Topological anomalies from the path integral measure in superspace

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

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aUT-03-07YITP-SB-03-11TopologicalmeasureanomaliesinfromsuperspacethepathintegralKazuoFujikawa1DepartmentofPhysics,UniversityofTokyoBunkyo-ku,Tokyo113,JapanPetervanNieuwenhuizen2C.N.YangInstituteforTheoreticalPhysicsStateUniversityofNewYork,StonyBrook,NY11794-3840,USAAbstractAfullyquantumversionoftheWitten-OliveanalysisofthecentralchargeintheN=1Wess-Zuminomodelind=2withakinksolutionispresentedbyusingpathintegralsinsuperspace.WeregulatetheJacobianswithheatkernelsinsuperspace,andobtainallsuperconformalanomaliesasoneJacobianfactor.Theconservedquantumcurrentsdi erfromtheNoethercurrentsbytermsproportionalto eldequations,andthesetermscontributetotheanomalies.Weidentifytheparticularvariationofthesuper eldwhichproducesthecentralchargecurrentanditsanomaly;itisthevariationoftheauxiliary eld.ThequantumsupersymmetryalgebrawhichincludesthecontributionsofsuperconformalanomaliesisderivedbyusingtheBjorken-Johnson-Lowmethodinsteadofsemi-classicalDiracbrackets.Wecon rmearlierresultsthattheBPSboundremainssaturatedatthequantumlevelduetoequalanomaliesintheenergyandcentralcharge.1Introductionandbriefsummary

Supersymmetryandtopologyareintimatelylinked.Forexample,instantonsplayanimportantroleinthee ectiveactionforrigidlysupersymmetric(susy)models[1].TheDonaldsoninvariants,whichcharacterizetopologicalpropertiesofcompactmanifolds,canbecomputedbyusingaparticulartopological eldtheorywhichisobtainedbytwistingaEuclideansupersymmetricN=(2,2)model[2].Weshallconsiderherethesurfacetermsinthesupersymmetryalgebra[3]whichformthecentralcharges.

Thesupersymmetryalgebraofthekink,anN=(1,1)rigidlysupersymmetricmodelin1+1dimensionswithasolitonsolution,readsasfollowsattheclassicallevel[3]

{Qcl,Qcl}=2Hcl 2Zcl(1.1)

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

HereQclistheclassicalsupersymmetrychargewhichleavestheclassicalkinksolutioninvariantand,properlyextendedtothequantumlevel,shouldleavethekinkvacuuminvariant.HclistheclassicalHamiltonianwhichgivestheclassicalmassofthekinksolution K(x),andZclistheintegralofatotalderivative

Zcl= ∞

∞U( K) x Kdx(1.2)

whichisnon-vanishingbecausethekinksolutionhasatopologicaltwist( K(∞)di ersfrom K( ∞)).Theresultin(1.1)canbederivedbyusingsemi-classicalDiracbrackets.Inthe1970’sand1980’ssolitonswerestudiedindetail[4],andtheissuewhetherforsupersymmetricsolitonsZismodi edbyquantumcorrectionswasstudiedinseveralarticles,withcon ictingresults[5].Thekinkmodelbreaksconformalsymmetryexplicitlyandisnonintegrable,andhencemethodsusedforexactlysolublemodelswereofnoavail.SixyearsagotheissuewhethertheBPSboundHcl=Zclremainssatis edatthequantumlevelwasagainraised[6],andsubsequentlyinaseriesofarticlesbyseveralauthorsthequantumcorrectionsto H and Z werecalculated,whereby H wemeantheexpectationvalueofthequantumHamiltonianinthekinkvacuum,andsimilarlyfor Z .ItwasfoundthateventhoughZisclassicallytheintegralofatotaldivergence,therearenonvanishingquantumcorrectionsto Z whichareequaltothoseto H ,sothattheBPSboundremainssaturatedatthequantumlevel[7,8,9,10].Thenonvanishingcorrectionsto Z comefromanewanomaly,whoseexistencewasconjecturedin[7]andsubsequentlyfoundandevaluatedin[8].Thisresultwasinfactincon ictwiththeresultof[9]whereBPSsaturationandnonvanishingquantumcorrectionsto H and Z wereobtainedapparentlywithouttheneedfortheanomalousterminZfoundin[8].However,ashasbeenclari edrecentlyin[10],thiswasduetomanipulationsofunregularizedexpressions;consistentdimensionalregularizationindeedreproducestheanomalyinZ.TheexistenceofananomalyinZisthereforebynowbeyonddoubt.Qualitatively,thereasonisthatZisacompositeoperatorwhichshouldberegularizedatthequantumlevelby,forexample,point-splitting,andalthoughboththenonanomalousandtheanomalouscorrectionstothecentralchargedensityζ0(x)arestilltotaldivergencesinthisregularizationscheme,thespaceintegraloftheanomalouscontributionsnolonger ∞vanishes,beingproportionalto ∞U′′( K) x Kdx.However,thedetailsarequitesubtle;di erentregularizationschemesgiveunexpectedcontributionstoζµ,whichconsistofnon-anomalousandanomalouscontributions.Forexample,usingordinary(’tHooft-Veltman)dimensionalregularization,parityviolationduetomasslesschiraldomainwallfermionsintheextradimensionisresponsiblefortheanomalyinthecentralchargein2dimensions,whereasusingdimensionalreduction,theanomaliesresidenotinloopgraphsbutintheevanescentcountertermswhichrenormalizethecurrents[10].Usingthehigherspace-derivativeregularizationscheme,thereisanextraterminthecentralchargecurrentwhichproducestheanomaly[8],whileusingheatkernelmethods,subtleboundarycontributionsproduceanomalies[11].

