topological quantum field theory - edward witten

7/24/2019 Topological Quantum Field Theory - Edward Witten

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ommunic tions n

Commun. Math. Phys. 117, 353-386 (1988)

athematical

hysi s

Springer-Verlagl988

Topological Quantum Field Theory

Edward Witten*

School of N atu ral Sciences, Institute for Advanced Study, Olden Lan e, Princeton , NJ 08540, USA

Abstract A twisted version of four dimensional supersymm etric gauge theory

is formulated. The model, which refinesanonrelativistic treatm ent by Atiyah,

appears to underlie many recent developments in topology of low dimensional

manifolds; the Donaldson polynomial invariantsoffour manifolds and the

Floer groups

of

three manifolds appear naturally. The model m ay also

be

interesting fromaphysical viewpoint;it is in asensea generally covariant

quantum field theory, albeit one in which general covariance is unbroken, there

are no gravitons, and the only excitations are topological.

1

Introduction

One

of

the dram atic developments

in

mathematics

in

recent years has been the

program initiated

by

Donaldson

of

studying the topology

of

low dimensional

manifolds

via

nonlinear classical fie ld theory [1 ,2] . Donaldson's work uses

heavily theself-dual Yang-Mills equations, which were first introducedby

physicists [3], and depends

on

some important results originally obtained

by

mathem atical physicists, e.g. Taubes' theorem on existence of instan tons on certain

smooth four manifolds [4] (as well

as

hard analysis

of

instanton moduli spaces

[5]). Thus there have been many conjectures that Donaldson's work maybe

related to physical ideas in an intimate way. However, sucharelation has not been

apparentinDonaldson's detailed constructions.

This picture has changed considerably because of the work of Floer on three

manifolds [6]. Floer's work involves tunneling amplitudesin3 dimensions,

andhasbeen interpreted byAtiyah [7] intermsof a modified versionof

supersymmetric quantum gauge theory. (Floer theory has also been reviewed

in

[8].)Inthis viewpoint, F loer theory canbeseenas ageneralizationtoinfinite

dimensional function space of the supersymm etric approach to M orse theory [9 ].

* On leave from Department

of

Physics, Princeton University. Research supported

in

part by

NSF Grants No. 80-19754, 86-16129, 86-20266


2/34

354 E. W itten

The Floer hom ology groups are then the ground states of a certain Ham iltonianH

which is closely related to physical quantum field theories. The Hamiltonian

H,

which will be described later, contains anticom muting fields of integer spin. Since

Halso acts in a Hubert space of positive m etric, the spin-statistics theorem implies

that the theory must not be Lorentz invariant. It is easy to see that this is so (the

anticommuting fields do not form Lorentz multiplets, and there are no anti-

commuting gauge invariances).

Purely from the point of view of Floer theory, which is a theory of three

manifolds, a non-relativistic description is adequate. But one of the most beautiful

features of Floer theory is its connection with Donaldson's theory of four

manifolds. The Donaldson polynomial invariants of four manifolds were origin-

ally defined for a four manifold M without boundary. It has turned out that to

generalize Donaldson's original definitions to the case in which M has a non-

empty boundary

B,

one must define relative D onaldson invarian ts of M tha t take

values in the Floer groups ofB.This connection between the Floer and Donaldson

theories has led Atiyah to conjecture that the Morse theory interpretation of

Floer homology must be an approximation to a relativistic quantum field theory.

That conjecture was the motivation for the present work. We will find a relativistic

formulation which turns out to require a not entirely trivial generalization of the

nonrelativistic treatment in [7] . This generalization is described in Sect. 2. In Sects.

3 and 4, we describe from this point of view the origin of the Donaldson

polynomials and their connection with Floer theory. In Sect. 5, some explicit

formulas are worked out. Finally, in Sect. 6, we will discuss the possible physical

interpretation of this work.

There are many results on instantons in the physical and mathematical

literature which are important background for the present work. Instantons were

used to solve the U(l)problem by

c

t Hooft [10] and were interpreted in terms of

tunneling in [11,1 2]. The formal theory of deformations of instan tons, relevant in

Sect. 3 and later, was developed in [13 ]. In addition, many remarkab le properties

of instantons in supersymmetric gauge theories have been uncovered in the physics

literature. In particular, the ideas of [14, 15] may well be important for future

developments in Floer and Donaldson theory, perhaps connected with the role of

the reducible connections. Our treatment of Donaldson polynomials in Sect. 3 has

a close formal similarity with the arguments given in [16] to determine certain

correlation functions in strongly coupled supersymm etric gauge theories. Finally,

it should be no ted that many arguments in Sect. 3 and later will be quite

recognizable to string theorists. This analogy is in fact tantalizing and is further

pursued in Sect. 6. Introductions to the relevant string theory are [17-19].

2

Construction of the Lagrangian

Let us first recall the description of Floer theory in a non-relativistic quantum field

theory [7 ]. One begins with gauge fieldsA (x)on a three manifold Y.Herei =1... 3

labels the components of a tangent vector to M,aruns over the generators of a

gauge group G, andxlabels a point inY. Yis endowed with a m etric tensorg

tj

.We


3/34

Topo logica l Q uan tum F ie ld Th eo ry 355

wishto consider differential forms on the space

si

of all

gauge

connections on

Y.

*

Abasis for the one forms would be the A (x).

2

The A?(x)can be regarded as

operators on the differential forms on

si

[if is a differential form on

si,

then

A

a

i(x)

acts on

by >W (x) ( ]. Regarded thus as operators on differential

forms, the

A^(x)

anticommute,

{ A (x), A^(y)}=0.

They thus correspond to

second quantized fermi fields. Following physical terminology, wewilldenote the

A(x) as (x).

T h e exterior derivative onsi is

I ts adjoint is

where the

(x)

[vector fields on

si

dual to the

a

i

(x)'\

obey { ?(x),

b

j(y)}

=

0,

{

a

i(

x

%

b

j(y)}

=

Sij

ab

3

(x

y ) . One then considers the Chern Simons functional

\ ^ A / \ A A A ) as a Morse function on Y. Thus, as in finite

Y

dimensions [9], one introduces a real number

t

and defines

d

t

= e~

tW

de

tW

,

d* =

e

tW

d*e~

tW

to be the supersymmetry charges. They obey

df = O, d*

2

= 0, d

t

df + dfd

t

= 2H,

(2.3)

where

H,

defined by the last equation, is the Hamiltonian of the nonrelativistic

theory. Explicitly,

^ i ^ )

2

4 r ^

+

,T

ri/

,w], 24)

with B^SifrF*,F

ij

=d

i

A

j

djA

i

+[A

i9

A

j

'].

3

In thefirst two terms of (2.4), we

recognize the Hamiltonian of conventional (Lorentz invariant) bosonic Yang

Mills theory. The last term is a Lorentz non invariant coupling to fermions. (The

fermions have spin one, impossible if the coupling were Lorentz invariant.) As

described in [7], the ground states of (2.4) are the (rational) Floer groups of Y.

These groups are graded by an additive quantum number which wewillcall

U,

with

U =\

for

and

U

= 1 for

.

(In finite dimensions,

U

would correspond to

1

Th e relat ionof differential formsinfunction spacetoq u a n t u mfieldswasdescribedforsigma

modelsin[20], whichmay

serve

asuseful bac kgrou n d

2

T h a tis, foreach

x, i,

a nd ,we

view

A(x)as afunctionorzero formonjrf,andoA (x)is aon e

form whichis theexter ior de rivative of the zero form A(x).Thesymbo l

simply denotesthe

exterior derivative on thefun ction space

si

3

Our

gauge

theory convent ions are t h a t theco varian t derivative of a charged

field

is

D

i

= d

+\ _ A

b

~ \ .

U n d e raninfinitesima l

gauge

transform ation with =[ ,],the transform a

t ion

of

Ai

is A

t

=

D f i .

Weregard the generatoroft h e

gauge

group

G

asreal, skew sym m etric

matr icesinthe ad jo in t representat ion ,and any

field

with valuesinthe ad jo in t representat ion ,

when notdescribed explicitly by c o m p o n e n t s

a

,

is such a real skew symm etric m atrix.The

symbol Tr deno testhepo sitive definite C arta n K illing formon the Liealgebra of

G


4/34

356 E.Witten

th e

grading of the de Rham complex by dimension.) Because of instantons,

U

is

only conserved modulo a constant; for SU(2),the constant is 8.

2.1.

