Topological anomalies from the path integral measure in supe(4)

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

equations,butthese eldequationssometimesyieldanomalies.TheBJLmethodtakessuchanomaliesintoaccount,andthisiscrucialinourcase.Forreaderswhoarenotfamiliarwiththistechniquewegiveadiscussionofthismethodintheappendix.WethuspresentafullyquantumversionoftheWitten-Oliveanalysis.ThedeformationofthesupersymmetryalgebrabyanomaliesissuchthattheBPSboundremainssaturatedduetouniformshiftsinenergyandcentralcharge.Inouralgebraicapproach,weinitiallyformulateallresultsintermsofatotalsuper eld,anddonotdecomposethissuper eldintoabackgroundandaquantumpart.Onlyattheenddoweneedtousesomeprop-ertiesofthekinkbackground,namelythefactthatthevacuumisannihilatedbyoneofthesupersymmetrycharges(thetime-independentcharge).

Wewouldliketosummarizeourresultsbrie y.ItisshownthattheNoethercurrentforordinary(nonconformal)supersymmetry

jµ(x)= [ +U( )]γµψ(x)

2withU( )=g( 2 v0)containsinthepresentformulationanapparentanomaly(1.3)

µjµ(x)=h¯g

2πγµψ(x)]α; µ(x)=0 µJ(1.6)

andtheassociatedtime-independentsupercharge

Q= α 0,α(x)dxJ(1.7)

whichgeneratesordinarysupersymmetry.Thisconservedcurrentcontainsthefollowinganomalouscomponent¯g µ(x))anomaly= h(γµJ

Divergenceswithrespecttoθinsuperspaceleadtoanomaliesinx-spacewhichareoftheformγµjµinsteadof µjµ.Similarly,theanomalyintheenergydensityisduetothetraceanomaly,whichitselfisduetoanotherθdivergenceinsuperspace.Fromasuperspacepointofview,θdivergencesandxdivergencesareequallyfundamental.1