ThediscoverythatananomalyispresentinthecentralchargehasledtoprecisecalculationswhichrestoretheBPSboundatthequantumlevel,butraisesprofoundquestionsconcerningtopologicalsymmetriesatthequantumlevel.In[10],apreliminary

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

analysiswasmadeofordinaryandconformalmultipletsofcurrents,andordinaryandconformalmultipletsofanomalies;inparticular,aconformalcentralchargecurrentwasidenti edwhosedivergencecontained,inadditiontotermsduetoexplicitsymmetrybreaking,theanomalyinthecentralcharge.Thusthecentralchargecontainsananomalyandisitselftheanomalyofanothercurrent.

Inthisarticle,weintendtostudytheanomalystructureofthecurrentsofthesuper-symmetrickinkmodelinsuperspace.WeusethepathintegralformulationofanomalousWardidentities[12,13],andthepresentworkextendsasuperspaceanalysisofconformalanomaliesinQCDin3+1dimensions[14][15].Asuperspaceapproachtotheanomaliesinthecentralchargeandthekinkenergywas rstgivenin[8].Inthatarticleseveralregularizationschemeswereusedtoevaluatetheone-loopcorrections,inparticularahigherderivativeregularizationschemeinsuperspace.Weshallstartwithapathintegralapproach,andcomputetheJacobiansforgeneralizedsupersymmetrytransformationsus-ingasuperspaceheatkernel.Thiswillleadtoamultipletofanomalieswhichcontainsthetraceanomalyandthecentralchargeanomalyinadditiontothesupersymmetryanomalyandotherterms.Crucialinthisapproachisthatcarefulregularizationoftermsproportionaltothe eldequationoftheauxiliary eldyieldsnonvanishingcontributionstotheWardidentities.Intheliterature,somearticlesdealdirectlywiththecentralchargecurrentwhileinsomeotherarticlesthecentralchargeanomalyisobtainedfromasupersymmetrytransformationoftheconformalanomalyinthesupercurrent.WeshallderivethecentralchargeanomalybothdirectlybyevaluatingtheJacobian,andbyasupersymmetrytransformationoftheconformalanomalyinthesupercurrent,andshowthattheresultsagreewitheachother.

Webeginwithalocal(xandθdependent)supersymmetrytransformationofthescalarsuper eldφ(x,θ)ingthequadraticpartofthesuperspaceactionasregulator,weobtainaWardidentityinsuperspace(cor-respondingtoahierarchyofWardidentitiesinx-space)whichcontainstheone-loop µ(x),inadditiontoexplicitsym- µ(x),T µν(x),ζanomaliesincertainquantumcurrentsJ

metrybreakingterms.Theuseofasuper eldformulationofheatkernelsinstrictlyd=2Minkowskispace-timetoregularizetheJacobiansinthepathintegralformulationmani-festlypreservesordinaryrigidsupersymmetryatallstages.Asubtlepointistheproperidenti cationofthequantumcurrents.ThenaiveNoethercurrentsarenotconserved, (x)µwhichareconserved,so µ(x),T µν(x),ζbuttheWardidentitiescontainthecurrentsJ

thesecanbeusedtoconstructtime-independentcharges.Ifweweretosubstitutethe eldequationsintheseconservedquantumcurrents,wewouldobtaintheNoethercurrents,butthisisnotallowedatthequantumlevel.Infact,thecontributionsproportionalto eldequationsproduceanomalies,aswealreadymentioned.

