I

Chiralanomaly (OutlineConsiderz(ATthecaseconst.

J==

the Dirac
to===

-

PR4
=

simplicity·for& (iFrDaYn

S(iTA4fermion 4iTA4rdeW(A o]eProjection

Upfor details

note

JiFRDA Pr dix-.
components

anomaly

with values

only,

inV

where.

In deW(A) /A[SaiDpE JR).

we can.

iTPr4
....

iF@41) diX+iF@Pr4)dix+R

toof computation ofNow let us compute-

VECO3-r

A is coupled,)@Fh84nBEOt

For the purpose
considerVconstJO584 ePr

where=(ATROtRe===VR

see the additional:=O=o -by"Corentz" inv.

↑2-ideW(A]la=o>Jax OeJ"Jany=+fJA=.

j)

comeJJain (JAm EJ j").

,Odav ebik=Jai Optacus=

Ja & talks

by Lorentz
M-PR4(3)FM)/diy:9319(e) Sirs (vPrPr-~JA

i9 Fall 2trtrues (V192** PR4(a)51)Vre3!

quadratically divergent⑪

Pauli-Villars regularization

Introduce

3

regulators

I:I.

:iistatisticsCo=1)

originalThis replaces[ (9)fermibosefermi(=-1)

bose(G1)(G=-1)

regulators⑧ by:=its(UP33

It turns

Thenoutthat

integral istheafter some computation,

[Y9)

,

convergent--hence

JAEIA]

notmatchwith

If fact

thiscan becancelled

toWCAT,

XWCA]

W(A)

=

*191

the.

=

0by adding

for the claimed formula.

alocalcounter term

In fact

Jaxtr (A d""(C DOY An )+with

does,

[gui.

This does*59)]=/ diEalkeigtryleces) 9# O.

/(8592-99) (log - + log()=GdeW(A)(a=tilio

findwe+anditjob,

C+= logta D=t

#We

alsomayadd

E W(A)
which--DJanDatak) End=

dabasically(die-08)An]0.

for some E

GW[A]

1
Apal

Jaix (GAn EJal) Gfor

is=-W(A] + XW(A] +*'W[A]=and,*/azo

02=E/314T)"

compute-ddeW(A)(Alet us,*WIA]

Considerthe above
Cand

Next=

E(aix r (An(5+de DWCA)

has

We shallfor=i(p +9)x=

o=.

GWIT,

daei9

computed=aA

=

dae e

in-G GoW[A] 10,

andti(E((e el) (85g2-gng"),l (logg25ipisxlog)-2E)

5JaxEPr4)diyiF. Pr4 /diz iPAPR4Sonn

&==-:

.

<+:(axfalk)try (eeg(Kexchange ;+g(k m).

=trs)9) Pr HaI-=+rs)rPm-I-k+-"Pm,=+ terms=/59(0492-k+m-(p)+ m(I- +m1

VP PR-(p)+ m~Sig( (i):=k+mPr

20 Pm

1VP PRMRV him).

-,mU
I-54(p 9)+ mI-(9)I(p)+j( 9),;? ) try (Pr(/ Vit U9)#((k 9/2 1)(k 1= )((k p)2 12)+2997)

that vanish--++159p2 2pp9)

as+:0,9/1 :+--20.

+**9,Pr7

6&=-(dix(a()[i(+9)

r(edee(t"++(dxfa(k)e-=+we-=:*(p),,in+(dis edid,acLA).

,,,/11=54 ( 2))(4b 9) 4+ (9 < p)th(e(ee])[(02 59) (log - E
E+- [tro leade e) exure 9 Pr
(5p2 pp) (logp-G +,the* + #, 2

Gd, de WIA] la=>[-setCancel-i(p + q(x-If(9): (+=192-gg)
,

and

wea

modulaClog1)]]

8"p = pops terms

findo2) tr /Ende e) Ex-9, Pr,iS tr (EdAndA)=log()Jtr (Ed)AdA

O(A3) terms

.

+=A)

.

