Topological anomalies from the path integral measure in supe(18)

A fully quantum version of the Witten-Olive analysis of the central charge in the N=1 Wess-Zumino model in $d=2$ with a kink solution is presented by using path integrals in superspace. We regulate the Jacobians with heat kernels in superspace, and obtain

Ifonesetsvµ(x)= µv(x)in(4.20),onegeneratesthedivergenceofthecentralchargecurrentbuttheproceduregivesnoinformationforatopologicalcurrent.InanalogywithU(1)gaugetheory,weareconsideringthechangeofvariableAµ→Aµ+aµinsteadofAµ→Aµ+ µatogeneratethecurrent.

Itisimportanttorecognizethatalloperatorsappearingontheleft-handsidesoftherelationsin(4.7)havehighermassdimensionsthanthoseofthecorrespondingoperators µ(x)andζµ(x)are,respectively,dimension2andontheright-handsides.Forexample,ζ µ1operatorssincethecouplingconstantgcarriesaunitmassdimension.Similarly,γµJ µandjµandγµjµare,respectively,dimension3/2and1/2operators,thoughbothofJ µandTµare,respectively,dimension2and1aredimension3/2operators.Also,Tµµ µνandTµνaredimension2operators.Inthissensealltheoperators,thoughbothofT

compositeoperatorsontheright-handsidesof(4.7)aresoftoperators.Thissuggeststhatonlythe“hard”operatorsgenerateanomalies.Inthenextsectionweprovethisstatement.

5SupersymmetryalgebraofthequantumoperatorsIntheprevioussectionwegaveadirectderivationoftheanomaliesbasedonpathintegrals,butwealreadymentionedintheintroductionthatonecanalsoobtaintheanomalies µanomalybymakingsuccessivesusytransformations.InthissectionwefromtheγµJ

implementthissecondapproach.Sincethisinvolvescommutatorsofcurrents,weconvertthepathintegralrelationsintooperatorrelationsbyfollowingtheBJLmethod.Webeginbyconsideringthevariation

δφ(x,θ)= ¯(x)Qφ(x,θ).

Thechangeoftheactionde nestheNoethercurrent

δS=

where (5.1)d2x( µ ¯(x))jµ(x)(5.2)

(5.3)jµ,α(x)= {[ (x)+U( (x))]γµψ(x)}α

2withU( )=g( 2 v0).TheJacobianfactorfor(5.1)givestheanomaly,andweobtain

theidentityh¯g µjµ(x)=

2πγµψ(x), µ(x)=0. µJ(5.5)

ItcontainsthecontributionsfromtheactionandJacobian,appearsinalltheWardiden-tities,andthisimplies,asweshallsee,thattherelationsamongvariousGreen’sfunctionsobtainedbyglobalsupersymmetryarenotmodi edinformbynon-trivialJacobians.