Relativistic

Generalization

I n

trying to find a relativistic version of this picture, ourfirstproblem is to decide

what the supersymmetry algebra willbe. In (2.3), there appears the generatorHof

time

translations. There is a natural notion of time translations as long as we work

o n

YxR

1

(Yis space andR

1

is t im e ) .However, the Donaldson theory applies

t o

a general (compact, smooth) four manifold. On a general four manifold, there is

n o

natural notion of time translations, so one must work with a smaller

supersymmetry algebra in which Hdoes not appear. This means that we cannot

keep bothd

t

anddf in the intrinsic, four dimensional, theory. We must keep

just

o n e

supersymmetry generator, say

d

t

,

which we

will

call ; it

will

obey simply

Q

2

= 0. [We willsee how to retrieve the algebra (2.3) if one specializes to a four

manifold

Y

x R

1

.] Obeying

Q

2

= 0,

Qwill

be rather similar to a BRST charge, and

we will suppose that it

plays

a BRST like role of identifying physical states;

physical states

will

be states obeying

Q

= 0, modulo those of the form

= Q .

4

I twill

t u r n

out that the physical states in that sense are

just

the Floer groups. In this

way, the negative norm states that one might have expected in a Lorentz invariant

theory

with anticommuting

fields

of integer spin

will

disappear.

5

Despite this

BRSTlike role of the supercharge , I have no idea how to obtain the BRST

invariant Lagrangian considered below by

gauge

fixing of a

gauge

invariant

Lagrangian. It

will

be argued in Sect. 6 that such a Lagrangian would have to be a

generally covariant one of a new type.

Trying to extend (2.4) to a Lorentz invariant theory, the next problem is to put

t h e

fieldsA , , *into Lorentz multiplets. Clearly, thegauge fieldA*is in a four

dimensional picture part of a (Lie algebra valued) one form

A

a

a

,

= 1... 4. (Tangent

indices to a four manifold M willbe denoted , , .)As for and , wewilltake

these to be a one form

\p

a

a

and a self dual two form

a

a

(thus

a

a

=

a

a

=

\

a y

y a

) .

Also,wewillsupplement these with a zero form

a

ofU= 1. The rationale behind

these choices is that the

{

9

a

,

a

)

multiplet is known to play a role in four

dimensional instanton moduli problems analogous to the role of(

i9

f

) in Floer

theory.

6

Let us try to make a supersymmetric theory from these fields. The only

reasonable (scale invariant and

U

conserving) Lagrangian that we can write is

J

F** i D +

i(D

a

) ^ .

(2.5)

As for the supersymmetry transformation

laws,

the only reasonable try is

(2.6)

with a constant anticommuting parameter.

4

BRST quan t i za t ion

of

gaugetheories

was

origina ted

in

[21 ,22] . Th e ro le

of

the BR ST charge

in

identifying physical states emergedin [23]

5

A

sim ila r phen om enon occu rs

in

string theory

in the

no ghost th eorem . This aspect

of the

analogy between D on aldson theoryandstring theo rywassuggestedby D.Fr i edan

6

See [13, 2] for

backg round ;

abrief

sketch appears

in

Sect.

3below


5/34

Topological Quantum Field Theory 357

Onequickly

sees

that (2.5)isinvariant under (2.6)ifthe

gauge

groupisabelian,

butin thenon abelian case thereis anuncancelled termofthe formTr \ jp

a

,

a

~\.

Thereis no way toavoid this exceptbyadding more fields. Heuristically,weshould

expect tohave to addmore bosons, becausewehave addednewfermions (and

0

)

to thesupersymmetric non relativistic theory. (The additionof

A

o

to thenon

relativistic theory goesin thewrong direction, because itimplies aconstraint,

r a t h e r than being a physical propagating field.) To

guess

what new

fields

are

required,one maynote thatthepropagating modesofA

a

have helicities (1,1),

while thepropagating modes of

( , , )

have helicities (1, 1,0,0). Sincethe

supersymmetry parameteris tocarrynospin, supersymmetry willrequire that

t h epropagating modesofcommutingandanticommuting

fields

should havethe

same helicities. Thus,

we

need

two

helicity zero commuting modes,

and to

accommodatethemweintroducetwo newspinless

fields

and

(in theadjoint

representation

of the

gauge

group).

A little experimentation leads to theLagrangian

*xT r

*

F

a

F +

\ D

a

D i D

a

xp+iD

a

2 . 7

This Lagrangian

is

invariant under

the

fermionic symmetry

A

a

=

Oi

,

=O, =2i , = ^[ , ],

(2.8)

a

= D

a

,

aP

= (F

a

+K ^ F * )

This action is also invariant under global scaling if thescaling dimensionsof

(A, , , , , ) are (1,0,2,2,1,2), and preserves the additive

U

symmetry if

t h e Uassignments are(0,2, 2 , 2, 1 , 1 , 1 ) .

Letus nowwork out thealgebra obeyedby thefermionic symmetry.Let

E

( )

denote

the

variation

of

any

field

under (2.8).

Let

T

( )

denote

the

variation

of

in agauge transformation generatedby aninfinitesimal parameter(thegauge

field A

a

transforms as

T

(A

a

)=D

a

,

and charged

fields

transform as

T

( )

=

[ , ]).Thenone canverify thatfor all ,

(

,

,

) ( )= T

( ) , (2.9)

where

a

=2 '

a

. (2.10)

So

the

commutator

of two

supersymmetry transformations

is a

gauge

transfor

m a t i o n with infinitesimal parameter

a

.

In

verifying

(2.9) for

= A, , , , ,

oneneednot use theequationsofmotion,but for

=theequationsofmotion

must

beused.

The

Lagrangian (2.7)is notquite uniquely determinedby itssymmetries.The

reasonforthisis as

follows.

Let usdefinealinear transformation

{Q,

} ofthe space

ofallfunctionalofthefieldvariablesasfollows,{ , } isdefinedbysayingthatfor

any functional

&,

thevariation

(9

of

underthefermionic symmetry (2.8) is

= i {Q

9

}.

(2.11)


6/34

358 E. Witten

[ H e r e{Q,

V)

is simply a linear transformation on a suitable space of functionals of

A, , , , , . The rationale for writing this transformation as {Q, }and not

merely as

Q )

is simply that in the Hamiltonian framework, this transformation

really corresponds to the graded commutator of with the superchargeQdefined

later.] Insofar as it is true that

Q

2

=

0,

{Q, )

is

Q

invariant and can be added to the

Lagrangian without spoiling the fermionic symmetry. Actually, since the proof

t h a tQ

2

= 0

uses

the

equation of

m o t io n ,

we are only entitled to add { ,

)

to the

Lagrangian ifdoes not appear in .Also,sinceQ

2

is only zero up to agauge

transformation,

we must pick

to be

gauge

invariant. In practice, there is one

choice of that respects all the symmetries, namely = Tr([< / > , ~ ] ) .Thisgivesus

t h e possibility

to add to the Lagrangian a new term

i [ < M ]

2

J 2.12)

with 5 an arbitrary parameter. Some of our later considerationswillbe simplified if

we add (2.12) with 5= 1, and thus the Lagrangian we actually use for many

purposeswillbe

[j

\

D

a

D *

i D

a

+ i D

x

f

(2.13)

T h e

following

remarks (though not strictly necessary for understanding this

paper) may be helpful for readers acquainted with conventional supersymmetric

gaugetheories. The above construction of a four dimensional supersymmetric

Lagrangian, starting with the non relativistic version, adding

fields,

and adjusting

couplings, undoubtedly seems rather ad hoc. There is, however, a simple way to

relate the output Lagrangian (2.13) to standard physical constructions. [The

argument

will

explain the form of the Lagrangian (2.13), but does not quite explain

why it is supersymmetric on an arbitrary four manifold.] Consider the usual

N = 2

supersymmetric

gauge

theory in

flat

Euclidean space

R*.

The rotation group of

R

4

is

SU(2)

L

x

SU(2)

R

.

The

N = 2

Lagrangian has a

global

internal symmetry which

we willdenote as SU(2)

I

xU(l)

u

. UnderSU(2)

L

xSU(2)

R

xSU(2)

I

xU(l)

u

, the

fieldsoN = 2supersymmetricYangMillstheory transform asfollows.Thegauge

fields

are

(1/2,1/2,0),

(2.14)

t h e

spinless

bosons are

(0,0,0)

2

(0,0,0

2

, (2.15)

a n d the fermions are

(1/ 2,0,1/2)^(0,1/ 2,1/ 2

1

. (2.16)

[ H e r e

the three numbers in parenthesis denote

SU(2)

L

x SU(2)

R

x SU(2)j

repre

sentations, and the superscript is the U() charge.] Suppose now that

while


7/34

Topological Quantum Field Theory

359

remaining in

flat

four dimensional Euclidean space, we consider an exotic action of

th e

four dimensional rotation group, replacing

SU(2)

L

x SU(2)

R

by

SU(2)

L

x

SU(2)'

R9

where

SU(2)'

R

isthe diagonal sum

ofSU(2)

R

and

SU(2)

I

.

