Remarks on axial anomaly in d=2ndFor=

Dirac fermion 4 with values in2na,
rep V of GDaleieva) DaleieVa4)
BAT OP=&theory where G

In theweaction

may,-Surla]Then ,=

the axial

The axial

shifts·exp(zi)a (()]+In d P=a+gangedas,aiSe Outr ( )") .
formula says

anomaly
4-Eitr4,- Generat

P + ze.

VG simple,

there

is

a

canonicallynormalized Theta term

-SolA]

part of thethe Theta term

considerrotation&is=i) +(F)"the standardusingtriXY
NoteThenWeSpThe=--↑But

this=-.

)...

Ex Y,.

.

ey.

,.

Y

0 + 4T-S=

Dirac fermion

amass ,mIm/e=gargimv4inducesthe,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

Cresp4din

Cresp Oftheta parameter
Or,m)

+with

i' =

(0=>

by&X .

iFrDaYm-iFate

phase ofbytrg(XY:=consider

maydefinedShift isthe0·)tr(XY)

that,...

trace".

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

②formChern-Simons& the density ofdenotedIt

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

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

OFAidentity==DadA

DaFA=,0.

followsn+:(DA(GA) E)

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

M

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)

Chen-Simons form

Want r(A"],,--The expression forthe methodto,,Gane ,v

be

found

from

chen

canfindWany v.

,[see Zumino'sFor.

,

they,v.

areo=X2v(g A)=Xv(9 A)=,,by extending

Les Houches Lecture

lowns ,

No v (9 A]:,(2n-2) form & 2n 2 v [9 A]someinvariant

gauge

Wint v(A) + Wen,v (ging] + & Genz v(9 A)=,for

NOTis-[ + (dgg A)-r(dgg(AdA++i+t(Adgg (2dAA + A3)++A(dgg'(Y)& Integrality
SupposeAgidg+SynCA)

and(Ford > 2Thenfor.

FA,SpaOu

For d 4=Chau (a]eThis

G~

that

simpleandandCFor

thisSo(A]isreason.

at1orOgidg)

Sv Or(A]is,einvariant

simply connected,=underinvariantOr+.

Spptr((Fa))-FA-0sumeans

Pu + 2/Kv>-&

implies A

co0thatso,

finite.
near=Kr.

someunderis

lastersas2

P-P + 25

calledis·Theta angle

D

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

~=] Gag'dgonSiatlarge R

Di SiG==i J

War (59]d 1-Si

Note that

Way v (gdg]*=,

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

class

isa

multiple ofanG

whose

cohomology

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

gaWar(gg)=S (SWar)IExamples·Wiv/gag]
For GSiUll = 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 Other

windingnumber(6 V d),.

,of theSanWat,

mapsuchaninterpretation

is:1999]

interpreted as "Winding number"org Snotin-

canthe samestraightorSUES
beway,even.
absente9.

d.

=2G,+X

simple.
=Chu/A]eo:

axial

Thus

However ,d 4,

We

is

not

have

asimplymayof 9 92:-+meansG:..

Then theOn 0 + 2.

=

S'- GaSU(n),finite group).

USp(n) Spin(ul),Thenmeasures

definesmap,(eg GSptFA)")

integer valuess ·2πandFrep V(TT G isconnected

the standard "tr" .

SgcitringThis

non-trivial,

usefor= 0absent.

issimple simply connecteda=anomalymayFrep .
V,

Wvlgng],weG

when·Exeg0

thewinding number

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

cantake all possible

periodicip of O is strictly.

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

be

Wen((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,

inan

Zumino's Les Houches Lecture
additional.
note.

Note

SpdWaCE:consistency,A]satisfies

theandbecondition

anomaly (upIndeed,constant

Wp(G A)=&x(e**, A](t,=++r((d(AdA dAA A3))++(d(AdA [A3)]+=+nothing but the Pd chiral anomalyis,o+r(de(AdA dAA A3)]+=Thusmultiplication).

a-thecandidate forato=

ThiscanWess-Zumino

factord 6=d=Why ?

Pofaxialchiral+d)d) -)uptoFyait.

anomaly seems to be related
anomalyto.

(We'll come back to thisinamoment.

-)

A

practical use

Considera4-dimensional

R-handed fermioninL-handed fermion

innoawithrepresentation VRof G

representation Unofabsent

betoChrs()-.

ExerciseaG-anomaly

The condition ofChr 6 (A)theory0=is.

Show that the standard modelgaugeG=VR=

k=anomaly.

In.
Ghas

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

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)

inD

=2n + 2a+2) =D=

Dirac fermion with

2n + 2L*- Eri E*=

Let us consider

a**

mFor

complex mass

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

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

R in dbr:

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

METr

2MZ

=(-a)
(b) +=+single

=

br

=

R-handedsupportedat-=0masslesszm())= Miz 4m(x) -)

=b)

==.
0fermion=0

D

Now consider

Thereis~·Thea

R-handed fermion

Effectiveaction

phase of mass

axial

anomaly~-Dirac fermion

2n+ 2Soo(a)inarep V of G.

.
m = 2 ME

with·=

=

=supp

atzo
=·arg(z) -arg(z)=Spo(A) with D0 -arg(z)=Chenz20

repV

Whv (Alo]arg(m)

Theta termiSin,v[A]-max &d Wanti v(A],=i)darg(z) Wenn v(A],

*IR x

Its

-gauge variation

ESor(A) =im>

In DID'stoisi)

deWenvA
darglz -IR x (DIDE)&Wan v(E A]&=limi S

be precise,darg(z) Wan v(E A]

IR2XID2=,,,

ziti

=The0

D

====2n+ 2de)/d

WanvCE A.

System (Dirac fermion)

-has

gauge

noanomaly

Wav (Alo) - SoolAs)deW(Alo) + 2πi((Wanv(E A)),:

Chiralanomalyof G

indeed has

obtainedviaind=2nR-handed fermion

oin

rep. Vdensity-IT Wen v(E A)

descent.

.

,: