I

Remarks on axial anomaly in d=2ndFor=Dirac fermion 4 with values in2na,
rep V of GDaleieva) DaleieVa4)
BAT OP=&theory where GIn theweaction,-exp(zi)a (()]
+may

SvoCA]=

VThen ,

the axial

The axialwith a·In d P=

gangedas,aiSe Outr (E)") .
anomaly4

rotation-Citro 4&Shifts&

G simpleformula says,

thereis-,+a- FeP + ze.

Vcanonicallynormalized Theta term-SolA]=

part of thethe Theta term

considerconstantaisi) +(F)if Un+ I

2"the standardusingtriXY
NoteThenWe

SpThe=--↑But

this=-.

)...

Ex Y,.

.

ey.

,.

Y

0 + 4T-S=

Dirac fermion

amass ,mIm/e=gargimv4

inducesthe,shift

System with complexi

System withM=

complex musa

FmYm+,

Frm Me*arginscan beabsorbed

argins Vati
IOn-Or + argim)-O + 2Tv argins

theta parameter.

Gsiple)
.

.

Ou

Cresp

4din

Cresp Oftheta parameterOr,m)+withi' =(0=>

by&X .

iFrDaYm-iFatephase ofbytrg(XY:=consider

maydefined

Shift isthe0·)tr(XY)

that,...

trace".

+&arg(m)+2Tvarg(m)

3②form

Chern-Simons& the density ofdenotedIt

ch 2n V (A)byand& chanv(A]In fact

it,is=exact ,

ch(A)
2n v

=a.

,

[A9]

written.

tr(eE),F)"]

ofand hence.

=chan v(A).

,as&Want v [A]
,

,

forie

In-form part

invariant,ch24,

thet[=gange0is=,

closedisanomaly

ChyCA]

Chern character

theisaxial

C2R-1) form

Wen-1 vCA]

-called theChern-Simons form,The expression

For anyvariation

Str(Fa")whereandbefoundA

A + JA ,can+(JFAF")

as=n +=

ndtr (JAF")

weBianchi

used

OFA

identity

==

DadA

DaFA

=,0.

followsn+:(DA(GA) E)

d +r(0) = tr(DaO),

M4

For a one-parameter family ofAt
garge potentials

trFa+ Fa -trE==t A-,

Sj'doftEndt=~dSjdWan,v(A](i) n)jd + (AF)+=

Fat;

More(AF)+ n +=:A=

dA + + A5=tdA + t2 A2

explicitly

W v(A] =,Wa r(A)=,=W5 v(A]=,a++A=A(2) a +(AltAA+(i) +ro (tAdA + A3)++ ((jd + (A(tdA + AYY++-t AldAS+t AldAA" + ArdA) + +PAS--=

i+( A(dA)" + AdA++2A3dA+ A5)

5

Chen-Simons form

Want r(A"]NOT,,-The expression forthe method

toMana v,

.

canbe

foundfromchen[see Zumino'sfindWany v,,

theyv.

areo=x2v(9 A)=Xv(9 A)=,,,Les Houches Lecture

low
ns ,

Xo v (9 A]

by extending,,For:,(2n-2) form & 2n-zv(g A)

someinvariantgauge

Wint v(A) + Wen,v (ging] + & Man-z v(9 A)=,foris-[ + (dgg A)-+r/dgg(AdA+i+t(Adgg (2dAA + A3)++A(dgg'(Y)6& Integrality
SupposeAgidg+SynCA)

and(Ford > 2Thenfor.

FA,

SpaOu

For d 4=Chau (a]eThis

Gthat

simpleandandeFor

thisisreason,at1orOgidg)

iSv Or(A]is.

einvariant

simply connected,=underinvariantOr+.

Spptr((Fa))
iSo(A]FA-0

sumeans

Pu + 2/Kv>-&implies A

co0thatso,

finite.
near=Kr.

someunderislasters

as2.
P-P + 25

calledis

Theta angle

D

SpaChar(Az =im S charladDi*da v(A),*((x( RY>=

7~=] GaG

[onSiatlarge R

Di Si=W=ing'dgd.

v(AS]Way v(A) + Wa,v(5ag] + dav (9 A)=,,--for A=!