Itis

easy

to

see that under

SU(2)

L

x

SU(2) '

R

x

/ (l), the bosons transform

as

(l/2

?

l/ 2) (0,0)

2

(0,0)

2

(2.17)

an d

the fermions

as

(lAl^e i)

1

^^)

1

. (2.18)

These are precisely thefieldsappearingin(2.13). The fermions

a

,

a

,

and

transform as( i , ^ )

1

,(0,1)~\and (0,0)~

,

respectively,

while

the bosons

A

a9

,

and

transform

as

(i,i), (0,0)

2

, and (0,0)~

2

.

As

long

as

the four manifold M

is

flat

Euclidean

space, the Lagrangian (2.13)issimply the standard

N = 2

Lagrangian

with an exotic actionofthe rotation group.

T h eglobal

supersymmetries

of

the standard

N = 2

model transform under

SU(2)

L

xSU(2)

R

xSU(2)

I

xU{l)

u

as

(h )~

)

+1

S o t h e

y transform

u n d e rSU{2)

SU{2)

R

as i ) ~

*

0(0,1)

1

0(0,0)

1

. The Lorentz singlet supercharge

t h a t

we have consideredissimply the (0,0)* component.

T h u s ,from this standpoint, it is obvious that (2.13) is supersymmetricifM is

R*

with

flat

metric. It is crucial andlessobvious that (2.13) is supersymmetric for M an

arbitrary orientable Riemannian four manifold. In

verifying

supersymmetry, one

sometimes meets commutatorsofcovariant derivatives, and the Riemann tensor

might appear. However,inverifying supersymmetryof(2.13), thereisonly one

p o i n tatwhich one meets the commutatorofcovariant derivatives. Thisis in

computing

(Jr(D

a

a

)) = $ Tr([D

a9

D

] '

a

).

Since thecommutatorof

covariant derivativesishere acting on the spin zerofield ,the Riemann tensor

does not appear, and alliswell.Ido not know whether twisted versionsofother

N 2 supersymmetricfieldtheorieswillsimilarly be supersymmetric onageneral

four manifold.

2.2. Some Useful

Formulas

We conclude this section

by

working out certain formulas that

will

be useful

in

later sections of this paper. First, we would

like

the formula for the supersymmetry

c u r r e n t

which, according to Noether's theorem, generates the fermionic symmetry

(2.8).The recipe for finding the supersymmetry currentisstandard. One considers

a

transformationofthe form (2.8) withan anticommuting parameter thatisnot

necessarily constant. Since the variationofthe Lagrangian would be zeroifis

c ons t a n t ,it must be proportional to the derivative of ,and so has the general form

e=


8/34

360 E.

Witten

Now, the variation of the Lagrangian (2.13) under (2.8) has the form (2.19)

regardless of the behavior of the fields

(A, , , , , ).

If, however, the Euler

Lagrange field equations are obeyed, then if is stationary under arbitrary

(compactly supported) variations of thefieldsand in particular under (2.19). Thus,

th e Euler Lagrange equations imply vanishing of (2.19) for arbitrary compactly

supported ; this must mean that the Euler Lagrange equations imply that

D

a

r = 0,

(2.22)

as one can

verify

directly. This enables us to construct a conserved charge. Given a

homology three cycle

Y

in M, the integral

2 2 3 )

or equivalently

Q

=

\*J,

with *

J

the closed three form dual to the current

J

\

J

d e p e n d s o n l y

on the

homology

class of Y.

N o w ,

wewouldlike to c o m p u t e thee n e r gy m o m e n t u m t e n so rof thet h e o r y.

T h i sis

defined

int e rm sof theva r i a t i o nof the

Lagrangian

u n d e r a c h a n g ein the

m e t r ic t e n s o rg

a

ofM . Thed e f in i t io nofT

a

ist h a t

u n d e r

an

infinitesimal c h a n ge

of

t h e m e t r i cg

a

+g

a

+

g

a

,

the

a c t i o n changes

by

T

aP

.

(2.24)

There is one subtlety that must be noted here. The antisymmetric tensor

a

is

subject to a self duality constraint

=i ysg

yr

g

d

'

r

s'

(225)

which must be preserved when computing the variation of the Lagrangian with

respect to

g

a

.

To preserve this condition, an arbitrary change g

a

in the metric

must be accompanied by

Xa

=^yS^'g Xy'S' ~U^garK ySg

7

''g^Xy'5' (226)

I t is then straightforward, although slightly tedious, to compute the energy

momentum

tensor. One finds

T

x

=T r

I

(

F

a

Ff

X

g

a

F

F +

\ (D

a r

D

x

)

+ (D

D

)

+ D

D

a

ga

D

D )

/ A w ) 2i

a

g

a

)

2 . 27 )

The

singlemost important property of

T

a

is, of course, that it is conserved (in the

covariant sense) if the equations of motion are obeyed:

DJ

= 0. (2.28)


9/34

T opol og i ca l Quan t um Fi e l d T heory 361

Equation (2.28)

follows

by aformal argument similar tothat which gives(2.22).

U n d e r

acoordinate transformation x

a

=u

, withx

a

coordinatesonMandu

a

an

infinitesimal vector field,

the

metric changes

by g

a

={D

a

u

+D

u

cc

).

This

coordinate

transformation induces some change

in{A, , , , , ).If the

Euler

Lagrange equations

are

valid,

the

action

willbe

invariant

to

lowest order.

The

change

of

the action

is in

fact

= ]/ g( g

a

)T

a

= jS]/g(D

a

u

+D

u

a

)T

a

.

This

vanishes

for

arbitrary compactly supported

u

a

if and

only

if

(2.28)

is

obeyed.

N o w

we

wish

to

discuss scale

and

conformal invariances.

F or the

trace

of

the

energy momentum tensor

one

finds

g

a

T

a

=

x [_ D

D

2D

+

2W[

,

]

+2i l , \+

[


10/34

362 E. Witten

with

K

= Tr

(F

a

+

F

a

h^F )

+

i

Tv(

a

D

+

V/

D

g

/

,V 0

)+

h

Tr fa [0, A]). (2.34)

Now, one might

guess

from (2.28) and (2.33) thatD

a

a

would vanish. Rather

one

finds

= 0,

(2.35)

with /

/?

=

U

ct

an antisymmetric tensor defined by

U

a

= \

Tr

\_(F

a

F

a

) ]

+

y

Tr

y

D

+\Tr ([0, A]

/

*). (2.36)

Equations

(2.28), (2.35), and (2.36) together imply that

0 =D

a

T*

=D

a

({Q, }) ={ ,D

a

}

= {Q,D

a

U

}= D

a

({Q,

U*

}),

(2.37)

so { ,/

/?

}must be conserved, even though U

a

is not. It iseasyto check this; in

fact

{Q,U

} = * W D

y

R

(2.38)

with

R

defined in (2.31). [I do not know why precisely the same object

R

appears

in

both (2.30) and (2.38).]

F r o m

(2.38) it

follows

that D

({ ,

U

a

}) = 0,

as expected.

As preparation for the next section, we will require one more formula of a

similar nature. Let

, ]). (2.39)

T h e n

one computes (with the aid of the

equation of motion) that

{Q,V}=


11/34

Topological

Q u a n t u m Field Theory 363

pathintegral formalism which wewilluse in the next section. There are several

points ofviewone might adopt.

Onepoint of

view

is to go back to the form (2.7) of the Lagrangian, perhaps

with the addition of the topological charge term. [Recall that (2.7) and (2.13) are

equivalent, differing only by a BRST commutator. We have introduced (2.13) only

because of its more obvious relation to N = 2 supersymmetry and the higher

symmetry it has when M =

Y

x

R

1

see Sect. 2.4.] In this version, the scalars only

appear quadratically; they can be integrated out by Gaussian integration, and the

indefiniteness of the kinetic energy does no harm.

Another possible viewpoint is that instead of regarding andas independent

real fields, one canview

as a complexfieldand set

=

A*. Then the( ,

)

terms

in(2.13) become positive definite. This is what one would get if one takes twisted

N

= 2

supersymmetry literally. All of the formulas givenabove still go through in

this viewpoint.

8

The drawback of this approach is that if is complex and

=

* ,the Lagrangian is not real, and it is not obvious that the Donaldson

invariants

will

come out to be real numbers.