7

mon-Su,By computing

Summary
weS

have

acA]
(A)

seen

=(=Jeth)EdAdA)+isconsistentalA]ala)
We may==O(AY)++O(t))

d local (A].

moduloThis(dAdA)

with+ (FE))

JTn(ed(AdA [A))+"computer ~On,tofix the higherBut

thatgeneral.

is

not

orderterms

necessarystructureofif---O(A3).
weanomalies.

usethegThe general structure of anomalies①Theanomaly

AECA PJ

local

is.

i.
e.

,

Sax polynomial of derivatives of (E A P),,,

becauseit

non-trivial&comesfromregularization procedure

only for divergent diagrams- local(Already discussed)

modify the action by
theThereanomaly AcCA PJ
,

is

a

freedomis,definedThus

only modulo.

JedezTaP-derde Dap=Green Dad(we've been considering right action)de Ae(A P1-de9e(A d),.

tolocal counter terms.

de local functional of [A PJ③

which is,,=.

Thus(e ,)Wess-Zumino consistency condition(A P),9

Let usconsider thea (A)(i)=

caseofchiralanomaly with V 10).
=[A3)c(trun(Ed(AdA↑+

Constant

Satisfies
(ii)

Itis

WE

thealsotheconsistency condition (Exercise.

unique solution tothe WE

condition(of course modulo de local (A7) with the"initial condition"

a (A)
Thus,

ourenough=x=ismosStr(EdlAdA + A)

the relation

Ga.

proveaCA]

VIj

O(A3)

+computation for mor
toIn view of(Strm(EdAdA)

,acA)

this=

betweenalso

GCA] a acos(0 A) for·confirms(t (F).

10

Fujikawa’s method
alternative ,an. . .

Axial

Dirac fermion

Axial

ina(Note rightanomaly@

compute anomalies.

.

:& (Feiers) &(4):toV of G

representation

Jax =i) A4=

method(GungangedanomalyS

direct=action

4 (eiers)"4(Det energ OF 04= (Deteter( (==o2 Trliet)

divergent---A

regularization

ac(A)=NoteDa:Thus:

2=Da-eFr(c Ve )Dahas/since.

UNE-UM andD=-realeigenvalues

andcanprovide

regularizationaDr.

hence.

.

Il

theUseofplane

basis

wave

Thia()

fixes

eof V-valuedSpinors

the space( se : CVtoikX
=

basis

serSCS,evaluate the trace :GlA)

2 Tr( ==>

2/dix(@h(ui(x) EVYui,2(d+=jikx

eix+

gix gitx··=>

re-E/)

=-j

*

x((x)

inxgihx(Sitr (vsVOS

yn(ikp==+bu Ap)+,yap(ihp (n Ap)(iku++Vip =* Er v3+,

Lik++ Af)/ihr(ikn & Ap)(ihr++&Cr0*, V-+Ar)=5Ar) - Ceu)+d+Gr + Ar)+++1*+=*

unFruFr=:12X+)Af) (b AM)(fp+ike-C-C

Magike=

As

Life~

41
1

=

OnlyG(A)the=+if0is

S&3

least two

attonecessaryPll-+m+2with

51

=

1m =0

-

powers,survive(X/12)e (For/12)

terms-·2ortryNs-).
itsreordering

2msurvives

2(ax((x))
Use

1m+ 2(12)e:.

zu F+.

kY12+h((X + Fr)/12)"

trgV Ms)-(X * V Fi)/12.

of
Gui Fur/12·A-*-e=·+exp)-eih yein) -=Ey-=a.
the limit 1 + 00

tros(v)*/2)2)F+

13=>

Jaix((x)+re((Fr F(x) (VV--emvex

=
=

Note:

diE()t (F)tr(FanFa)

The axial

rotationit, VsgitsC.

theisadditiveala] +,itiI

integrates

finite axial rotations& (iv)=(A)
totheAbeianisgroup(E + (2) V5,einfinitesimal formula aandThus.

