,q g&,qxa
= aA
= 0, = cab$xb. (6g)
z _T_
One _
may suspect that the third
line in (6.9) d oes not describe. a gauge transforfrom the gauge invariance of
mation
of x; but, in fact, it does as can be verified in (6.2).
23
the first expression
The invariants
of the theory
will be discussed in section 8. We now proceed case of the higher genus Riemann surfaces.
to explore the more interesting
7. Higher
genus Riemann SO(2,l)
surfaces and theory
negative curvature to a
the topological
Riemann metric positive that
surfaces of g 2 2 can be described by a constant metric). As will soon become obvious, constant. The corresponding
(hyperbolic (X = +l)
this corresponds symmetry
cosmological
group is
of SO(2,l).
owing our discussion F o 11 SO(2,l) gauge theory.
in section 3, we proceed to construct the same procedure that we

the topological
We follow
used in the last section. First, connection. &D,e; we fix Fo,p to zero so as to project For SO(2,l) this implies onto configurations with a flat
the conditions,
c"@a+~p + det(e)
= 0 and
= 0. Since &d,wp constant
= det(e)R,
we, in fact, gauge fix the curvature Lagrangian is
to a negative
(R = 1).
The resulting
_..._ . .
l(l) = &(2,1][@[email protected]]
=&(2,1)[if(@d,wp [email protected][(s(aawp  i(?(capdaGp
i xa(Datii
+ det e) + [email protected]], + irabeaaepb) + B,D,eF) (7.1)
+ cabcap$,,epb)
i ~ab&epb)],
where &(2,1) fields );/,xa, i
A = i4(2,1) and B,
denotes the SO(2,l)
topological
variation,
and the ghost 6. Under the (as in eq
are defined in the same way as in section
ghost symmetry, _
S transforms
again like the gauge field under SO(2,i) L.
&(2,1)&
= = aa6 + Eabeaabb9
ic
3.11), namely,
A &(2,1)&z
* &(2,1)!&
D,qS" I Eab&orb,
(7.2)
24
and the auxiliary
,z.. .. _
field B transforms
as,
&F~,IJB
= i[Q,x]+
ii =` cab&,& .&q2,1~B~ = cab&b j&$b). (7.3) b(2,1)
Similarly,
the second stage of gauge fixing
is:
~(~1 = L&+~~
=
&%T~2,1~[3D,~a
+ xB1,
 @Pa ,Sa] + B2  ix[Q]. (7.4
= iAD,DV
 iqD,W
In terms of the "component" L(2) =i&[xa,affJ + &[i;i(a,@
fields the Lagrangian
is:
+ XaD,D"~"], + eabPp eaa$pb) I V(D&Z + $,b:qg) k i2  eabeL$a)], + Bd"
(7.5)
+ &[icab(&?/Jab?,iQ
iEab(2#aXb? + ixaxb)],
A
A
where we used &(2,11X .. . . term in the Lagrangian
= 2~, which
` 6T(2,1~X = c, was introduced
6Tc2,r)Xa = qa The additional in the last section for ISO(l,l),
now takes the form
(7.6) = ifi' ab($qafb d 2Gvadb) i gi(d2AaAa i2qjar$a (EabdaAb)2],
using that &(2,r)q
= [@,A]. to write the total Lagrangian, terms: l = l(O) +
We can combine these expressions f(l) I __T.
_ _r* lkin =fi[(@&yp)2 + i~[(C + (+d$$] @aa$/? + Xaaa+z)
+ p)
+ p),
and collect the kinetic
+ (Il"aaGa
+ rlaaatiZ)l
V7)
25
The gauge interaction ,c. J&, _[ip, ([A",
terms can be written
as LCgauge= &(J&,Aa)
where
Ap] + 2d["Apl)] + @[x,&I) + $[Q,aa~] + [a% Xl + [[C @)I,Xl)
 i([rl,V]
ae~] + ecb(w"epc  e:wp) ={ cabePa[2d'
ieab(rla$f ·k EaPXa$pb

i [a"($aAb)

k (W"da +

e&&]))J 
{+cab[wP(w"ei
+ + Eab[(+$," P(fJAa)
eacwp)ec,
+ +
ePaccdeFepd

2(wPi3iae;l
ePad["wpj]
$"rla)
@(%+pa ebdezcjcXa
Xa9zp)
A(W"4a ez$)]}% P8)
iY(id,)
and the other interactions L
id =&[iEab(Xa$hb$C
are,
+ ~i$~$f) iCab(2$aXbg + $XaXb)], (7g)