I n

Floer and Donaldson theory, reducible connections (that is,gauge fields

which are invariant under a non trivial subgroup of thegaugegroup) arewell

known to cause many difficulties. In the present framework, these problems show

upin zero modes of the( ,

)

system for reducible connections. [In other words, for

reducible connections, the Laplacian

D

a

D

a

,

which is the kinetic operator for the

( ,

)

system in the linearized approximation, has a non trivial kernel.] The proper

treatment

of the

( ,

)system, which is not yet clear, is bound to interact in a non

trivial way with the proper treatment of reducible connections in Floer and

Donaldson

theory.

3. Path

Integral Representation

of Donaldson

Polynomials

I n

the last section, we formulated a version of supersymmetric YangMills theory

thatpossessessome fermionic symmetry on an arbitrary smooth orientable four

manifold M. This makes it possible to use techniques of quantumfieldtheory to

describe invariants of four manifolds. As wewillsee, a natural description of the

Donaldson

invariants willemerge.

I n this section, wewillsee what can be obtained by formal manipulations of

Feynman

path integrals. Of course, a rigorous framework for four dimensional

quantum

gaugetheory has not yet been developed to a sufficient extent to

justify

all of our considerations. Perhaps the connection we will uncover between

quantum

field

theory and Donaldson theory may

serve

to broaden the interest in

constructivefieldtheory, or even stimulate the development of new approaches to

thatsubject. Though our considerations in this section and the next onewillbe

purely formal, we will see in Sect. 2.5 that even without having a rigorous

construction

of the quantumfieldtheory, one can extract from it concrete and

8

It is not necessary to worry about whether the variation (2.8) is compatible with

=

* .The

supercurrent(2.20) is still conserved by virtue of the equations of m o t i o n if

is complex and

= * , and this is what c o u n t s


12/34

364 E. Witten

rigorous formulas that are relevant to Donaldson theory, together with a recipe for

proving some of the main properties of those formulas.

Proceeding formally, we will find path integral representations for certain

topological invariants.

8a

The integrals considered

will

be integrals over all

th e fields (A, , , , , ) considered in Sect. 2. The integration measure

3>A'g> '9i 3> 2i will

be abbreviated as

(9X).

9

The integrals we

considerwillbe of the form

Z ( W ) = j(^X)exp( J27e

2

) W, (3.1)

where


13/34


Also, recall t h a t the energy momentum tensor is a BRST

c o m m u t a to r :

T,

= { , } (38)

with

a

defined in Eq.(2.34).Therefore, the change in Zu n d e ra change in metric is

Z=S(@X)[exp{ &'

1

cP'le

2

)' \0

f

=

0. (3.9)

T h i s

shows t h a tZ is a topological invariant.

10

Beforetrying to evaluate this invariant, let us observe t h a t it is for similar

r e a so n s i n d e p e n d e n t

of the gauge couplinge. I n d e e d , the variation of Z with

respectto

e

2

is

Z

= \ [ X)

exp( JT/ e

2

) = 1 j J

i X)

e

P

(&'/e

2

)

JS?

(3.10)

wherewe are borrowing from Eq. (2.40) the fact t h a t '={ ,V}.This shows t h a t

Z

is

i n d e p e n d e n t

ofe

2

as long ase

2

0.

1

1

Therefore , we can evaluate Z by going to the limit of very small e

2

, whereupon

th e

p a t hintegral isd o m in a t e dby classicalm in i m a .To find thesem in i m a ,

n o t e

t h a t

th egauge field terms in if' are

=1J ]/ g r(F

a

+ F

x

)(F^ +

F ).

(3.11)

M

T h i s

is positive semidefinite, and vanishes if and only if

F

= F

x

,

(3.12)

t h a tis if and only if the gauge field is anti self dual. Therefore, the evaluation ofZ

d e p e n d son expansiona r o u n dsolutions of(3.12),known as in s t a n t o n s .(It would

1 0

We ignored in (45) a

possible

dependence of the measure

{ X)

on the metric g. Taking account

of this dependence is really the problem ofshowingthat the crucial equationT

a

= {Q,

a

} is true

q u a n t u m

mechanically. Making this completely rigorous is one of the

tasks

of constructive

q u a n t u m

fieldtheory. In this paper wewillrestrictourselvesto essentially classicalconsiderations

1 1

An attempt to go to

e

2

= 0, by writing the whole factor exp(if'/ e

2

)as a BRST commutator,

failsfor two reasons. First, without the exp(

/e

2

)

convergence factor, the integration by parts

in

function space used to prove that


14/34

366 E. Witten

be tiresome to call them anti instantons; solutions of the opposite equation

F

a

=fF

a

willplay no role in this section.)

Let us therefore make a few remarks about instantons. Depending on the

choice of the manifold M and the bundleE,instantons may or may not

exist.

If they

doexist,

then (for a generic choice of metric g) the instantons have a moduli space

J

(smooth except for some relatively mild singularities) whose formal

dimensiond(J) is

given

by a certain topological formula [13, 2] .

l 2

If the

gauge

group G is SU(2)

9

this formula is

d{J) = S

Pl

(E) MM) + (M)), (3.13)

whereP(E)is the

firstPont

ry agin number of the bundleE,and (M)and (M)are

the

Euler characteristic and signature of M.

If we do manage to find an instantongauge fieldA,we can look for a nearby

instanton A + A.The condition for A + A to obey (3.12) is

O

=D

A

D

p

A

a

+

a

D' A

.

(3.14)

I n addition, we are interested in requiring Ato be orthogonal to the variations in

A that can be obtained purely by a

gauge

transformation. This is conveniently

achieved by imposing the

gauge

condition

0

= D

a

A

a

. (3.15)

Let n be the number of solutions of (3.14), (3.15). These solutions describe

infinitesimal instanton moduli, so at a generic point in moduli space,n=d(M)(at

least if conditions are such that the formal dimension equals the actual dimension).

Now let us look at fermion zero modes in the instanton

field.

Theequation

gives

D*

~ D W*+e

a yS

DV = 0, (3.16)

while the

equation

gives

D

a

= 0.

(3.17)

These are the equations we have

just

seen, so the number of

zero modes is the

numberwe have called n.(This relation between the fermion equations and the

instanton moduli problem was the motivation for introducing precisely this

collection of fermions in Sect. 2.)

For

a generic

SU(2)

instanton, there are no

( , )

zero modes. This is so precisely

whenn=d(Ji). The general statement, governed by an index theorem, is that the

number

of zero modes minus the number of( , )zero modes is equal to d(M\

Recall that the Lagrangian (2.13) has aglobalsymmetry Uat the classical

level.

has

U=

+ 1, and

,

have

U=

1. As in [10], the number of

zero modes

minus the number of , zero modes is the net violation of Uby the instanton at

thequantum

level;

wewillcall this number AU. Thus

(3.18)

1 2

Theformal dimensionequals theactual dimension under ce r t a i n

conditions

n o t e d later


15/34


(The

meaning of this statement is that the integration measure

@X

is not invariant

under

U

but transforms with a definite

weight A U.)

Equation (3.18) holds for any

gauge group G; of course,

d(Ji)

must be computed using the appropriate

generalization of (3.13). It must be borne in mind, though, that the appropriate

d(J)

is the formal dimension of the instanton moduli space, which equals the

actual

dimension only if G and

E

are such that the generic instanton is not

invariant under any subgroup of G; if G is larger than

SU(2),

this is only so under

certain

restrictions on

E.

Let us now go back to our problem of computing Z. Z vanishes unless M, G,

and E are such that d(J) 0. Otherwise, U + 0, and the partition function

vanishes because of the fermion zero modes.

To further

simplify

the discussion, we

will

suppose that in addition to the

formal dimension

d(J)

of the instanton moduli space vanishing, the actual

dimension also vanishes. In fact, we

will

assume that the moduli space consists of

discrete, isolated instantons. (Standard quantum field theory methods could,

however, be used to deal with a more general situation.) In expanding around an

isolated instanton, it is enough in theweakcoupling limit to keep only quadratic

terms in the bose

fields = (A, , )

and fermi

fields = ( , , ).

(The

weak

coupling limit is adequate because we have seen that Z is independent of coupling.)

The

quadrat ic terms are of the general form,

M

= ]/g( A

B

+i D

F

)

9

(3.19)

M

where

B

and

D

F

are certain second andfirstorder operators, respectively.

13

The

operatorD

F

is a real, skew symmetric operator. The Gaussian integral overA

B

and

D

F

gives

Pfaff(D

F

)

3 .20 )

Here

Pfaff

denotes the Pfaffian of the real, skew symmetric operator

D

F

.

(Recall

t h a t ,up tosign,the Pfaffian is the same as the square root of the determinant.)