=(A)

/Et F

=ae(A].

anomaly formula for,4)

EF84i=DATP4

exp(i)+G (A]e(FE))

14

Chiralanomaly

VV

For the purpose ofaS

=thecomputing

Dirac fermion=V 90,values

with(dix(i) ↑DARY

anomalyWe.

may

considerVinDar;

O=+AirChira rotation :-I49(9Pr=843843

A

regularization:=>++gP)Pr) &F O4

Det (gPr + gP)8584

Tr(EPr-EP1)=5(Pr=+=Tr(EV)---divergent.

:-id(A)

4+==,Det((Pr g"Pc) 19Pr Pc)] 8484

=i@P1)4

Det (Pr + 9Pc] : Det (9Pm=:+Tr(EVs eSimilarYar/)computation(tr(Ed(AdA [A))++deloc(A).

15

Axial anomaly in a general even dimension

Gr pry-25=,(V !.. 2)2=( )

aS
Axial

=represented

MU = -d...

Dirac fermion=2non

onrIP;

Va

with=

Ver

1,values

ina==

D2-U"Va.
rep V of G(ax (i)πP

anomalyActivate)a (A)S"dVa :=↑d=:

DE/122/x((x))tos(vieiY)same"=j2irLeVane]-=id (A]computation(Exercise (2)- + (i)
dIR

16

(d 2nAxial anomaly and index of Dirac operator
The axialanomalya (A)hasforze=constantabove)>-Tr(Vae-Y]

Rechnical
To avoi

IRd byvalues

withneeds

Insteadsubtleties

betoainof plane wave basis

By Compactness of X
theSuppose,eigenvectors

Dy↑ 114(1 = (4=,X20

and=abundle Eon Xmanifold).
,

weofcanuseaof DEisbasis

Da ?
discretesquare normalizable .

Di DaX Y.

x4)

considerandthe Spectrum
arebetter to replace

isvector

aspin

consisting of eigenvectors

andit,

compact manifold X

afermion ↑:as

topological meaning.

a7 X

a==14 DE4)↓=

10#Day(AY DAY)
,,

=0.

=

11 YAYIl?.

17Also

VanDa

as

,

=

Of course
, Da:

Van Y

If 4

Suppose X20.

left handed# (Day) =2 y.e

Tthere.

righ handed

Summary:

isspinors

LetSxCE)For X30,.

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

valuesin.

correspondence betweeneigenvectors.

Ebe thedecomposition

theDa eigenspaces

intoSY(E)

Vat1

Vati-:

then,

Furthermore,.

xyxY

SY(E)

ADa?

eigenvectors of DE withSCE) = SxCE)

=with

X

left handed

withcommutes

DawithOne-to-one

aaLet

of=

Van,

also

areeigenvalue

the sameis

Da Voelcommutes

DayandDAY-

bethe R-L decompositionSALE) SYIE) isalinear isomorphism.

18

Tr (a) = Tres(a)e-Then

TrsEs(at)and==>

Toe(oTry , (Vane/)==Note:SYCE)
So (E)

1)+dimS(E)=:+ifTres -1)
dimSY(E)-X 30

Trsoles Nav
dimS(E)-

dimS(E)=Ker(Pa

SY(E)

-

SYEl)

=Ker (a : S'(E)--SP(E)(:.

-adjoint of[m (Da===>

X/12

:

Da SP(E):

S (E) +SYE))&

SY(E)/[m ( e SPCE):Coker (a S & (E)
:>++-

SYEl)

SYE)

SY(E)

19

:

Tr

S(E)
=(VE/)
dim Ker/Da SP(E):index (a : SY (E)=:Thus foraa (A)=seeconstant &2E

thisS'(E)dim Coker (-We can-·-:SR(E) -S'(E))

S' (E) (,index (A : SR(E)alsobyS'(E))-mode expansionof 4 5.

Da(Teievan Daleiera4)=

LiedinSCE)-ziEdimSCE·Dat DAY

20

Atiyah-Singer index formula-index (a:Su(E) +SCE))=

SchlE)

Ex

wherech(E)x==1tr)eFz)

Cher charactor of EzP)X)

A-roof genus of X

-+..

.

-someofIn the flat space,power

seriesPontjagin classes of TX

Ex=1andit,

reads

St(( F )")thesameexpression

asfoundby Fujikawa's method.