Topological anomalies from the path integral measure in supe(6)

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

TheoperatorsTµν(x)andζµ(x)arespeci edby

Tµµ(x)anomaly=0,ζµ(x)anomaly=0(1.15)

µandcorrespondingtotheabsenceofananomalyinγµjµ,see(1.5),butalthoughbothζ

ζµareareconserved,Tµνisnotconserved

νTµν(x)=0,

similartothenon-conservationofjµ.

IntermsofTµνandζµ,thesupersymmetryalgebrareads

i{Q,Q}= 2(γ)Pµ+ α βµαβ(1.16) dxh¯g

1 (x)(γ5)αβ(1.17)π

whereweused 01=1and

Pµ=

Z= dxT0µ(x),

dxζ0(x)= H=P0,

dxζ0(x).(1.18)

Weseethatthesupersymmetryalgebraintermsoftime-independentchargeshasthesameformatthequantumlevelasattheclassicallevel,see(1.10).Thisagreeswith[8],whoseanalysisisbasedonthisobservation.In(1.16)wehaveusedchargesPandZ;Z explicitlyappearsontheright-handside,isfreefromanomaliesandtheanomalyofZ

µbutPstillcontainsasuperconformalanomalythoughTµisfreefromthetraceanomaly.

(Inotherwords,T00andT11haveequalanomalies,andtheanomalyinT00doublesthe intermscontributionin(1.14)proportionaltoF,ing(1.6)towriteQ

ofQ,alltheanomalytermsin(1.17)cancelseparatelyifonesplitso theanomalyfromPµ.Inthisway,alsointermsofQ,PµandZthequantumanticommutatorhasthesameformasclassically.However,QandParetime-dependent,sotheyarepysicallylessrelevant.)Thesetwoalternativewaysofwritingthealgebragiverisetothesamephysicalconclusion,namely,uniformshiftsinenergyandcentralchargeinthevacuumofthetimeindependentkinksolution.BothmaintaintheBPSbound,sincetheydescribethesamealgebraonadi erentbasis.

2Themodelandthesuperspaceregulator

Webrie ysummarizesomeofthefeaturesofthemodelwhichdescribesthesupersym-metricmodel;theN=(1,1)supersymmetricWess-Zuminomodelind=2Minkowskispace.Themodelisde nedintermsofthesuper eld

¯(x)+1φ(x,θ)= (x)+θψ