Gravitational anomalies
A RFT

gmay

be formulated

A naturalSuvdkds.

=

under

covariantarewhetherquestion isI e.

.
preserved

iscovariancegeneral gravitational backgroundaon.

generalthewhether correlation functions

diffeomorphisms

spacetime

whether the partition function

,

e.

g.

invariantisz(f 97 Ez(S].
*breakdown of thisAWe
a

may

would

also consider

gravitational anomaly.

anomaly of internal symmetryinageneral gauge a gravitational background.
Computation ofinas

thecaseanomaliesof gauge

anomaly toandthe fullthe W]

isorderobtained

consistency condition

succinctly described.

in

thein

thecomputation

the 1st non-trivial
answerbe done incananomaly

The perturbative (one loop)

tothebein

thesameway

flat spacetime :determines thetheby finding

background,a

The result issolution
mostlanguage of descent.

Letconsiderus

withvalues inaTheanomalyis

I'd oftheHere ,

/GJaz

aweof

in terms

a=zn.
group G

the d-form descentpolynomial

sinh(R)) Jane(tredet)Rstandsfor.
component of 0

the d+ 2 form

regard the Levi-Civita connection Tp=(pr)as,DL(d IR) gange potential (with curvature Fr R)

and=,, FA) as
regard [CRLie algebra
angiven=dimension din

representation V ofinvariant

I(RE)whereR-handed fermionaGotinfinitesimaltransformation E

au e (9 A)
,.

where

Qu=

angl(d IR)=,invariant

polynomial of the. Then thediffeomorphism (U=anomaly forUn) and gauge

is=-Es (4 As[d((Qu2),(r) (drU")=.

,,Thisderivedbecansteps

twoinStep 1

To the first non-trivial order in the fields (ideviationhow.

flat metric So

from the=,it matches with the result

darthure.

and,theAp),[AG-W] of the perturbative

computation .

Step 2

It satisfies the WE consistency condition.

(This is not-so-trivial

and will be,

explained

laterinthis note.

)

Remarks

DThe above [CR FA),ofthed+ 2

in

axial

is

ty density

anomaly of the Dirac fermion in dimension

thatsurprising

is-2[ (10 E) (P A)]·that

theCallan-Harvey anomaly inflow argument,

thenotTherenothing but

general gravitational and gauge background.

aIn view ofitisisyields

also
the,,densing of chiral anomaly

theinageneral background.

anargument

based on index

above

formula[AG-G, ASZ].

theory②Fermionsingravitational backgrounda

Orthonormal frame field

formulated

Using
its inverse
ore
expressed

isbye Thesan+

In this frame.

isege= Was:

The above result

gauge

choiceby the metric,

ise.

g.

so,

=that

metricbe

Of course

canwork withweinwritten

ZCe A]

whether

Cs0adetermined.

....

Thenthings areFor example.

Theninvariance under,,theas/g A].

anarbitrary choice of,,,thezene)which the partition function say,the frame ,aswritten=

the frame is(and A)

canGruof [AG-W] where a

di + Chapt

partition function

,

is :

Catin the setupCa.

by thedetermined justambiguity

madeisconnection

SO(d) gauge potential

SO(d)-wisean

C ("vielbein")the Levi-Civita,(Wis is determined by eve as the metricThereusuallyarethere isanewOrthnormal frameisquestionchange:Ca Hes0aisR

anomaly for that

anI'descenttheFw=

preservedis

andforthe[CR FA),

It:

eFp - eg:

Thed-form descent is of,[CR FA),,

thanofwhereinvariantanasgl(d iR) 5,

hence [(Fw FA),

thethere,in terms

given

rather,Indeed.

regardedis(Note=issame

polynomial for sold)Fro

notor=.

[CEr Fz).

)
.

form

In ((0 E) (w A)],

where ⑰
For this,isthe,,

aninfinitesimal SO(d) frame change

WZ

consistencyisautomaticfollows directly from the descent equation

Thereisathegeneral covariancelocal frame change

andit.

in[AG-G, BZ]

between theanmaly ofprecise statement given

concerning the relationship.

theanomaly of the.

End of Remarks

otherTwo

gravitational anomaly

withsystems

often consideredantisymmetric tensor field
which appear

The

Spin 3/2 fermion

R-handedarewithandself-dual fieldstrength,

parts of supergiarity

aswhich areorsuperstring theory

anomaly is
au (g)

I'd

whereinvariant=is-2) [d(Qr T],

thed-form descent of thegl(d IR),

polynomials

1)[s(R)-feet)( - tde+ ) atacoh(n)-d+22p(R)These

are=non-trivial

To the lowest orderresultsin[AG-W]

Thebelow.

sameinonly in dhar,).
4h + 2 /ke *they match with the perturbative

The WI

conditionaremarks

=

as

in

will

bechecked

Spiny fermion

apply.

.

WI

consistency for diffeomorphisms

The W]

consistency condition for diffeomorphisms is somewhat
different from the case of internal
And the.
symmetry

proof requires more than justthe,Diff(X).

on. The diffeomorphism
oppositegroupissignacts

f (fz(s)Xnaturally from the left (f ofu) (n),if rector fields

Thus,of the Lie algebra of Diff(X)is( 1) times

(v w]
,

On the19,Aother=-,,=

onthefeig),(X),field bracket:.

of Diff(X) on fields

action,(fof) grector[V w] vie,

handthe,

...

