Topological anomalies from the path integral measure in supe(20)

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

whereδsusyφ(y,θ)standsforthevariationofφ(y,θ)

δsusyφ(y,θ)=δ2(x y)Qαφ(x,θ).(5.14)

WethenapplytheBjorken-Johnson-Low(BJL)analysistoreplacetheT productbytheTproduct8

µ,α(x)φ(y,θ) + δ2(x y)Qαφ(x,θ) =0 i µ TJ(5.15)

andobtaininthelimitk0→∞theequaltimecommutator(seeappendix)

0,α(x),φ(y,θ)]δ(x0 y0)=δ2(x y)[ i[J

2π( (y)γν )β(5.18)

whereweused(4.19).Inthepathintegralframework,thisrelationisderivedbystarting νwith j(y) =Dφjν(y)eiSandconsideringthechangeofvariablescorrespondingto(local)supersymmetry

µ(x)jν(y) =δ(x y) δsusyjν(y) .i µ T J(5.19)

Thelocalvariationsoftheactionandthemeasuregivetogethertheleft-handside,justasin(5.13).TheBJLanalysisthengivesrisetothecommutator.Theoperatorsappearingherearegivenby9

ζµ(x)= µν ν (x)U( ),

2πγµ 01= 1,

ψ(x)φ(y,θ) =0.Ifonewouldkeepthe

derivativeoperatorinsidetheT-product,thisconditionisspoiled.SeetheappendixforanaccountoftheBJLprescription.9Bynotingthecompletenessof(1,γ5,γµ),thesupersymmetryvariationofthecurrentjν(y)isex-pandedas

δ jν,β(y)= 2Tµν(y)(γµ)βα α 2ζν(y)(γ5)βα α 2vν(y) β

Bymultiplyingthisrelationby ¯γρ, ¯γ5and ¯,respectively,wecanprojectoutthe3componentsaboveµbynoting ¯γ = ¯γ5 =0.Thevectorcomponentvνisshowntovanishon-shellbyusingsafe(i.e.,¯(x)γµδSanomaly-free)relationssuchasψ¯(x)=0,exceptforthetermexplicitlywrittenδψ

in(5.18).¯gψ(x)φ(y,θ) .Itshouldbereplacedby µ Th2πγµ