The

important point now is that

D

F

and

B

are related by supersymmetry. A

look back to Eq. (2.8) shows that a classical field configuration in which

F

a

+ F

a

= 0

and

, , , ,

vanish is invariant under supersymmetry (the

requirementF

a

+F

a

=0 is needed to ensure

a

= 0). Therefore, supersymmetry

relates the bosonic and fermionic excitations about such a

field

configuration. To

be precise, for every eigenvalue of D

F

D

F

=

(3.21)

with

=0,

there is a corresponding eigenvalue of

A

B

,

A

B

=

2

.

(3.22)

(More

exactly, as D

F

is a skew symmetric operator, its eigenvalues occur in

complex conjugate pairs. Each such pair corresponds to an eigenvalue of

A

B

)

At

least for

M = S

4

,

this relation between bosonic and fermionic eigenvalues is a

A

B

is an elliptic operator acting in the directions in field space transverse to the gauge orbits


16/34

368 E. Witten

standard result in the theory of instantons [25]. For the particular supersymmetric

theory we are considering, the argument goes through for generalMin the same

way (since we have arranged so that supersymmetry holds for general M).

T h e

ratio of determinants in (3.20) is thus formally

=

+

^ (3.23)

with the product running over all non zero eigenvalues of

A

B

(or equivalently, over

all nonzero eigenvalue pairs of

D

F

) .

The reason for the uncertainsignon the right

h a n d side of (3.23) is that although up tosignthe Pfaffian is the square root of the

de t e r m i na n t ,thesigndepends on a choice of orientation, which we must now

discuss.

In

fact, for anygiven gauge fieldA,there is no natural way to determine thesign

of Pfaff

D

F

,

since there is no natural way to pick an orientation (or equivalently, to

fix

th e

signof the fermion measure). One may aswellpick a particulargauge field

A

= A

0

,

and declare Pfaff

D

F

(A = A

o

) >

0. Once this is done, there is a natural way

to

determine thesignof Pfaff D

F

(A = A')for any othergauge fieldA'(withA

o

and

A'

being connections on the same bundle). One simply interpolates from

A

o

to

A\

via (say) the one parameter family ofgauge fieldsA

t

= tA

0

+

(1

t)A\0 t^ 1, and

requires that the sign of Pfaff D

F

(A

t

) changes sign whenever D

F

has a zero

eigenvalue for A = A

t

. This uniquely determines the

sign

of Pfaff D

F

(A),but one

must still ask whether the assignment is consistent whether thesignthat onewill

obtain

this way depends on the choice of an interpolation from A

o

toA'.It is

equivalent to ask whether Pfaff D

F

willchangesignwhen followed continuously

a r o u n d a non contractible loop in the spacestfj ofgauge

fields

modulogauge

transformations. Physically, this is the question of whether the theory we are trying

to discuss has a global anomaly, in the sense of [26]. If so, the theory under

investigation is inconsistent, in the usual sense of quantumfieldtheory, and one

c a n n o texpect to learn anything of interest by studying it.

A priori, the Pfaffian ofD

F

must be regarded not as a function onstfj but as a

section of a certain real line bundle ,which we may call the Pfaffian line bundle.

T h e

issueis whether the Pfaffian line bundle is trivial (orientable). Precisely this

question has arisen in Donaldson's work. For Donaldson, it was important to

know whether instanton moduli space

J

is orientable. If we denote the highest

exterior power of the tangent bundle ofJ as ,

then

for Donaldson theissuewas

whether the real line bundle was orientable. The two questions are related

because, thinking of J as a subspace of

J / / ^ ,

the restriction of to Jt is

canonically isomorphic to , at least if conditions are such that the formal and

actual dimensions ofJ are equal. (This is so because the kernel ofD

F

corresponds,

u n d e rsuch conditions, to the tangent space of moduli space.) Donaldson actually

proved orientability of J by using index theory and certain topological

arguments to prove that

is

always

orientable, and thus his results show that there

is never a global anomaly that would prevent a consistent determination of the

signof Pfaff D

F

.

In our problem, we simply pick one instanton and declare that (3.23) is + 1 for

this instanton. Once this is done, there is awelldefined way to evaluate (3.23) for


17/34


any other instanton; for the /

th

instanton it equals (

I )

1

,

where w

f

= 0 or 1

according to the outcome of the process sketched above. The contribution of the i

th

instanton to Z being ( 1)% we have finally

Z= (324)

This is a familiar formula, originally introduced by Donaldson (who motivated

the

definition of the n

i

in a slightly different but equivalent way). D onaldson

showed on topological grounds that if M, G, and

E

are such that

d(Ji) =

0, then the

right hand side of (3.24) is a topological invariant. We have argued for the same

conclusion by using the equation

T

a

=

{Q,

a

} to prove that Z is a topological

invariant, and then evaluating Z to arrive at (3.24).

Equation (3.24) is only the

first

of Donaldson's invariants. More generally,

when

d(J)>0,

D onaldson defines certain more subtle analogues of (3.24), which

have had rather dramatic implications for the study of smooth four manifolds. We

would like to bring these within the framework of quantum field theory.

When d(Ji) >0, the non vanishing path integrals will be of the form

Z{0)

= J

( X)

exp(

5'le

2

) 0

, (3.25)

where

&

must carry a

U

quantum number equal to

d(Ji\

so as to absorb the

fermion zero modes. Let us determine the conditions on for (3.25) to be a

topological invariant.

The variation of (3.25) under a change in the metric is

Z(&) =I(SIX)exp ( /e

2

) [ ^ +

g

&

(3.26)

where

g

is the variation of with respect tog

a

(ifg

a

appears explicitly in the

definition of

\

and / =

jSf'/ e

2

.

Thefirst term on the right hand side of (3.26) vanishes if {Q,$} = 0, for then

0 .

3 .27)

[Notice

that we are using (3.5).] The second term in (3.26) vanishes if

has no

explicit dependence on

g

,

or more generally if

= {Q, }

for some

.

l

obeys these conditions, then

Z(G)

is a topological invariant. However,

Z( )

will vanish if = {Q, ] for some , since then

Z( ) =({Q, })= 0. (3.28)

Thus, topological invariants will come from operators such that {Q, }=0,

modulo those of the form

(9= {Q, },

and with

obeying the extra condition


18/34

370 E.Witten

g

&= {Q, } (

g

being

the

change

in

under

a

change

in

g

a

).

In our

actual

examples, wewillalways have

g

=0.

Looking back

to the

supersymmetry variations

in

(2.8),

it is

easy

to

find

operators that obey these criteria. The spin zero

field

isBRST invariant, does not

depend

explicitly on the metric, and (being the only localfield

of

scaling dimension

zero) cannotbewritten as{Q, }.Ofcourse,

itself

is not

gauge

invariant,but

invariant polynomials

in

such

as

Tr

2

,

Tr< />

4

,

etc., aregaugeinvariant

as

well

as

obeying ourother criteria. The numberofindependent invariant polynomialsis

equal

to

the rank

of

G ;they correspond

to

the independent Casimir operators. For

G

= Sl/ (2), thereisonly one, which we maytake to be

W

0

(P) =

2

(P) .

(3.29)

H e r e

P

denotes

a

point

in

M;

we

are emphasizing

by

the notat ion that

W

o

is a

local

operator

that depends

on the

choice

of a

point

P.

N ote that

W

o

has

(7 =

4.

We can now define some new topological invariants.Letthe manifoldMand

th e bundleE

be

such thatd(J)= Ak.Pick kpoints

P

l 5

. . . , P

f c

on M, and

define

Z{k)=\ { X)e

1

W

0

(J\)

=

< W i ) . W

0

(PJ).

(3.30)

=

1

T h e n Z(k)

is

independent

of

the choice

of

metric

on

M

by

virtue

of

the discussion

above.

It is

also independent

of

the choice

of

points

P

l 5

. . . ,

P

fe

, since the choice

of

k

pointshas no

intrinsic significance independent

of a

choice

of

metric.

While

this argument

shows

that Z(k)

is a

topological invariant,

it is

very

illuminating

to

check more explicitly that

Z(k)is

independent

of

the choice

of

pointsP

l 3

. . . ,P

k

.To this aim

we

differentiate W

0

(P)with respect

to

the coordinates

x

a

of

P,

and

find

^ (

1

) . (3.31)

Thus,

although

W

o

is not a

BRST commutator,

its

derivative

is. I t

follows

from

(3.31) that

fiw p

W

0

(P)

W

0

(P')= { d =

, J ,

(3.32)

P OX I P )

where W

is the

operator valued

1

form W

= v(

a

) dx

a

.

