Topological anomalies from the path integral measure in supe(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

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