r]acb
+
2Gva4b)
+
$[J2xaxa

i2badu

(Eab+aAb)2],
The derivation tensor is similar _..._ . . and therefore
of the Noether to the derivation
current
JtRST
and the energymomentum abelian theory (4.124.14)
for the topological
the same conclusions
hold here also. The BRST with a parameter
algebra is closed
up to an SO(2,l) can be realized
gauge transformation directly
@ = c$J + qSaPa. This
from the transformations:
@c2,l+x
=D,@ 
G(2,l)e:
G(2,l)W~
=
=
D,q5"  eabeab$, aa + Eabe$$b,
cab (&kab cab+a+cvb, cabXa$b, eab&b. #b&x),
~&,1pL
&(2J)X
a _T_
.
=[vkY] I ~;(2,,)~: I J ` =+ [@,X1
G(2,1)@
=
=
(7.10)
s^;,,,,,2
=
@(2,4$=?
=
L.
_
In general the BRST
algebra on any operator
[h(2,1)EI > &(2,1)E*P
is closed as follows,
k~2P% 01.
=
(7.11)
26
Any explicit A. ,; symmetry..
calculation This
will require a further
gauge fixing FadeevPopov
of the SO(2,l) procedure. of the theory.
gauge In the
is achieved by the standard on the topological
next section we will elaborate
observables
8. Topological
Invariants
of Moduli
Space
to the topological construction a description space is We have of
We will now discuss the quantum invariants of moduli space.
field theoretic
approach
Before proceeding
into an explicit our theory provides
the observables, of the topology equivalent
let us discuss why, in fact, of the moduli space of curves. metrics
In this context,
moduli
to the space of Riemann
modulo
diffeomorphisms.
chosen a gauge group whose action on the metric For g 2 2 the group SO(1,2). Since ISO(l,l) is SO(2,l);
is just that of diffeomorphisms. and for g = 0 it is quadratic Casimir and
for g = 1 it is ISO(l,l),
d oes not have a nondegenerate topology,
consequently
Mr has no interesting
save for a torsion
group of order 12
117' will from now on only consider g 2 2. we The structure of a projection _..._ . gauge fixing of our theory .is that of a principal 7r : E + B. SO(2,l) bundle consisting From our
The fiber is the gauge group SO(2,l).
condition
we see that: E = {flat SO(2,l) connections over Fs} now also show that would condition E is be (i.e.
where Fg is an oriented equivalent
surface of genus g. on Fg.
We will Thus,
to the space of metrics to the moduli
B = E/S0(2,1)
homeomorphic vanishing
space. Recall that the gauge fixing of SO(2,l) + wiae;l = 0,
field strength)
on the generators D[,eFl =a[,ej$
was such that: (8.1) G
* _T_  *a_
d[,wpl We can define a covariantly
=  det(e), L. conserved metric
by g,p = ctcpa.*
From the first
* It should be noted that this is to be distiguished from the worldsheet metric in the quantum field theory. 27
relation i ,c
above we can determine
wa as a function
of e: . Then, substituting
the
expressionfor
wa(eg) into the second relation fiR=&j =+
above will give us, as shown earlier, R=1. (8.2) But we know that negative curvature and
This is the condition all Riemann metrics (i.e.
that we have constant
negative curvature. by constant
surfaces of g 2 2 can be described hyperbolic metrics
). For g = 0, the gauge group is SO(1,2) the verification
we have positive provides
curvature.
This then completes of the moduli the dimension space. of this This
that our theory
for a description
Let us now compute fermionic modes} operator  {number
space using
the index of Q,
of the zero are
in our Lagrangian. of (x, Q) zero modes}.
is simply:
{number
Since \k,
is a vector
and (x,r))
scalars, we have that the dimension Dim(SO(2,l)) sult.
of moduli
space is: is, of course, a wellknown re
x (2g  2) = 6g  6.
This
There is one unfortunate
fact that we must note. Because we always have basis for of
(x,~,J) zero modes, the XP, zero modes do not form a good coordinate moduli _..._ . space. In TYM polynomials Witten was able to construct an explicit
mapping moduli
the Donaldson
into differential
forms on the instanton This was possible
space
after an integration analogous moduli
of the nonzero
mode fields. deformations)
because his
KP zero modes (instanton
could be used to parametrize there are no (x,~) to choosing a 9
space, since he considered
the generic case in which
zero modes.
In general, the x zero modes are the obstructions moduli space.
which coordinatizes For the moduli
of Riemann
surfaces we know that the quadratic with
differentials
are good coordinates. * _=_ 
Since we must be content we can even though
our KIXca and. x, we will mapping onto the
glean whatever differential
information
the explicit
forms on moduli
space will not be so easy. of our observables, or topologically
Let us now proceed with the construction invariant correlation functions. This will 28
be very much along the lines of ref.
[I]. i . ;
Witten
showed that manifold.
the observables
will be a ring of differential
forms on degree of the
the background beginning BRST with
These will be a sequence of forms of increasing from the invariant polynomials
the zero forms constructed
scalars:
J w(4n1)
1 7 2
wt4") =TT(@2)n, 0
=53(2n
s 7 [email protected]),
c
sW(4n2)
=73(2n
FapB2"Ids"
A dxp
(8.3)
A dxp),