Wealsoderivethesupersymmetryalgebraatthequantumlevel.Thisrequiresafullquantumoperatorapproachwhichincorporatesanomalies,ratherthanasemiclassi-calapproachbasedonDiracbrackets.Fortunatelythereexistssuchanapproach:theBjorken-Johnson-Low(BJL)methodwhichautomaticallyincorporatesthee ectsofsu-perconformalanomalies.TheBJLmethodhasbeenwidelyusedforcurrentalgebrasinthe1960’s[16].Itallowsonetorewriteresultsobtainedfrompathintegralsintooperatorrelations.Togofromthepathintegralresultstooperatorresultsonemustuse eld

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

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).

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

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

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

anti-commutewitheachother.Wehave

¯µψDαφ(x,θ)=ψα+θαF+(γµθ)α µ +(γµθ)αθ

and

¯+2ψθF¯+2(ψγ¯µθ) µ ¯(x,θ)Dφ(x,θ)=ψψDφ

¯[FF µ µ ψγ¯µ µψ]+θθ

¯µθ=0.whereweusedθγ

Wenextnotethat

¯) 2+1φ3(x,θ)= 3+3(θψ

2(2.9)(2.10)¯)(ψψ¯).Wethuschoosetheaction(θθ

dxdθL(x,θ)=

2dxd2θ[1

32gφ3(x,θ) gv0φ(x,θ)]

=dx{1

2 2Thedeltafunctionisde nedbydθ1δ(θ1 θ2)=1andgivenby

1δ(θ1 θ2)=(θ1 θ2)(θ1 θ2).

¯ThepotentialVinL=T VisgivenbyV= FU+gψψ where

2U( )≡g( 2 v0).¯)=1(θθ(2.13)(2.14)(2.15)

Weuseacouplingconstantgwhichisrelatedtothecouplingconstantλusedinotherarticlesby µ0m0,v0==g=2λ

2µ

istherenormalizedmesonmass.

Toapplythebackground eldmethod,wedecomposethe eldvariableasfollows

φ(x,θ)=Φ(x,θ)+η(x,θ)(2.17)

whereΦ(x,θ)isthebackground eldandη(x,θ)isthequantum uctuation.Wethenconsiderthepartsofthe(super eld)Lagrangianwhicharequadraticinη

L2(x,θ)=1

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

orequivalently

L2(x,θ)=η(x,θ)Γ(x,θ)η(x,θ),

1Γ(x,θ)=

(2π)2eik(x y)exp[ iθ1¯kθ2]

1p2 i δ(x y),

µ.i

Thisequationyieldsthefollowingimportantrelations

1

3(2.25)1p2 i =δ(θ1 θ2),ThesymbolT denotes(covariant)timeorderinginthepathintegralapproach.Ithasthepropertythatitcommuterswithordinaryderivatives

1¯ 1JyieldsZ=h¯[ 2DD) ¯ 1δ(x dµexpi2DD)

y)δ(θ1 θ2).Weset¯h=1inmostplaces.

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

wherethesecondrelationisderivedbyperformingtheintegral

1(θ1 θ3)pθ2] d2θ2in

2¯D(θ1)δ(θ1 θ2)= exp[ iθ1pθ2],

D2(θ1)δ(θ1 θ2)= p2δ(θ1 θ2).

Usingtheseresults,weobtainthefollowingequations

Γ(x,θ)= 1

D2 1

12gΦD+(gΦ)2,

2gDΦ 1¯DD,(2.28)4

Γ2(x,θ)δ(θ θ1)=[

4p2 1

2gΦD+(gΦ)2]2δ(θ θ1).(2.29)