We now see

that

( W

0

( P ) W

0

(P

f

) ) 'Y \W

0

(P

J

) \

=

(IQJJW^YlWoiPj^

=0.

(3.33)

H e r e ,

of

course,

we

have used

the

fact that

{Q,

W

o

} =0,

an d we

have again used

(3.5).

T h e

key

equations

so far

have been

= i{ ,W

o

}

,

dW

o

=i{Q,W

) (3.34)

ero

form and one formonM,res]

ization.

One

finds recursively

dW

1

=

i{Q,

W

2

}

, dW

2

=

{Q,

W

3

}

,

with W

o

and W

beingazero form and one formonM, respectively. This process

has

an

important generalization.

One

finds recursively


19/34


with

W

2

=T & + i F),

(3.36)

W

3

=i ( F), W^ \ {F F).

I n

these formulas, , ,andFare regarded as zero, one, and two forms onM. W

k

for 0 ^ k ^ 4 is a

k

form. Notice that

W

k

has

U = 4

k.

If

y

is a /c dimensional homology

cycle

on M, consider the integral

I()=\W

k

. (3.37)

y

This integral is BRST invariant, since

1

=O.

(3.38)

I n addition, up to a BRST commutator, / depends only on the homology

class

of

y.

F o r

if

y

is a boundary, say y =

d ,

then

(

3 3 9

)

This formula is to be seen as the generalization of (3.32) from zero

cycles

(points

P

a n d P)

to

k

cycles. Itsaysthat if y is trivial in homology, then

I(y)

is trivial in the

BRST sense.

Now we are ready to propose quantumfield theory formulas for the general

D o n a l d s o n

invariants. Let M, G, and

E

be such that

d(J)^0.

Pick homology

cycles

... y

r

of dimensions k

... k

r

, such that

=

1

This formula ensures that

\ W

{

has

U = d{J).

Then let

Z (

u

. . . ,

r

)=

\ (@X)exp( J?

f

/e

2

)

JW

k

= I J . (3.40)

This integral is a topological invariant by virtue of our standard arguments

[including a use of (3.5), (3.38), and (3.40) toshowthat (3.40) depends only on the

homology classesof the yj. Of course, if the group

G

is other than St/ (2), we can

write similar formulas beginning with W^ r

4

or other invariant polynomials

in .

This corresponds precisely to the fact that a vector bundle with a rank

r

gauge

group has r essentially independent characteristicclasses,each of which can be

used in principle in constructing Donaldson invariants, though so far the

interesting applications have come from the second Chernclass.In Sect. 5 we

will

extract from the quantum

field

theory viewpoint some explicit formulas for the

D o n a l d s o n invariants as integrals over the instanton moduli space.

4. Hamiltonian

Treatment

and FloerTheory

I n the

last

section, we worked on an arbitrary four manifold M, with no preferred

t i m e

direction. As a result, there was no natural Hamiltonian formalism, and we


20/34

372 E. Witten

have used Feynman path integral techniques, manipulating the BRST chargeQin

away that is familiar in string theory. In this section we

will

specialize to the case

M = Y x R

1

,

with

Y

a three manifold and

R

corresponding to time. In this

situation, we

will

discuss the Hamiltonian formalism. We

will

in the process see

how to recover the results anticipated in the nonrelativistic treatment of [ 7 ] .

1 4

In

the Hamiltonian formalism, one constructs a Hubert spaceH,a, Hamil

t on ian H,

and a fermionic charge

Q

obeying

2

= 0, [


21/34


Evidently then, we can find an operator obeying { , }= 2H simply by choosing

Q = 2$d

3

x

00

.

(4.4)

Let us now show that [if, ]

=

0. We recall from Sect. 2 that

D

a

a

= D

a

(U)

a

with

(U

=(U)

C

. So

i[H, ] =

= 2

j

d

3

x

^ = 2 f d^D^A

01

l/

Oi

) 0. (4.5)

at Y dt Y

H e r ewe are using the fact that

U

00

= 0 (since /

/?

=

U

a

)

and that the integral of a

t o t a l divergence over the compact manifold Y is zero.

With

H = ^{Q,

}, the fact that [if,

Q]=0

means that [ ,

2

] = 0. It is actually

true in the case at hand that

2

=0. To see this in the most transparent way, let us

write out the Lagrangian of Eq. (2.13) (without the topological term) in a 3 + 1

dimensional language. Thus, with

M= Y

x

R

1

,

we have

JS?= \dt

f

d

3

xl/

~ lXiad ~ [

i9

j [

09

0

]

[ , ] [ , ]

2

. (4.6)

Z

Z Z Z

, ] [ , ]

2

.

o

H e r e

(parametrizing

R

1

)

is time, 0 denotes the time direction,

zj ,

fc = 1,2,3 run

over a

basis

of the tangent space to Y

i=

=

oi

, andF

Oi

=^

ijk

F

jk

. We have taken

7 x #

x

with signature ( + + + + ) in writing the above. It is

easy

to see that (4.6) has

a symmetry under >

t

together with

+ , ^ , i^ Xi,

4 . 7

Let us denote this operation as

T.

It is

easy

to see that

T

2

= ( 1)

F

(the latter being

t h e operation that changes the

sign

of all anticommuting fields).

Since

T

(mapping > ) is a time reversal symmetry, it will be realized in

q u a n t u m

field

theory as an anti unitary operat ion. This means that the Floer

groups, rather than

just

being complex vector spaces, have a real structure. (Of

course, they actually have an integral structure, but this is not evident from the

q u a n t u m field

theory viewpoint that we are developing here.)

Now, the explicit formulas for Q and may be determined from = J J,

Y

Q = 2$

00

(with

J

the conserved supercurrent found in Sect. 2). One finds

y

= Tr[(F

Oi

+F

Oi

)

i

D

o

Drfx,

o

[ , 0]/ 2],

. (48)

Q

= J Tr

[ ( F

Oi

F

Oi

)

Xi

+

o

D

o

t

D

t

+ ,

]/2].

Y

We see that under ,

Therefore, the fact that

2

= 0 implies that also

2

= 0.


22/34

374 E. Witten

Beforediscussingthe

Q

cohomology, a few comments are in order. In the best

of worlds, a quantumfieldtheory has a Hamiltonian that is bounded

below

and a

H u b e r t

space with a Lorentz invariant and positive metric. In the case at hand

these properties do not hold. The indefniteness of the

( ,

)kinetic energy means

t h a tthe Hamiltonianwillnot be boundedbelow.The Hubert space inner product

willbe indefinite because of the indefmiteness of the D

o

o

andD

0

\p

i

t

terms

16

.

T h eformer of these problems was already discussed at the end of Sect. 2, where it

was pointed out that it can be avoided if one sets =2*, and for the present

purposes wewillaccept this.

Let us temporarily postponeworryingabout the positivity of the norm and

review the standard Hodge theory argument relating the cohomology to the

ground states of the Hamiltonian (which is positive semidefinite since we are

setting =

* ).

Since [ , i/] = 0, we can represent the cohomologyclassesbyH

eigenstates. Given such a class

(Q =

0) , if

H =

with

+

0, then since

H =

\ {Q,Q] we get

=

Q\^Q ) so

is trivial in cohomology. Hence the

\2 )

cohomology classes are zero eigenstates of H. Conversely, if H =

0,then

0

=

K \H\ }

=

i\Q\ >\

2

+\Q\ >\\

so

Q\ ) = Q\ } = 0.

In particular,

represents a Qcohomology class. And if + 0, thisclassis not zero. For as

[H ,

Q] =

0, if

can be written asQocwe can assumeHoc

=

0, but as we have seen

Ha = 0implies Qa =0 and so =0.

Clearly, the proof that cohomologyclasses

give

quantum ground states does

n o t

depend on positivity of the scalar product, but the proof that quantum ground

states are annihilated by

Q

and

Q

does. Since in the theory of interest, the natural

Lorentz invariant scalar product is not positive definite, some discussion of the

validityof the above argument is required. Let usfirstdescribe the computation of

th e

space of quantum ground states for small coupling.

F o rsmall coupling, the quantum Hubert space is straightforwardly construc

ted by expanding around the classical minima of the potential. Because of a term

vFijFij

in the energy, a classical minimum corresponds to 7^ = 0, that is, to a

flat

c o n n e c t i o n .

Once we pick a

flat

connection, we must choose andsoD

t

=D

t

=

[(/>,]= 0, to set the scalar contribution to the energy to zero.

