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SLAC-PUB-4760 October 1988 CT)


The Topology

of Moduli

Space and Quantum

Field Theory*


MONTANO Stanford


Linear Accelerator Stanford,





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


introduced space. examine

the cohomology

of Yang-Mills quantum


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 DE-AC03-76SF00515. + 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.


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 Yang-Mills

for investigating


theories in 4-dimensions

and nonlinear


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 well-known 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


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


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


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


to two dimensional This


it as a topological gravity

gauge theory with

of SO(2,l). cosmological

is the isometry We will of

group of 2-dimensional

a positive


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


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


to investigate


one has to study the topological

gauge theory

space is M,.

We will

discuss the reasons for this in section Yang-Mills, 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


two dimensional


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


this zweibein

` a negative be be treated

that the zweibein related

and spin connection

as independent

only by the gauge fixing


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



Section 3 will be devoted to the fundamental TQFT' s emphasizing


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


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


we will summarize

the work and comment

open questions


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 non-linear field theory moduli provide

dimensional, equations.

to solutions of a quantum

a local gauge symmetry theories

over the relevant space, instantons The moduli

. In Yang-Mills an important

on 4-dimensional nonpertur-

tool for computing

bative effects.

spaces must then be understood

in order to complete

the final integration.

_-..._ . -.



the topology

of these spaces has recently topological realization

been dramatiYang-Mills 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-


where the gauge symmetry

is that of world-sheet

one has an analogous structure. (i.e. the metric) simplifies

The integration

over the gauge conspace of

into an integration

over the moduli


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-


has shown that topological moduli space.

field theories may provide a fruitful 4

new tool for studying

Mathematicians .?. principal G-bundles


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


In this section,


these two approaches.

Readers who are only 3. In section 8, we will


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


to describe the structure A principal G-bundle

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 G-bundle over some manifold G-bundle 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.


The observables of TQFT' s z- _P_ _of instanton -lore moduli

are the generators of H* (M) ,-the cohomology


; --

space. These are also elements of H*(A/$). which mathematicians of G-bundles; 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


dimensional ,c. rings: __ -

Lie group G, we have the following


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


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].


field theoretic The details


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, ", Q-21 ` + H* @A, $2) (2.2)

where /ci E H2i(BAg). Recall that diffeomorphisms. related!"' M, is the space of metrics As in Yang-Mills theory, for genus g Riemann H, (BA,) and H, (M,) surfaces modulo are intimately

Indeed, their rational


are isomorphic:

fL(BA,$l) __T_ ix-






Mumford Mumford

gave a construction invariants.

for. the classes, Ki!141 We will

refer to them of



They are elements of the stable cohomology


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


classes do not necessarily observables. Indeed,



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


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


space of curves. gravity un-

is to obtain a topological candidate quantum

theory for the moduli is obviously

a natural

for such a TQFT



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


used to describe two dimensional The zweibein vector


are the metric

and spin-connection. 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,


(60 (v)), 7

Weyl resealing

(6~ (p)), local Lorentz

transformations ,z..

(6~ (A)),

and the additional




which allows the metric the fields g,p,

to be gauged away locally. as follows:


Under these transformations,

eaa and wa transform

GDg,p = &up + Dpv,,

= 0,

6Dez = epaD,vp,

6Lei = babe&, bLWa = -&A,

c?DW~= -EabDaDcyVb,


~wgap = PSaP,

6Thp = kp + opa,


= 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 Chern-Simons


action for the group ISO(2,l). First,

analogy, we want to analyse the ISO(l,l) gauge transformations _-..._ . -. given in (3.1).


we want to compare the to those

of ecva and We , the gauge fields of the ISO( l,l), gauge fields by +


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.