!

0Wa,v(59]
d 1-Si

Note thatWay v (gdg]*=,

where Wan v (G)isg Wa y v(G)-aclosedd-l formon,

class

isa

multiple ofanG

whosecohomology

integral class*(Way v(G)]e KvH (G 2).
,,ThenS

gaWar(gg)=S (SWar)I

g

Examples·Wiv/gag]
For G

Si

Ull = S'=

Wi 1519]=·tr(g9)=()x

Cs v/59]

theV,

==D()

charge 1 representation

Sgi5dgS'- Will ES !
winding number of the map 9:

tr (9d933=.

For G SU12)ES"V=,=fundamental rep D2,Sga 53 (5'99] Stratag=,=theFor Otherwindingnumber(6 V d),.

,of theSanWat,

map

such

aninterpretation

is:v1999]

interpreted as "Winding number"
or

g S

not

in-canthe samestraight

or

SUES
beway,
even.
absent

9e9.

d.

=

2G,+X

simple.
=Chu/A]eo:

axial

Thus

However ,

d 4,

We

is

not

have

simplynon-trivial

mayof 9 92:-+means

Gdefines

On O + 25.

,

S'- G
afinite group).

=.

Thenmeasures

Then themap(eg G SU(2)SptFA)")

integer valuess ·StrictlyandFrep V(TT G isconnected

the standard "tr" .

Sgcitring
Thisa,

usefor= 0absent.

issimple simply connected

a=anomalymay

Frep .
V,

Wvlgng],weG

when·Exeg

0thewinding number

H (6 2) = Tz (G) = X,

cantake all possible

periodicity of There angle

is.

10

Anomaly descent(In what follows

Chantz(A]

·is

gauge

de Chantz(A]Chantz[A]Wanti (A)==is0dchantz(A),&Want [A]
,

not

gauge

deWant(A]Wan(E A)
,

-,

its

infinitesimal

exactis.

The derivation

also

beWen((E En] A],,&Wan (Er Ez A],,anomaly descent.

befoundpostedascan-,,called themay,de Wan(t A)

iIt

exact

0

invariant , but,=isandsatisfies& Wan(En A]This.

.

dWan(f A]==simplify expressions)toclosedinvariant ,gauge transformation·writing "V"

omit

we,

in

an

Zumino's Les Houches Lecture
additional.
note.

11NoteSpdWaCE:consistency,A]satisfies

theandbeconditionanomaly (upIndeed,constantWp(G A)=Xc(e**, A](t,=++r((d(AdA dAA A3))++(d(AdA [A3)]+=+nothing but the Pd chiral anomaly

is,o+r(de(AdA dAA A3)]+=

Thusmultiplication).

a-thecandidate for

a

to

=

This

can

Wess-Zumino

factor
d 6=

d=

Why ?

Pofaxialchiral+d)d) -)up

toFyait.

anomaly seems to be relatedanomalyto.

(We'll come back to thisinamoment.

-)

12

plays

The descent

Supposewehavebackgroundonfat(A)aA

arole

anomalyin

moregenerally·system of differential forms

anddependtransformation parameter Egauge

fac (A] face A] faiCEEn A],,:. . .

,,,,thatdegree) which obeys the descent(the subscript shows the formequation

defatz(AT=fath(A]=dfaulA]0,

defin (A)=0,dfa (A],=dfa(E A],,

defalez Al-dezfn(EA] fallEE) Al-,,=df (E A],!

Suppose there is

ad-dimensional

The partition function

Xwith

backgrounddeZxlA]
Note :

the=onA

Ex(A]theory with anomaly fd

closedavaries:d-dimensional myd

asiSX fale Al.

anomaly satisfies.

theWE

consistency condition.

13We

d+1a,

andDextending AEx(1)to.

zx(A] e: =withbounday X:Spfin (A)↑iSyfaCt A) e

Ex (A)+&YA-==

Y and putting-GeEx(A)=Y

dimensional manifold
X A

Thencancelling the anomaly by choosing

may consider

iSyfan LA]

,

zx(A]

eiSyf(A)(i) def(Az)

Ex(Asi) Syfale Al-Special)
,~↑
=0X.