,

is

aright action

just like the.....

AS)(A3)9 ?

=

Eden W(A)-dede W(A],

Thus,

=

=regarded as elementsarealgebra bracket**G(X)

Liethe) + (f"S f A...

.

itselfThe condition

Firstthe descent equation

actionofjust like

Greens WCA],
Lie

have

wedu duz W(9 A -),-,,Juzdu W(g A ---devivueW(9 A=,Second,in the presencediffeomorphisms

dt and Ge*ji A=)

devive W(9=of internal,A,...

).

,gauge symmetry G
donot commute.

the infinitesimal transformations

bedA

LvA ,Their compositions-,,G-transformations

andLet,,de + (A E]=,

areGitjA = (d (A) ↑ [A]=+of acts onlyon

the fields

GA = A) =G ((VGAGUA=dx"t (Ap (2)(v +& Opt=*dx+,+[OyAm E],(++29(dyt (Ay z])+,(Ap OgE],(Ued An du(Ap(dr(29d =) (Am Upd E) E]),+Give Am,,++,(GrAp,,E]

(A=(GA t)+,=

(GG s:TheWE condition for the simultaneous differ agauge,

diff

dr = dotransformations

de+,

is

[du E druien] W(g A]
,,

,

=

( (E) (d + )W(S A)=>(litdi) WA+.

-(12)-↓diTree,A)-])W(It i=+,d +),-==jaif Unl(-j+vO-CViva),,Ga,+) + vt ,(Entu) + VE,--v,e) W(9 A).

VienW(9 A),.

The proof①Letusfirst consider the case of pure diffeomorphisms.

thoughEven(suchas

there

theminau/t]
I'dwhereinvariant

background fieldssystem)

gravitational field g.

focus on theomit

Spin fermion

forA

be othercanthe notation .

=-TheIe.

may justwe,.

we

simplyanomaly formula is2) [CO ↑],isthed-form descent ofpolynomial [CR)ad + 2 form

gl(d IR)of,,and= dry"The effect↑wherethis*g+f gof==gYtheon(M" o MIM2 NYcof

Mir=of Ifwe

M"f T, M + M OpM*.

connection isIM"OnM)"

denotereads↑=+Levi-Civita-(f* = Tit,

Infinitesimally,
GTp

Gotp

=+(p Qu],

Gar-n~

gait t
Thus

+jGs

M,

I w N + Gl(Ow 4).

Jr[n(Ow i) = G,,Note that

get

only

acts

jain(@w T)=([a(@ P] is,

For example,andLoOw

linearinQ.

.

Thus

[d(GrOw P],.

=U, Qu==+r(@Lv(dPd41)

UG,=.

Sw

Lo [d (Ow PJ)Gc[(Qu + ]

Thus

-on

[vir(@dPdP) - Tr(Lr@dPd4)

On the other hand,

integrationnot

CrT(Ow T]

Gtr(RTR)=

Note thatN,,

Since

on,0.

disappears afterdo) [ SQw +](NEW-,

S2-[(w@w=,+] + IQw 4])-,[TOr↓

Crews

The GL(d IR) descent equ,,-Crews

STh[-r@w wou

=+[!

7=>+w

:=,,,,[Q , w7)"-&(w -]",&U dow-d, W d Ve
Y+Fequal !

IDu (Wid, "=aud , U" + Widdot"-drVJW"- Udde WM/ [ (@cw,2).

-V90, WM)+].

draw/p]-JoQu/t]
This,[Qu Ow] T)

UpddrW" + Wed, drun

( 3)*=++[d((Qu Ou M

=is=Acurs(P).

nothing but the WZ consistency condition .

&of simultaneous differ a gauge transformations

The caseThe.

anomaly formula is
Qu e (4 A)

Variation=,.

on-2) [d((Qu E) (4 A)].
,.

,(T,A) isdu eGi=,G((d) XG+G(Q e),Gu ez [a((@ E_) (P A)],,.

,[ ((Duz f) (t A)) G TdL(Qu E) (T AS]
+=

,,,.

,.

-& In ((Qua Ev) (P A)]·-vanishes after.

-V=The GL(d 1) x D.

,.

integrationGuen) [((Qu En) (t A))·:

Th (NiQue V.Ez) (4 A)],.

,(1702)-,,descent

equation

StarVien) + (VnOv Unti) + (CQri G) (Ove ful) IP Al],(- U .

+u

UnQu + (Qui Qua],-⑮ Cruis.

-,,,,

Vitze + (EE)),·Ove Que (T A),,=>

Thisis&[Ua,V. ]-,-GutzAve (4 A]

Vien + Ve + (tu].

(P A],.

nothing but the WE consistency condition.

References[AG-W] Alvarez-Gaume and Witten, “Gravitational anomalies,”

Nucl. Phys. B234 (1983) 269.
[AG-G] Alvarez-Gaume and Ginsparg, Annal. Phys. 161 (1985) 423.
[ASZ] Alvarez, Singer & Zumino, Commun. Math. Phys. 96 (1984) 409.
[BZ] Bardeen and Zumino, Nucl. Phys. B244 (1984) 421.
[GSW] Green, Schwarz and Witten, Superstring theory: 2,(Cambridge University Press, 1987) Ch.13.