They obey the following


[J, J] = 0,


(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


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


(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


~~~ = [Do, Dp] = D[ae$Pa

+ d[awpl J


given in

where 1 1 denote anti-symmetrization. (3.4) F,p transforms as follows:

Under the gauge transformation








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


the de-Sitter

and anti-de-Sitter constant,

spaces, respectively. X. For the de-Sitter


by introducing

a cosmological


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:




XEab J

[J, J] = 0,

[J, Pa] = EabPb.


The invariant operator __-


form which is consistent


a non-degenerate

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.


As will be clarified

later, we will be


only in X = 1 ; so from here on we will laws of the gauge fields are

--dava + iiEabe,*b

discuss only this case. The




42,l)C q2,l)wa

= =


- eabeaavb.


The field-strength,

which is now

Fcyp = [Da, Dp] = D[,$l

Pa + ($+~pl

-t- Eabhaepb)J,



under SO(2,l)

as follows:


6(2,1)Fap = (bcabDiaepjb

- Eabd[,wpjvb)pa

-I- EabDc*epavbJ.


The "topological" the "topological" simply _-..._ . -. The difference approach that


or SO(2,l)


are now different The former

from are



given in (3.1).


= 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.


of space-time

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 non-trivial 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


of the ghost fields, for otherwise

the gauge field could be completely i . ,cthe instanton Euler number invariant invariant number



For example,

in TYM the

must be left invariant,

and-in two dimensional


must be left invariant. Lagrangian.

We, thus, prefer to construct In two dimensional gravity

a topological the natural

as our original

is the Einstein-Hilbert

action: IQ = d2x&jR. (3.13)


This can be re-expressed 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


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


4. Topological

.. _


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 two-dimensionalmetric. 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, two-dimensinal Lorentz


the first Chern number cl = & s F. Considering (as is often done for instanton applications)

a non-compact



one has,





d2xPPF aP =

J WC2 121=00 f 6C




where the boundary QED instantons) breaking!16' --..- . -.

6C is a circle at infinity.


results for cl (i.e.

are known to exist for scalar QED with spontaneous symmetry QED instantons. For _

There are, however, no pure 2-dimensional

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 gauge-fixed. 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 non-trivial

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.



= eoI, while maintaining

This is done following 12

the procedure we introduced

for other topological


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 Faddeev-Popov, with E a constant + 9,


and the BRST anti-commuting

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 anti-ghost, as follows:


field 5 transform

iTI", ` &+

= =

$, B,

i,,$ A

= -





we expect an additional under the of a gauge

Since we have gauge-fixed Iocal symmetry. ghost symmetry: field. _-..._ . -.

only one degree of freedom, to check that

It is straightforward $TrG, = ;aa$

(4.2) is invariant


is the U(1) transformation by:

We fix this additional


c(2) = &,,

The BRST transformations

= 8Tl[-i&qp]

2 of the ghosts i,

= l&j,&

2 71" and 4 are:

- ;+&p.


&1X = 2ij


= 0


= 0.


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


i-C 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


after eliminating

the auxiliary

fields, we

U(1) gauge invariant


+ Lt2) =;c"pFap

+ f(6%,wp

- C)" - i~~ap&~p (4.6)

The anti-commuting ten as :

part of the Lagrangian

(on a flat background

) can be rewrit-



+ I?ac#





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


7' = 01 and r"=

-- ..- _ -.




= 26' Ip

(4.8) Writing eq.

We then see that (4.7) in holomorphic

b is just the usual 2-dimensional coordinates we get,

Dirac operator.

gT $J = g+a&+ + g-a&L

where d, = ai + idc, $J* = $1 f ;+e, gh = 21 f ;gc.


The anti-commuting for ??;and the two the

zero modes will then be the 2g (anti)h o 1omorphic constant Index(a) _-


zero modes for the scalars, = 2 - 2g which is just



we then have that

the Euler number. theory.

We will come back to this


when considering For the calculation

the SO(2,f)

of the invariants gauge symmetry. 14

of the theory, To maintain

we obviously

have to


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


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


currents Noether

for the action current which

in (4.6) without follows from the


the topological latter

action is given by:


= [email protected]






The energy-momentum

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)