c
J
+ 2n(2n  1)
J c
(p2n2Q,XlQdxa
where we have dWN = {QB~s~, is then the structure Mathematicians the form theory _..._ . . S' wk lk
WN+~},
{Q, WO} = 0, and WO # {&,A}. invariant
This
of the generators
for the topological
polynomials.
refer to these as characteristic where rk E Hk(C) choose a combination and wk
classes. We see that they are of E H"(C). In the quantum field
we must
of these classes such that
the fermionic
zero modes are absorbed. number will
This involves choosing a product of a differential
of classes with ghost surface The
6g  6. The ghost number
form on the Riemann form on moduli
correspond
to the degree of the differential as mappings
space.
observables
can be considered
from forms on a Riemann invariants,
surface to
forms on the corresponding previously, identified
moduli
space. The Mumford
as mentioned
are elements of the even cohomology with the even ghost number with Wi2).
groups of M,.
They will then be ICI E H2(Mg) to:
observables
. For example,
will be identified
It is known that lcfg' # 0; this corresponds
393
*
_T_
K:""
MM
+ (Tr n i=l L.
mq. J c

(8.4
limit where it is
_
The expression identical
above can be computed
in the weak coupling theory
to the computation
in the abelian 29
(done in section 4). This is
an exact result i . ` verifiesthat
because the theory
is independent
of the coupling
constant.
It by
&fge3 # 0. The Euler character They found it to be: x(h) SP  29) =  qg _ 1) zeta function.
of moduli
space was computed
Harer and Zagier!"'
(g > 1)
(8.5)
is two dimensional should not be too
where c denotes the Riemann using quantum field theory.
This should now also be computable field theory
Since the quantum
and a weak coupling difficult. work. In summary mology
expansion
can be used, the computation
It appears to be related to a zero point energy. We leave this for future
our observables
describe the following
mappings
between coho
elements on the Riemann
surface and those on moduli + H2"(Mg),
space:
Wo(2n) :H' (C) W,(2n` ) W,(2"2) :H' (C) :H2(C)
+ H2"` (M,), + H2"2(Mg),
(8.6)
_..._ . .
9. Conclusion
In this paper we have constructed topology of moduli space. a quantum principle field theoretic description of the
The guiding
has been the study
of moduli
space topology theories
by using topological
gauge theories.
These are gauge invariant onto certain classical
whose Lagrangians
are assembled by gauge fixing of the manifold space).
configurations. tions modulo
They probe the topology gauge equivalence
of classical configura
(i.e. moduli
For the study ,of the moduli * _=_ _be the symmetry This is SO(2,l) gaugefixing of M,.
space of curves we chose the gauge group to of a Riemann surface of a given genus. for g = 0. Then by description
group of the metric for g 2 2, ISO(lii)
for g = 1, and SO(i,2)
the connection
to be flat, we obtained
a field theoretic
30
The previous ,L.
attempts
at obtaining theory
a topological
theory
for M, were based
on develop.ing a topological that moduli
of pure gravity. [` We have seen in this paper I by an SO(2,l) gauge theory on a curved
space is described naturally background. several open problems.
twodimensional There remain space topology,
Since so little compute
is known
about moduli
it would be fruitful
to explicitly
some of the invariants. field there
This should not be too difficult, theory
since we are dealing with a two dimensional There is also a question as to whether
about which much is known.
are any global anomalies.
We don' know what effect torsion t field theory.
groups in moduli to extend
space will have on the quantum this work to obtain a description that a supersymmetric
It would also be interesting
of supermoduli
space. We would naively expect SO(2,l) gauge theory work. will do the It may lead
version of a topological extension
job. This should be a straightforward to interesting consequences.
of the current
Acknowledgements
_..._ . .
:
for introducing We thank us to the problem addressed
We would like to thank E. Witten in this work preprints. and for fruitful
discussions.
J. Harer for sending us his for useful conversations
We are grateful
to M. Peskin and R. Brooks
and for reading
the manuscript.
_. I _Ye_

2
31
REFERENCES
* ,c.. 
.
1. E. Witten,` Topological 117, 353 (1988). 2. R. Brooks, malization
Quantum
Field Theory,' Commun.
Math.
Phys.
D.Montano,and in Topological
J. Sonnenschein, Quantum
` Gauge Fixing
and RenorMay
Field Theory,' SLACPUB4639,
1988, to appear in Phys. Lett. B 3. L. Baulieu Topological and I. Singer, ` August 1988; talk YangMills presented Symmetry,' at Annecy 1988. IASSNSHEP87/7, preprint, Mtg. on
PARLPTHE88/18, Conformal 4. E. Witten, February
Field Theory ` Topological 1988.
and Related Topics, March Sigma Models,' IAS preprint,
5. D. Montano
and J. Sonnenschein,
` Topological B,
Strings,' SLACPUB4664,
June 1988, to appear in Nucl. Phys 6. M. F. Atiyah, ` New Invariants
for Three and Four Dimensional of the symposium
Manifolds,'
to appear in the proceedings _..._. Heritage of Hermann 7. S. Donaldson, Manifolds,'
on the Mathematical
Weyl (Chapel Hill, May 1987), ed. R. Wells et. al. of Gauge Theory to the Topology Invariantsfor of Four Smooth
` An Application
J. Diff. Geom. 18 (1983) 269; ` Polynomial Oxford preprint.
FourManifolds,' 8. M. Gromov, Math.
` Pseudo holomorphic
curves in symplectic
manifolds,' Invent.
82 (1985) 307. M. Pernici, preprint, Verlinde, and E. Witten, ` Topological Gravity in Two 4 Superstring
9. J. Lambastida, Dimensions,` ,IAS __. _=_  ..10. E. Verlinde,:H.
IASSNSHEP88/29, ` Multiloop Calculations
June 1988 in Covariant
Theory,' Phys. Lett.
192 B :95, (1987). of the Moduli 32 Space of Curves,' preprint
11. J. Harer, ` The Cohomology
12. A. Belavin i . ;JETP
and V. Knizhnik,
Phys. Lett.
168 B (1986) 201; Sov. Phys.
64 (1986)214. Homology (1982). an Enumerative Geometry of the Moduli Space of of the Moduli Space and the Mapping Class
13. E. Miller,` The Group,' preprint 14. D. Mumford,`
Towards
Curves,' preprint 15. E. Witten, preprint, 16. C. Callan,
(1982). Gravity as an Exactly 1988. and Massless Fermions in

` 2+1 Dimensional IASSNSHEP88/32, R. Dashen,
Soluble System,' IAS
September
D. Gross, ` Instantons
TwoDimensions,' S. Raby,
Phys. Rev. D16: 2526, 1977. ` Instantons in (1+1)D imensional 1978. of the Teichmuller modular group,' J. Abelian Gauge
A. Ukawa,
Theories,' Phys. Rev. D18,1154, 17. D. Mumford, d' Ana1. Math., 18. J. Harer _..._ . ` Abelian quotients
18 (1967), 227 . of the moduli space of
and D. Zagier, ` The Euler characteristic (1985).
curves,' preprint
*
_T_ ie.
33
Information
33 pages
Report File (DMCA)
Our content is added by our users. We aim to remove reported files within 1 working day. Please use this link to notify us:
Report this file as copyright or inappropriate
737844