Topological anomalies from the path integral measure in supe(5)

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

isthenderivedbyusingtheThesupersymmetryalgebrafortheconservedchargeQ

BJLmethod.WerewriteWardidentitieswhicharederivedfrompathintegralsandwhichcontainthecovariantT timeordering,intermsofWardidentitiesattheoperatorlevelwhichcontaintheTtimeorderingsymbol.Weobtain2

α,Q β}= 2(γµ)αβP µ 2Z (γ5)αβi{Q

where

µ=P

=Z (1.9) 0µ(x),dxT

0(x)= dxζ =P 0,H

0(x).dxζ(1.10)

TheBJLmethod,unlikethesemi-classicalDiracbracket,incorporatesallthequantum µareconserved µandζe ects,inparticularsuperconformalanomalies.TheoperatorsTνquantities µ=0 µ=0, µT µζ(1.11)ν

butcontainsuperconformalanomalies

¯g¯(x)+h µ(x)=F(x)U(x) g ψψTµ

2πF(x),

h¯g

µ (x)γµ.(1.12) µ(x)γ5= µ (x)Uγµ+γµζ2π

Wederivetheseequationsfromthepathintegralformulation.Intheseequations,Tµµ(x)and(γµζµ)containonlythetermswhichexplicitlybreaksuperconformalsymmetry,asweshallshow.Thesearisefromthesuperpotential,are“soft”(theyhavelowerdimen-sionbecausetheyareproportionaltothedimensionfulg),andtherearenoanomalouscontributionstothesequantities.

Therelationsin(1.5)and(1.6)canbecombinedtogiveasimilarresultasin(1.12)

µ(x)=γµjµ(x) γµJh¯g

¯g µ(x)=ζµ(x)+hζ

24πF(x),Thesymbols(γ0)αβand(γ5)αβdenotethematrices(γ0)αγ(C 1)γβand(γ5)αγ(C 1)γβandareinourconventionsequalto iandiτ3,respectively,seeSection2.Theindicesαandβareequalto+or ,andforα=β=+one ndsthatthequantumanticommutatorhasthesameformastheclassicalrelation(1.1).