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")


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



first search for a BRST of another must operator

0 which

0' namely: ; independent. cycle

= 0, but

This operator

also be metric


= &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


We can now create the chain of higher

0 = i{Q, W,$2n)}, dW;2n) = i{Q,W,

(2n- 1' )

(2nd29' ,

[email protected]")


= @l

= &n-1$ = &n-l ,

dVV(2n-1) =~ ;{Q,W, dw:2n-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 k-dimensional ---I;" =< 5 >,

case (n = 1) we get to TYM , in the

I1(I) = f 4 and ii

= Jx F -


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


for < Wc > the (c, c) zero modes will invariants,


Just as in the case of the Donaldson values of operators


are those expectation

which can absorb the zero obeying this condition surface. This operator explicitly.

modes of the anti-commuting 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, ]


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) +


and the 6i are anticommuting the canonical


We normalize

by choosing

basis, ai, bi, such that

Wj = Sij.


_-..._ . -. Then, the period matrix

a; of the Riemann

Wj =


surface, C is given by:





Using the Riemann



for closed l-forms,


/P1~P2=~~~P1fP2-fP1fP21, c

(4.19) bi bi ai _

i=l a, I L.



is simple to show that,


det[ $t"' A $!"` = det[ wi A Qj] 3I s J a c c



where, i





Wi A Uj





Hence, we have that,


J 4 A 4) = det"/2(lmTij) l c



Det' jJ

is just the determinant Since the theory structure,

of the Dirac operator is topological


all periodic

boundary on the -

conditions. complex

and thus does not depend

we have that, det' f?) det' 1/[email protected]

- [det(lm



For g = 1, we know that the above determinants

are given by:

det' $

_-..._ . -

=(~)1/2~29~` ~~)T) =72 [q(r) 12,

1 = fi2jq(~)12,


det' lj2a2 which verifies

the general relation

given by eq. factor.



(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


In fact, the only observables

and the Chern number.

c a- _T. -L.


5. The Problems

i ,z.-

of a Pure Gravity


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


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,



c1 = f!&[~~"Pd,wp ]

_-..._ .

= kCpd,wp

- det(e-` )~Ppc7' D,D~~p7.



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


= [email protected])

not a




scalar, 8~~4~ = er$E4 b. This,

in the introduction, invariants. Different

source of the inability gauge-fixing

to derive nontrivial


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


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


is a scalar, here it is a vector.

because the BRST is a vector.


is closed up to a diffeomorphism

Since both the topological ants on Riemann moduli

U(1) and the TG theories cannot



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


and g=l




theory. space


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 two-dimensional that the only homology field theory

Minkowski groups of Ml


are torsion (unless

groups!"` * it is through

We do not expect global anomalies).

a quantum

to probe torsion

Thus, interesting


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 - i-c 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,


= e:P, Chern

+ waJ, number,



fZ(O), the first

is invariant

under the topological


of the gauge field, A,,



= *a



= ti$

A b(l,l)Wor

= &Y-


The strategy

for eliminating theories,

the redundent


degrees of freedom

is the

usual one for topological while maintaining %

namely gauge fixing the topological 1ocal gauge symmetry. having


the ISO(l,l)

Since we are to describe Fcyp = 0. This -

we gauge fix to configurations that [email protected],wp

a flat connection:


= 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,




` @d,wp


[email protected],D,e~)


--...- . -.


- i&%3

a' + ~aDcr$F i- EabXaGaepb) &/3 variation. From here on exThe anti-ghost x =





denotes the topological

pressions written xaPa + jiJ

in group multiplet as follows:

will always be traced.



&r(i,i)Xa &-(1,1)X = B ==+{

= =

Ba, B*



1, < Pa,Pb We know >= that babe the ---

Note that


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.


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


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 non-degenerate Since the topological an additional ISO(l,l) ghostly

fixed, we expect to have algebra will close up to an of Q under


gauge transformation

that the transformation

this symmetry

will be like that of a gauge field under ISO( l,l),



By transforming it is invariant,

Xl?, = iD,@


&y1,1)& &ii =

&y,,l,~: = i(DaqP -I- cab&ab)a


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.


a world-sheet

since our gauge-fixing

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 world-sheet

or simply metric). can-


to the two-dimensional

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 world-sheet