Weusetheheatkernelexp[(Γ/M)2]ing(2.29)we nd

(exp[1

[ 212gDΦ 1

M

M

2Γ(x,θ)}|θ,x 2

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

=

= d2xd2θω′(x,θ) x,θ|exp{d2xd2θω′(x,θ)

y→x,θ1→θ12 D+gΦ(x,θ)]2}|θ,x

= ×lim

2exp{d2k

[212D+gΦ(x,θ)]2}δ(θ θ1)δ(x y)(3.3)dxdθ212MDgΦ(x,θ) 1

(2π)2

×limexp{θ1→θω′(x,θ)e ikx4p2 1

2

1gΦ(x,θ)D+g2Φ2]}eikxδ(θ θ1).1WerecallD=

2¯x,θ)D+1 (

(2π) ikxeexp{214p2 1

2gΦ(x,θ)D+g2Φ2]}eikxδ(θ θ1).(3.7)

Passingthefactoreikxthroughtheintegrandreplacesp→p+k,andtheoperatorDismodi edasfollows

1(γνθ)kν][D+i(γµθ)kµ]e ikxDeikx=

=11(kθ)D (kθ)(kθ)

¯kD+1=D iθ

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

Replacing

weobtaintheintegral

M2kµ→Mkµ(k+22(3.9)1

M2

g2¯kDθ2 d2k41

M2kpM2) 2 i¯)]gΦ(x,θ)k2(θθ i¯kDθ¯)]+k2(θθ

usingthataccordingto(2.28)

θ1→θ2k].By4limD(θ)δ(θ θ1)= 1(3.11)

whiletermswithoutDactingonδ(θ θ1)vanishforθ1→θ,andnotingthatonlythetermsintheintegraloforder1/M2orlargersurvivewhenMtendstoin nity,onecancon rmthatonlythetermstosecondorderintheexpansionsurvive.Infact,thesecondordertermscompletelycancelbecausetheterm

¯)D/M2k2(θθ

fromthecrosstermsofD/M2and1

2¯)k2DD/M¯)D/M2.¯(θθ= k2(θθ(3.12)2

(3.13)Wethusneedtoevaluateonlythe rstorderterms

d2k

4k2}[ gΦ(x,θ)D]δ(θ θ1)=i

ig

(2π)2exp[ k/4]=2

πΦ(x,θ)].(3.16)

Notethatthiscalculationremainsvalidforgeneral(non-derivative)interactionsdepend-ingonΦ;ifonehasapotentialV(Φ)insteadofgΦin(2.19),onemakesthesamereplacementin(3.16).

Inthespiritofthebackground eldmethod,onemayreplacethevariableΦ(x,θ)bythefullvariableφ(x,θ)totheaccuracyoftheone-loopapproximation.The nalresultfortheresultoftheJacobianofthepathintegralinMinkowskispaceisthusgivenby

lnJ=i d2xd2θω′(x,θ)[g

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

Forexample,fortheclassoftransformations

¯Dφ+c(D¯ )φ= ¯Dφ+1δφ(x,θ)=

withaconstantc,oneobtainsfortheJacobian

ig

)2g1¯D( Φ(x,θ))+(c 2¯ )φ)(D(3.18)

2

2

where

L=1¯β)(D¯αφ)Qβφ+ ¯αQαL](Dα 2gφ3 gv0φ.(4.2)(4.3)3

Anytransformationofφ,whetheritisasymmetryoftheactionornot,leadstoacorrespondingWardidentity,butusingalocalsupersymmetrytransformationhastheadvantagethatoneobtainsahierarchyofWardidentititesinx-spacewhichcontaintheWardidentitiesforordinaryandconformalsupersymmetry.Theseare,ofcourse,theWardidentitiesweareinterestedin,andweexpectinthismultipletofWardidentitiesalsoto ndaWardidentityforthecentralchargecurrent.

Forconstantsuper elds α,theactionisinvariant,butforlocal α,thevariationofSisporportionaltotheNoethercurrent.OnethusobtainsthefollowingWardidentityforcorrelationfunctions

i

g2¯β)(D¯αφ)Qβφ+ ¯αQαL]φ(x1,θ1)...φ(xn,θn) (Dα

= i

2¯g¯αφ)Qβφ]+QβL= hDα[(D

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

Herewewriteh¯explicitlytoindicatethat¯weareworkingattheone-looplevel.2θθ ¯Using1

µ andQβφ=ψβ+Fθβ θ2(DαD

βαφ)Qβφ QβV,and1

µ+1

2(γµJ µ)(x) T µµ(x)θ+1

2(ψγ¯ν

ψ)γνθ

δ(θ)µjµ(x)

