Topological anomalies from the path integral measure in supe(7)

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

whereθαisaGrassmannnumber,andθαandψα(x)aretwo-componentMajoranaspinors;¯=θTCwithC (x)isarealscalar eld,andF(x)isarealauxiliary eld.Wede neθ

thechargeconjugationmatrix,andtheinnerproductforspinorsisde nedby

¯≡θTCθ=θαCαβθβ≡θ¯βθβθθ

withtheDiracmatrixconvention

γ0= γ0= iτ2,γ1=γ1=τ3,C=τ2,γ5=γ0γ1(2.3)(2.2)

Thechoiceγ1=τ3hascertainadvantagesfortheevaluationofthespectrumofthefermions;weshallnotevaluatethisspectrum,butstilluseγ1=τ3inordertoagreewiththeliterature.Weusethemetricηµν=( 1,1)forµ=(0,1),hence(γ0)2= 1but2γ5=efulidentitiesare νµγµγ5= γνandγµγ5= µνγνwith 01=1.Sinceinthisrepresentationthechargeconjugationmatrixequalsτ2,theMajoranaconditionψ τ2=ψTCreducestothestatementthatallMajoranaspinorsarereal.Wefrequently¯, ¯µ and ¯5 (thesigninthelastrelationusetherelations ¯ψ=ψ ¯γµψ= ψγ¯γ5ψ= ψγ

isoppositetothe4dimensionalcase).

Thesupersymmetrytransformationisinducedbyitsactiononthecoordinatesinsuperspace,φ′(x′,θ′)=φ(x,θ).Onehas

θ′=θ ,

¯µ x′µ=xµ θγ

leadingin rstorderof to

¯µ ,θ+ )=φ′(x,θ)=φ(xµ+θγ

¯µ µ (x)+ ¯(x)+1φ(x,θ)+θγ¯ψ(x)+θ F(2.4)

2¯)¯(θθ γµ µψ.(2.6)

Intermsofcomponentsoneobtainsfromδφ=φ′(x,θ) φ(x,θ)

δ = ¯ψ(x),

δψ= µ (x)γµ +F(x) = (x) +F(x) ,

δF= ¯γµ µψ= ¯ ψ(x).

Thesuperchargewhichgenerates(2.4)

Q≡ ¯¯α (2.7)

+η¯γµθ µ¯ θα