If we pick a

flat

connection that is irreducible (there is no subgroup of the

gaugegroup G thatleavesit invariant), the conditions in the last sentence require

= = 0. If, in addition, the

flat

connection is isolated (no zero modesoA ),life

isverysimple. There being no bosonic zero modes (and, by supersymmetry, no

fermionic ones), the quantizationwill

give

riseto a unique quantum ground state

for each isolated, irreducibleflatconnection. ThevalueofUfor this state (and thus

its dimension in Floer theory) must be determined by computing the fermion

n o r m a lordering constant. Perturbative corrections cannotgivethis mode a non

zero energy, since this would violate the invariance of the Euler characteristic [or

T r ( 1 )

F

] .

Instanton corrections lead precisely to Floer's considerations.

1 6

The reason is as follows. Consider a general Lagrangian with real anticommutingfields ,

whose

time

derivatives

enter the Lagrangian only via a term

P

F

=

J

dtiMijOCiDoOCj,

with

M

(j

a

constant

symmetric matrix. Quantization

will give

the anticommutation relations

{


23/34

Topological Q u a n t u m Field Theory 375

F o rconnections that are not isolated and irreducible there are bosonic (and

fermionic) zero modes, and it

will

be a more subtle problem to determine the

q u a n t u m ground states. This is the counterpart in the present framework of the

well known problems in Floer theory with reducible and non isolated

flat

connec t ions .Roughly speaking, for

flat

connections that are irreducible but not

isolated, there are zero modes of

A

t

but not of

and

.

The

flat

connections

will

t hen

form a moduli space of positive dimension, andjustas in finite dimensional

degenerate Morse theory, the evaluation of the Floer groups

willinvolve

the

cohomology of the space of

flat

connections. But for reducible connections, there

are and zero modes, and onewillmeet new phenomena, perhaps of a subtle

q u a n t u m fieldtheoretic nature.

The important question is now whether the quantum ground states are really

annihilated by g, as would followfrom the Hodge theory argument if the scalar

product on the quantum Hubert space

were

positive definite. In fact, it is quite

straightforward to see in perturbation theory that this is so. An isolated fiat

connec t ion

is annihilated by

Qclassically

[since the right hand side of (2.8) is zero

if the connection isflatand all otherfieldsare zero]. Expanding around an isolated

flatc o n n e c t io n ,Q

is quadratic (plus higher orders) in oscillators, and certainly

annihilates the ground state. The structure isjustas predicted by Hodge theory,

even though the Lorentz invariant inner product is not positive definite.

A natural way to explain this seems to be that one can define a modified inner

product which is positive definite but not Lorentz invariant and which perhaps is

th eappropriate one to consider in the Hodge theory argument.

17

If (| )

L

is the

Lorentz invariant scalar product on the quantumfieldtheory Hubert space Jf

7

,

o n ecan define a new inner product ( |)+ bysayingthat foru,

(u\ )

+

= ( u \ T v )

L

, (4.10)

where Tis the time reversal operation which was introduced in Eq. (4.7). The idea

behind (4.10) is that

while ,

for example, is self adjoint in the sense of (| )

L

, its

adjoint in the sense of ( |)+ is

o

= T .Quantization of the Lagrangian (2.13)

shows

that the canonical conjugate of

is

0

,

so that a positive metric on Jf

must be one in which

0

is the adjoint of?/. Notice that in the sense of ( |) +,Qand

Q

are adjoints of one another, as the Hodge theory argument requires [in the sense

of (I )

L

, , andQare each self adjoint]. Thus, it maywellbe that ( |)+ is the proper

structure for use in the Hodge theory argument. It might even be appropriate to

t u rn

this argument around in thefollowingsense. In showing that the nonlinear

q u a n t u m field

theory under study really does

exist,

the positive definite inner

product may be the right one to use. One would then introduce the Lorentz

invariant one via (4.10) at the end of the construction in order to achieve Lorentz

invariance.

4.1. Relation of Donaldson and Floer Theory

The next

issue

that we should

discuss

is the connection of Floer and Donaldson

theory. According to [7], to define Donaldson invariants of a four manifold M

1 7

Such a situation has also arisen recently in work by D. Olive on a Hodge theoretic

i n t e r p r e t a t i o n

of the no ghost

t h e o r e m

of string theory


24/34

376 E.

Witten

with boundary

B

one must

specify

a state in the Floer homology of

B.

(As explained

inthe introduction, this fact was really the motivation for the present paper.) In the

context of quantum

field

theory, the relation of Donaldson invariants to Floer

homology has thefollowing interpretation (which was anticipated by Atiyah).

I n

quantum

field

theory on a closed four manifold M, the nicest path integrals

are of the form

Z( )=\{9X)e~

I

G (4.11)

with / the action, andGa product of local

fields

(usually polynomials). X is an

abbreviation for the whole collection of integration variables. It is

well

known,

though, that if M has a non empty boundary B, the path integral requires a

boundary condition on

B.

Such a boundary condition may consist simply of

specifying

thevaluesof the

field

onB.More generally, one picks an arbitrary state

in

the Hubert space Jf of the quantum theory formulated on

B

(or more exactly,

thetheory formulated onBxR

1

as in our above discussion). IfX\

B

represents the

restriction of the whole collection of integration variables to

B,then

Jf is a certain

space of functional of theX\

B

,and a state in Jf corresponds to a functional (X

B

).

The

path integral with boundary conditions determined by is

just

Z(G, )=

J

( X)

exp (

/e

2

) G (X

B

).

(4.12)

Now we can ask, For what

and

is (4.12) a topological invariant? The

arguments of Sect. 3 show that we needQG = 0~Q . Thus represents a Floer

cohomology class. Moreover, with

QG = 0,

the arguments in Sect. 3 show that

(4.12) is zero if = Q for some A; therefore, (4.12) depends only on the Floer

cohomology

class

represented by

.

On the other hand, the interesting choices for

Gare precisely the ones that we considered in the case thatMhad no boundary,

namely

=\[\W

k

.

(4.13)

Herey

t

are certain cohomology

classes

on M, and the W

k

were constructed in

Sect. 3. Thus, in (4.12) we obtain Donaldson polynomials withvaluesin (the dual

of) the Floer groups ofB.

As an example (described to me by Atiyah and part of the inspiration behind

thepresent paper), letBconsist ofseveralconnected componentsB

t

Choose the

Floerclasses

on the

B

t

so that one may take

G = \ .

The connected components of

B

can be considered roughly as incoming and outgoing three branes (higher

dimensional generalizations of strings). In this situation, (4.12) can thus be

considered roughly as a three brane scattering amplitude. Further thoughts

along these lines are one route to certain speculations about the physical

interpretation

of D onaldson theory which can be found in Sect. 6.

There is a

slight

modification of (4.12) which is also significant (and related to a

recentaxiomatization of conformalfieldtheory [27]). Let us group the connected

components

ofBinto incoming three branes B

t

and outgoing three branes Bj.

Suppose we aregivena functional (X\

B

) of the boundaryvaluesof thefieldson

theB

t

.Then the path integral can be used to compute a functional of the

fields

on

theB

j9

by

{X')=

j

(X)exp( J?'/e

2

) (X\

B

).

(4.14)

( X \ X

f

)


25/34


[This formula requires some explanation. The integral in (4.14) is carried out over

all fields

X

whose restriction to the

Bj

is equal to some given field

X'.

The

d e p e n d e n c e

of the integral on

X' gives

a functional of

X'

which we are calling *F.]

Moreover,

Q = 0

if

Q = 0;

in fact, if

Q =

0, then everything on the right hand

side of (4.14) is g invariant. Thus,

+

is a morphism of the tensor product of the

F l o e r

groups of the

B

t

to that of the

Bj.

A specialization of this gives what is perhaps the nicest way, within the

framework of the present paper, to show that the Floer groups are topological

invariants (and depend, that is, on the three manifold Y but not on a choice of

m e t r i c ) .

Let M=YxR, with R denoting the real line, parametrized by a time

variable

t,

with oo *)V

1

..V

n

.

5 4

)


27/34


is an antisymmetric tensor with nindices otherwise known as an rc form on

t h e^ dimensional manifold J. Replacing (9by 0' and inserting (5.3) and (5.4) in

(5.2),

we get

\da

i

...da

n

dxp

...d

Wn

h

^...xp

i

= \ . (5.5)

Thus,

computing a correlation function

Z(&)

in the weak coupling limit amounts

t o integrating the non zero modes out of so as to get an rc form on instanton

moduli

space.

N ow

suppose that

is a product,

=

1

2

...

k

(5.6)

with

k

having U = n

k

,and n

k

= n.By integrating out the non zero modes from

k

any of the

r

,one gets an object

'

r

=

1

\ ..

inr

h

...

i

. (5.7)

H e r e

^

in

can be interpreted as an

n

r

form on

J.