=1

2π[ψ(x)+F(x)θ (γµθ) µ (x)+δ(θ) ψ(x)]

orincomponentnotation

(γµJ µ)(x)=(γjµ)(x) ¯hg

µ

2πF(x),

ζ µ(x) 1µσ

(ψγ¯2π σ (x),

5 ψ)(x)=0,

jµ(x)=h¯g

µ

2(γµjµ)(x) (Tµµ)(x)θ+(γµζµ)(x)γ5θ δ(θ) µ(Uγµψ)

≡=QV U((φ ()x,ψ θ))[FU( ) g ψψ¯]θ+ µ U( )γµθ δ(θ) µ(Uγµψ)

whereV= [1(4.6)(4.8)

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

µ(x)= [ (x) F(x)]γµψ(x)J

h¯g=jµ

12ηµν[( ρ )( ρ ) FU]

4

h¯g¯ρ ρψ+2g ψψ¯],ηµν[ψγ4¯[γµ ν+γν µ]ψψ+ µν(x)= µ ν 1T

=Tµν(x)+ηµν

2¯g¯µγ5 ψ(x)+hψγ

φ,andthus

yieldsthetraceanomaly.Theparameterl(x)generateslocalLorentztransformations(orchiraltransformationssinceγ5istheLorentzgenerator),buttheLorentztransformationis,ofcourse,anomaly-freeforourvector-likemodel5anditsgeneratorvanishesidentically θα

¯µγ5ψ≡0.ψγ(4.11)

¯ψ,anditleadstotheWardTheparametertα(x)generatesthetransformationδF=t

identitycontainingthegamma-traceofthesupercurrent.

Themostinterestingcasearethetransformationswithcµ(x).Theparametercµ(x)generatesthetransformationsδF= µνcµ ν andδψ=cµγµγ5ψ,andthesetransfor-mationsyieldtheWardidentity

µ(x) cµζ1

2π µν ν (x).(4.12)

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

ThelastterminthisWardidentityisananomaly,andthisanomalyconstitutestheanomalouspartinthecentralchargecurrentitself.Thisisanunusualpointthatmayleadtoconfusion:theanomalyisproportionaltothecentralchargecurrentitself.Inthisrespectitresemblesneitherthetraceanomalynorthechiralanomaly:thetraceanomalycontainsacontractionofthecurrentwhilethechiralanomalycontainsadivergenceofthecurrent.Infact,ithasbeenshownin[10]thatthecentralchargecurrentistheanomalyinthedivergenceoftheconformalcentralchargecurrent(whichisexplicitlyx-dependent,justlikethedilationcurrent).Actually,theanomalycomesonlyfromtheδFvariationandnotfromtheδψvariation,seefootnote4.WeshalllaterexplicitlycomputetheanomalyfromtheδFvariationseparately,see(4.19).Inthatcaseone ndstherelation¯ν ψtermin(4.12)withouttheψγ

¯g µ(x)=ζµ(x)+hζ

2ηµν[ F2 FU+

4πF1(4.14)

Thisrelation,andothersin(4.9),wasobtainedbyextractingcontractedcurrentsfromtheWardidentity,andbygeneralizingthecontractedcurrentsandtheanomaliesinthecontractedcurrentstothecurrentsandtheanomaliesinthecurrentsthemselves.Thisdoesnotdeterminethecurrentcompletely.Wenowprovetheseuncontractedidentities.Webeginwith(4.14)andclaimthefollowingrelation

F FU+21

δF(x)

= h¯g 1¯(x)δψ

=21¯ w(x)θ2¯(x,θ)w(x)θQφ

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

integralintheformof(4.4)givestheidentity6(4.15).Thisanalysis xesthemagnitude

µoftheWeylanomaly.AttheendofSection5weshowthatTµdoesnotcontributetothe

traceanomaly,sothetraceanomalycomesonlyfromtheconservedtensor. µandjµ.WecanshowthatThelastrelationin(4.9)tobeprovenistheonewithJ

µ(x) jµ(x) = γµψ(x)(F(x)+U( )) J

δSγµψ(x) = γµψ(x)2π

byconsideringthevariation

δφ(x,θ)=δ(θ)¯ µ(x)γµ (4.17)

