(d 2n)

Axial anomaly and index of Dirac operator
The axialanomalythe index

ofTo avoid

technical subtleties

I&itwitha

metric ga

connection

Thusthe=,SCE)

toIn Fujikawa's

basis

toconsider

isthe space

vectorbundle Ewith,,we(Vae/of spinors.
values in Ewith

wesuppress

dependence

shall

simplywrite

computation on

IRSevaluate the Tracebasis of SCE)

of non-compact

compact manifold X

aa

2 Tr

exampletoX.

on

simplify the expression

Forbelow.

instead,inSIE)
isproportional

isSpin structure) and consider

avaluesanomalya (9 A)To

andwithA>-considerbetter tois

constantaDirac Operator .

theDirac fermion

wherefor=.

we

Hereused,weconsisting of eigenrectors

on19, A)

Dfor
Xg A

theplane wave.

shalluseof $ ?

the.

Compactness of X

By

theandeigenvectors&Y

Suppose=Also

Of course

VanY:

D=-Suppose X20.

If 4

is

X(xy)
.eT.

there

righ handed

left handed=isax ya=0=D?

with?

eigenvectors of $2

also

11311.

commutes

withareeigenvalue=Van,(4 xy),

DY

X Voelthe same

DY=,commutes

D4and(4 D -4)=

10#and,DE X

X Yx4)

Van Xas,discreteis↓,X20ofsquare normalizable .

are.

↑ lly(x = (y:the Spectrum,

2With.

X

isright handed (Va+= 1)(Van=-1)

One-to-one

left handed

Furthermore,

.

x44Y

then,.

correspondence between

XX0.
eigenvectors provided,

Summary

Let:SCE) = SxCE)

subspacestoS (E)=of D2-eigenvectors

1=+orSALE) SYIE):

Therefore

SY (E)

S& (E)

according to Van

Xisand-1anand,Then.

letdecomposition,

for XJ0,.
isomorphism(V) = Trees (a) e
Tr

Sx(E)

(ati)

Toe(=o=(Vae-Y)
SCE)

==+1)

dimS(E)=>Tr

be thedecomposition,

ThSIE):

be theif+-aToes 1)-dimSY(E)X 30

Troles Nav
dimS(E)-dimS(E).

NoteSYCE):

So (E)=Ker(D SY(E):-SYEl)=Ker(D : SY(E)-SP(E)((Pr $3)~W(D Y2 41),&[m (D : S (E)=+SYE))SYE)/[m( SPCE)=-:Coker ($ S & (E)
:=>

:=.

++SYEl)SYE)(V)

TrSIE)

dim=Ker($ SP(E):-S'(E)

dim Coker (D : SR(E) >S'(E)the right

called the--The number

index ofThis

theaa (9 A),

operator $obviouslyisThus foronanconstant &=2E·hand side
:

SR(E)

+is

S' (E)

integer.

,

index (1) : SR(E) -S'(E)).

We can also& FD4

see

thischanges

directly by looking

Let us first definethe4measureofthe fields4Let<4SECE),

orthonormal basis (Hmaxial rotation

theunderhow theatmeasure.

by mode expansion.

SYSOCE) be,*dim So (E)=,ndimSo"(E)).

=

Let& Y yoofDSE) a SPIE) consisting of Deigenvectors.

&,

DY

Then

be the Orthonormal basis

By the discussion above

they form pairs

whereYXnis=(Y Yh)

AnYm,

andthe common

wethatS .
t.

DYn = Yu
-eigenvalue of X ?

3470944% (SPE)

3may assumeSYE)]

Ortho normalbasis.

The fields

4=(PR)=,(F,n)canbeexpanded

as= R4b4 + :4

↑n#LThen,thepath-integral measure is& O4 =definedbydbdb5o dbdddbdb
RL,

Under the axial rotation

4II-

giv4,

it Vd + L

-TeIl

Il

(i) Le (r) (withConstant E ,themodes transform as

giepR

(b)
I &Thus,the transformationof(5)>-the measure isbei:,zi).

&(Geirany (ieVa4)(b) debe (deb=~ de)d= eddbe
Zif-ZiE

dodddbdb(ezie)r(gzi) &TOP=.

-ezie(nm nu)-SincethisMr-4)

dimS(E) -dimSOCE)=index (D : S (E)&-SYE)),

reproducesa (9 A)
,for

=a=

2E

constant -.

·index (1) : SR(E) -S'(E)).

(An extension of) Fujikawa'sa (g A]=.

S

2method tells

Ech(E)us(TX)

Ecort2E)ch(E)A(TX)
whichmeansindex ( : SRIE) -SYE))
Thisisnothing but=(chIE) AlTX)

the formula fortheindexthatfollows from Atiyah-Singer index theorem

Thus ,

Fujikawa's Computation of

theAtiyah-Singer formula for the index of theDirac operator.

axial.

-anomaly rederives

RemarkTrgz(Vane-Y)We

mayconsider

Space ofHThis=IS(E)

system alsoG2==additionalhasVanF() 2H,

CH R]

andhence,systemis,=isadjoint Operators#I

called@ Obey,=-CH (1) F],

calledmechanics (SQM). &andgiven by.
Xa=

selfeigenvalue

with

H (1)

awith the,The three operators

Such

Hamiltonian H

and-X2=*

example of Witten index

anquantum mechanics

aStatesHistheisa2()7=0(2 (1)F are conserved)

supersymmetric quantum

calledthesupercharge

supersymmetry algebra-.

Whatwehaveonthem

andholdin

particular

any=

H

2=This quantity
isisclear),

S & M.

:
dimCo-dime=index (2 : 2=H

that this,Tre (1) * PH)
wheresoe(I havejust from the relation

follow

intentionally presented

It,TradimSECE)-dimSCE)

4

In

eigenvalues/ and eigenvectors

theREL correspondence for 130

of D

all

seen

*&F

zenKorb
isthe*-HF)

according to (1) F,

2==+1or-1.

Gen Korf.

Witten index of the S & M

very much recommended to.

read the originalE.Witten, “Constraints on supersymmetric breaking”,
Nuclear Physics B202 (1982) 253

paper

I

the system has

setof variables X

then

Lagrangian description witha,

Witten index

theEuclidean

andhas

Try=SoxX(z B)+Our
asystem (2Lagrangian

Witten index(Donee.

g.

in,& , ()F)

description
can,=.

n.

Se L(X(z) X(T)
,e

X(t)

=(SIE) D Van)
,and thecomputedbeLagrangian L(X X),

path-integral expression

a-Fe,

doeshavepath-integral for theexactly.

Alvarez-Gaume, Commun.Math.Phys. 90 (1983) 161
Friedan and Windey, Physica 15D (1985) 71

The answer

isSch(E)(TX)
Thusa,the.

Atiyah-Singer formula is also rederived fromphysics.

Historicallymethod(inthe,thisisgeneral case).

earlier

thanFujikawa's(