to the world-sheet 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 -




+ 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









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,*]


&,,q g&,qxa

= aA

= 0, = cab$xb. (6-g)

z- _T_

--One _-

may suspect that the third

line in (6.9) d oes not describe. a gauge transforfrom the gauge invariance of


of x; but, in fact, it does as can be verified in (6.2).


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


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)



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.


= = aa6 + Eabeaabb9


3.11), namely,

A &(2,1)&z

= [email protected]

* &(2,1)!&

D,qS" -I- Eab&orb,



and the auxiliary

,z.. .. _

field B transforms



= -i[Q,x]+

ii =` cab&,& .&q2,1~B~ = cab&b j&$b). (7.3) b(2,1)


the second stage of gauge fixing


~(~1 = L&+~~



+ 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


+ XaD,D"~"], + eabPp eaa$pb) -I- V(D&Z + $,b:qg) -k i2 - eabeL$a)], + Bd"


+ &[-icab(&?/Jab?,iQ

iEab(2#aXb? + ixaxb)],



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



The gauge interaction ,c. J&, _[ip, ([A",

terms can be written

as LCgauge= &(J&,Aa)


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 -


+ + Eab[-(+$," P(fJAa)


+ +






@(%+pa ebdezcjcXa


A(W"4a ez$)]}% P-8)


and the other interactions L

id =&[-iEab(Xa$hb$C


+ ~i$~$f) iCab(2$aXbg + $XaXb)], (7-g)











The derivation tensor is similar _-..._ . -. and therefore

of the Noether to the derivation



and the energy-momentum abelian theory (4.12-4.14)

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:


=D,@ -





D,q5" - eabeab$, aa + Eabe$$b,

cab (&kab cab+a+cvb, cabXa$b, eab&b. #b&x),



a- _T_


=[vkY] I ~;(2,,)~: I J ` =+ [@,X1











In general the BRST

algebra on any operator

[h(2,1)EI > &(2,1)E*P

is closed as follows,

-k~2P% 01.




Any explicit A. ,; symmetry.-.

calculation This

will require a further

gauge fixing Fadeev-Popov

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


8. Topological


of Moduli


to the topological construction a description space is We have of

We will now discuss the quantum invariants of moduli space.

field theoretic


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,


to the space of Riemann



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,


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).


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 world-sheet metric in the quantum field theory. 27

relation i ,c-

above we can determine

wa as a function

of e: . Then, substituting



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


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:


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 well-known re-

x (2g - 2) = 6g - 6.


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


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


are good coordinates. *- _=_ --

Since we must be content we can even though

our KIXca and. x, we will mapping onto the

glean whatever differential


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 . ;-


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


J w(4n-1)

1 7 2

wt4") =TT(@2)n, 0


s 7 [email protected]),





A dxp


A dxp),




+ 2n(2n - 1)

J c


where we have dWN = {QB~s~, is then the structure Mathematicians the form theory _-..._ . -. S' wk lk


{Q, WO} = 0, and WO # {&,A}. invariant


of the generators

for the topological


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


to the degree of the differential as mappings



can be considered

from forms on a Riemann invariants,

surface to

forms on the corresponding previously, identified


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:


. For example,

will be identified

It is known that lcfg-' # 0; this corresponds






+ (Tr n i=l L.

mq. J c



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


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)


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


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


between coho-

elements on the Riemann

surface and those on moduli + H2"(Mg),


Wo(2n) :H' (C) W,(2n-` ) W,(2"-2) :H' (C) :H2(C)

+ H2"-` (M,), --+ H2"-2(Mg),


_-..._ . -.

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) gauge-fixing 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


The previous ,L.


at obtaining theory

a topological


for M, were based

on 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.

two-dimensional 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 super-moduli

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


_-..._ . -.


for introducing We thank us to the problem addressed

We would like to thank E. Witten in this work preprints. and for fruitful


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_





* ,c.. -


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_T_ i-e.-



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