The process leading from

r

to

&

r

is just analogous to that leading from to &. Naively, we might hope that

G' = \ '

2

... O'

k

. (5.8)

I n general, there is no reason for (5.8) to be true, because in integrating the non

zero modes out of the product &

2

...&

k

> one might need to make

Wick

contractions

between

t

and jfor i j. However, it

will

often be the case that (5.8)

is

valid

to lowest order in e

2

, and this is all we need topologically. In the situation

u n d e r study here, fortune smiles and (5.8) is valid.

Equation

(5.8) is equivalent to the statement that the differential forms

and

( )

described above are related by

When (5.8) isvalidfor all products of interesting operators

a

,a particularly simple

prescription can be given for computing integrals

)

a i

a2

. ..

G

a

. (5.10)

One extracts from each

(by integrating out the non zero modes) a differential

form

( )

(of appropriate degree) onJ. Then

Z(

... 0

) = J

( i )

( 2 )

...

M

. (5.11)

ji

I n

our study of Donaldson theory in Sect. 3, the interesting operators were

^=\W

k

.

(5.12)

y

H e r e is a homology cycle on M, of dimension k

, and the W

ky

for k

y

= 0,...,4

are differential forms on M of degree k

y

defined as follows:

(5.13)


28/34

380

E.

Witten

W

k

has U =4k,so with each

iy)

we should associate a 4k

y

form

( y)

on ^ .

Equation(5.8) is

valid

for arbitrary products of the

iy

\ so the Donaldson

invariants are simply

Z(

{n)

...

{ys)

) = J

in)

...

{ys)

. (5.14)

T o obtain explicit formulas for Donaldson theory, we need only integrate the

non zero

modes out of the

W\

This is

easily

done. Whenever Fappears in (5.13), we may, to

lowest

order in e

2

,

simply replace it by the classical instantonfield.Whenever appears in (5.13), we

simply replace it by zero modewavefunctions.

18

Therefore, all that needs to be

done

is to integrate out from the W's.

I n doing this, the relevant terms in the action are

~ =

x

g

\^

2

D

a

D ~ l

x

, }+ . . . ] . (5.15)

Integrating out means computing the integral

(

a

(x))

=$@ @ Gxp ( /e

2

)

a

(x)

=

f

S S exP

[ ^2 l/g W

x 0

M r [

,

V

] + . . . .

(5.16)

(We have expanded exp [(i/2e

2

)J r [_ , j] to extract the term linear in ,which

is the only one that

survives

after integrating over

and

.

The + ... on the right of

(5.16) is irrelevant to

lowest

order ine

2

) The anddependence reduces to the

Gaussian integral

^

f |/ g

D

a

D J \ x) \ y)

(5.17)

which we

will

denote as

(

a

(x)

b

(y)}.

19

According to the rules of Gaussian

integration,


29/34

Topological Q u a n t u m Field Theory 381

Substituting (5.18) in (5.16), we learn that

(

a

( x ) }

= iI

d*y]/gG

ab

(x,

y) [_ M

*(y)]

b

(5.20)

M

N o te

that the factors ofe

2

have cancelled out, a crucial test. Equation (5.20), with

a

replaced by its zero modes, is the required formula expressing

a

in terms of zero

modes.

Replacing

in (5.13) by

( )

whenever it appears, we get our desired formulas

for differential forms

( y)

onJ corresponding to homology cyclesyon M. Ifyis a

zero cycle, say a point P, then

(5 .21)

I f

yis a one cycle,

t h e n

> . (522)

F o r a two cycle

Mi F ) . (5.23)

F o r a three cycle,

) .

(5.24)

And if

y

is a four cycle, say a multiple

s

of the fundamentalclass[M] of M,

y

isjust

a

constant function (that is, a closed zero form) onJt. The constant is equal to

s/2 J rF AF. In general a fc cycle on M gives an operator of U = 4k,

M

corresponding to Donaldson's map H

k

(M)^>H

4

~

k

(J).

Some readers may have considered the reasoning in Sects. 3 and 4 to lack the

precision of mathematics. However, we have arrived at perfectly concrete formulas

for differential forms

( y)

on instanton moduli space.

At this point, one may wonder what is involved in proving that the

i )

have the

necessary properties so that the

Z (

u

...,

y

r

) =

J

( y i )

iy2)

...

{yr)

(5.25)

jt

are topological invariants. There are really four steps.

(a)

One must show that the

are closed. This is not selfevident, but it can be

checked by a completely classical computat ion; there is no need for input from

q u a n t u m

field

theory.

(b) We must show that

i )

changes by an exact form if we change

y

in its

homology class. It is enough to show that if y is a boundary, say y=d ,then

{y)

= d

)

for some differential form

t

i )

on

J.

H e r ewe actually get some useful insight from quantum

field

theory. Ift

{ )

sch

t h a t

( r )

=

dt

i )

exists,there are many sucht

i

\ and we wouldlikea best choice, so

as to be able to push Donaldson theory to its limits and deal with the singularities

and

non compactness ofJ to the extent possible. Quantumfieldtheory actually


30/34

382 E. Witten

gives

a canonical but not entirely obvious formula for t

{

\ Ify = d ,then

(5.26)

I n the small

e

2

limit,

G

{y)

reduces to our differential form

{y

\ Q

reduces to the

exterior derivative

d

on

Ji,

and

W

k + 1

reduces (using the same formulas given

above) to a differential form

t

{ )

on

J.

Equation (5.26) is the desired formula

(c) One must show that under a change in the metric

g

a

of M, the Z(y

1?

...,y

r

)

are invariant. Here again quan tum field theory

gives

canonical formulas that are

n o t

completely obvious. Before writing formulas, let us express the problem

precisely. Consider a family of metrics on M parametrized by a parameter space

S.

LetX = M

x

S

be the total space of the family. Denote the fiber of

X

above a point

5G

S

as

M

s

.

Each

M

s

has a metric g

s

. The differential forms

(y)

are defined on each

fiber; let us denote them as

{y

\s).

To show that the

Z(y

u

...,y

r

)

are independent of

m e t r i c ,we must exhibit closed differential forms

( y)

on

X

whose restrictions to the

M

s

coincide with the

{y

\s).

This can immediately be done in a canonical way using

ourstandard formulas. Under a displacementoseS, the metric of

M

changes. Let

g

a

(P) denote the change of the componentsg

a

(P) of the metric of M at a point

PE M and in some

basis

of the tangent space to

M

at

P.

The

g

a

(P)

are closed one

forms on

S.

F r o m

our general formulas, under a change in the metric of M, the change in

is

M = ( JW

kv

r \ ]/gdg

aP

T

xli

) , (5.27)

where < ) is an instruction to integrate out the non zero modes. Using our favorite

formula

T

=

{Q,


31/34

Topological Q u a n t u m

Field

Theory 383

exterior derivative on the total spaceX isd =

iQ.Equation (5.28), together with

{Q,

{y)

} =0 and P =0 (which can be

verified

using the form of

given

in Sect. 2),

means that the differential form

{y)

=

( y )

iPis annihilated by d, and this is the

desired closed form on

X

whose restriction to the

fibers

givesback

{y

\

(d)

Finally, of course, to make rigorous assertions about integrals

j (vi)

(V2)

_

A

( y

r

) (530)

J

onemust know that the instanton moduli spaceexists,has singularities that are

nottoo bad, and behaves not too badly under a change in metric. Such questions

involve

hard analysis [4, 5]. On these questions the viewpoint of this section

offers

no

new insight, except the hope that analysis of the above formulas near

singularities ofJt maygivea new insight about the necessary criterion in not too

bad. Such a hope is supported by experience of physicists with instantons.

I n

principle, in Sect. 3 we

gave

formulas for D onaldson invariants as

correlation functions in quantum

field

theory which are

valid

regardless of

whether instanton moduli spaceexistsand what properties it has.

From

that more

fundamental

point ofview,the considerations of this section are

just

a recipe to

evaluate the correlation functions under favorable circumstances. But to make

thatfundamental point ofviewrigorous

will

indeed require considerable progress

inconstructive quantum

field

theory.

6. Physical Interpretation

I n

this concluding section, I would

like

to discuss the possible physical meaning of

the present work. Here lie many of the most intriguing questions.

The

fermionic symmetry that we have used is very reminiscent of BRST

symmetry. Its use is quite similar to the use of BRST symmetry in string theory. So

it is natural to think that in a suitable framework, this symmetry arises upon BRST

gauge

fixing of an underlying

gauge

invariant theory.

If so, that theory is of averyun

topological quantum field theory - edward witten

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