We would likeEx(A]of Y and extension ofA

totobeY.

independent ofthechoice14Solet,

usdifference

Thechoicemake another(YA') and.
compareis

Ex(A)/Ex(A] exp(iSyf(Az -i) fa(A])=,exp(iSpa(s)=YwhereAandY and I glued along X

isiss.t

Ely.

=

A

AlysandYA

=

=&Y

=-Y

A-=(, )X A,Stiles

We would likeLet us chooseboundary ↑=a=od+ 2 dimensionalOZ

andextend(mod2E).
manifold

EtoZ.

ZwithThen

Syfi(E) )0Zfan(E) ) dfilA] SfaCA).
===ZZ

15If fatz (A]=then0we,

have modified

unambiguously so that the anomaly

isthetheory.
absenthave to vanish for this :However , fatz(A]may notfatz(A)Ifweincan use

theafailA]-gan(A]-gaugeinvariantinstead ofGat (A)fin (A)=

Zx(A) eiSx((A] -9a(AT)the condition for the,Theorythewithmodification :Ex (A)
Thusgax(A]=sothatitis

cohomology class(faz(Aj)=0.

-ability

anomaly free

toismodify thevanishing ofof the top form :IfA

practical use

Considera4-dimensionalR-handed fermioninL-handed fermion

innoawithrepresentation VRof G

representation Unofabsent

betoChrs()-.

ExerciseaG-anomaly

The condition ofChr 6 (A)theory0=is.

Show that the standard model

gauge
G=VR

=

k=anomaly
.

In.
G

has

this case,SU(3) x SU(z) xU(l)((k 1,-,((1 2,1) (B 1,-,z)(B,,5)(B2,5)7*3,4-,5)7317

d=2n+2 axial anomaly vs d=2n chiral anomalychentz v(A]density of...

,Wan v (- A]

dof chiraldensity...

,,=anomaly

Dirac fermiond = 2n+2

descent

axial

rep V of Gin.

anomaly in

R-handed fermion

2nin

in

rep V of G.
.

Why ?
>
-D

Anomaly&Index(Callan-Harvey 1985]

inflowtheory(Alvarez-Gaume-Ginsparg 1984
afteryM:T=

Gamma matrices(b -)vm(0)12+panpane (b)128)

=(o)

in

&+d=2nAriyah-Singer 1984]=Samma[matrices

=(ii)

in

D

=2n + 2a+2) =

Is

D=

Dirac fermion with

2n + 2L*- Eri E*=Let us consider

a**mForcomplex mass

2M"(x=En=( &)Dirac equation

position dependentis( )+-=,(0am+Ermass(2)

Op br:

R in dbr:

Lin d,92,=2n=In:E))
+Ar

aGet

METr

2MZ

=(-a)
(b) +=+single=br=R-handedsupportedat-=0masslesszm())= Miz 4m(x) -)

=b)

=

=.
0fermion=019D

Now consider

There-·isR-handed fermion

aeffective

phase of massmTheta term

SoolA]

ina

rep V of G.

=arg(m)Spo(A)

iS

repV

in

suppatzo=Whv (Alo]actionThe-

Dirac fermion

2n+ 2.
m = 2 ME

with·==arg(z) -arg(z)
=DP

with

=-arg(z)chenvA

Do-&d Wanti v(A],=i)

darg(z) Wenn v(A],*IR xDIts-

gauge variation

ESor(A) =>Im DID's

toisim i)deWenvA
darglz -IR x (DIDE)&Wan v(E A]&=dbe precise"StrxIDs,darg(z) Wan v(E A],,,20ziti=TheD0de)==2n+ 2-/dWanvCE A.

System has

nogaugeanomaly:

Wu (Alo) - SoolAs(deW(Alo) + 2πi((Wanv(E A))

==

,

:

Thedensity

fermion-inofchiralrep.

V24 Wen v/E A).

,anomalyof G

isobtained

ind=

In

R-handed

indeedvia

odescent.