δF(x) = h¯g

ThecurrentTNµν(x)generatedbythevariationδφ(x,θ)=ξµ(x) µφ(x,θ)isgivenby Noether

δS= ( µξν)TNµν.Itreads

TNµν(x)= µ ν 1

¯[γµ ν+γν µ]ψ+1ψ¯µ µψ+2g ψψ¯],ηµν[ψγ

4¯[γµ ν γν µ]ψψ6=

=hg µν(x) ηµν¯T4 µν(x) ηµν[ F2 FU+1T4¯[γµ ν γν µ]ψψ2

whereweusedtheWeylanomalyin(4.15).Thelasttermismanifestlyantisymmetric,anditcan¯5γρ ρψ.Ityieldstheantisymmetricpart¯ρσγρ σψ= µνψγ¯[γµ ν γν µ]ψ= µνψ bewrittenasψ

ofthestresstensor,andisproportionaltothedivergenceoftheLorentzcurrent.Onemayshowthat¯g µTNµν(x)= h

µ 2π νF(x)),andthisimplies Tµν(x)=0.Inouranalysis,theconservedTµν,andTµνwhichis

µνandTµνaremanifestlynotconservedbutfreeofatraceanomaly,playabasicrole.NotethatbothT

symmetric.7Clearly,thenaiveequationofmotionF(x)+U( )=0cannotbeusedinthisderivation.

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

Ifonesetsvµ(x)= µv(x)in(4.20),onegeneratesthedivergenceofthecentralchargecurrentbuttheproceduregivesnoinformationforatopologicalcurrent.InanalogywithU(1)gaugetheory,weareconsideringthechangeofvariableAµ→Aµ+aµinsteadofAµ→Aµ+ µatogeneratethecurrent.

Itisimportanttorecognizethatalloperatorsappearingontheleft-handsidesoftherelationsin(4.7)havehighermassdimensionsthanthoseofthecorrespondingoperators µ(x)andζµ(x)are,respectively,dimension2andontheright-handsides.Forexample,ζ µ1operatorssincethecouplingconstantgcarriesaunitmassdimension.Similarly,γµJ µandjµandγµjµare,respectively,dimension3/2and1/2operators,thoughbothofJ µandTµare,respectively,dimension2and1aredimension3/2operators.Also,Tµµ µνandTµνaredimension2operators.Inthissensealltheoperators,thoughbothofT

compositeoperatorsontheright-handsidesof(4.7)aresoftoperators.Thissuggeststhatonlythe“hard”operatorsgenerateanomalies.Inthenextsectionweprovethisstatement.

5SupersymmetryalgebraofthequantumoperatorsIntheprevioussectionwegaveadirectderivationoftheanomaliesbasedonpathintegrals,butwealreadymentionedintheintroductionthatonecanalsoobtaintheanomalies µanomalybymakingsuccessivesusytransformations.InthissectionwefromtheγµJ

implementthissecondapproach.Sincethisinvolvescommutatorsofcurrents,weconvertthepathintegralrelationsintooperatorrelationsbyfollowingtheBJLmethod.Webeginbyconsideringthevariation

δφ(x,θ)= ¯(x)Qφ(x,θ).

Thechangeoftheactionde nestheNoethercurrent

δS=

where (5.1)d2x( µ ¯(x))jµ(x)(5.2)

(5.3)jµ,α(x)= {[ (x)+U( (x))]γµψ(x)}α

2withU( )=g( 2 v0).TheJacobianfactorfor(5.1)givestheanomaly,andweobtain

theidentityh¯g µjµ(x)=

2πγµψ(x), µ(x)=0. µJ(5.5)

ItcontainsthecontributionsfromtheactionandJacobian,appearsinalltheWardiden-tities,andthisimplies,asweshallsee,thattherelationsamongvariousGreen’sfunctionsobtainedbyglobalsupersymmetryarenotmodi edinformbynon-trivialJacobians.

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

Ifoneconsiderstherigidconformalsupersymmetytransformationgeneratedbytheparameter

¯(x)=a¯(x)x(5.6)

theactiontransformsasfollows

δS=

andoneobtainstheidentity

¯g µ(x))=γµjµ(x) h µ(xJ d2x[( µa¯(x))xjµ(x)+a¯(x)γµjµ(x)],(5.7)

π µ(x)ψ(x)=γµJ(5.9)

where(γµjµ(x))ingγµγνγµ=0instrictlyd=2oneobtains

µ(x))exp=(γµjµ)exp=γµjµ= 2U( )ψ(x)(γµJ

and

µ(x))anomaly= (γµJh¯g(5.10)

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

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

Tµν(x)= µ ν

41¯[γµ ν+γν µ]ψ 1ψ

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