From Here to Criticality Renormalization Group Flo

arXiv:hep-th/9112031v113Dec1991HUTP-91/A061FromHeretoCriticality:RenormalizationGroupFlowBetweenTwoConformalFieldTheoriesW.A.Leaf-Herrmann†JeffersonPhysicalLaboratoryHarvardUniversityCambridge,MA02138Usingnonperturbativetechniques,westudytherenormalizationgrouptrajectorybetweentwoconformalfieldtheories.Specifically,weinvestigateaperturbationoftheA3superconformalminimalmodelsuchthatintheinfraredlimitthetheoryflowstotheA2model.Thecorrelationfunctionsinthetopologicalsectorofthetheoryarecomputednumericallyalongthetrajectory,andtheseresultsarecomparedtotheexpectedasymptoticbehavior.Excellentagreementisfound,andthecharacteristicfeaturesoftheinfraredtheory,includingthecentralchargeandthenormalizedoperatorproductexpansioncoefficientsareobtained.WealsoreviewanddiscusssomeaspectsofthegeometricaldescriptionofN=2supersymmetricquantumfieldtheoriesrecentlyuncoveredbyS.CecottiandC.Vafa.12/91†([e-mail],@huhepl.bitnet,[e-mail])1.IntroductionThesetofconformalfieldtheoriesaretheinfraredorultravioletfixedpoints,orcriticalpoints,ofrenormalizationgroupflowtrajectoriesinthespaceoftwo-dimensionalquantumfieldtheories.Atthesefixedpoints,conformalinvarianceprovidesasetofconstraints,realizedbytheinfinite-dimensionalVirasoroalgebra,whichoftenallowfortheexactsolutionofthetheory,inprincipleviatheBPZbootstrap[1],orinpracticeviaexplicitrepresentationsoftheVirasoroalgebra[2].However,thesemethodsgenericallyallowustosolvethetheoryonlyatthecriticalpoint.Tobetterunderstandthestructureofthespaceoftwo-dimensionalfieldtheories,andthespecialroleplayedbytheconformalfieldtheories,wewouldliketobeabletocomputethecorrelationfunctionsbothonandoffofthecriticalpoint.Typically,thebestthatcanbedoneistouseconformalperturbationtheoryintheneighborhoodofafixedpoint,asdemonstratedbyZamolodchikov[3]andothers[4–8].Forexample,onewouldliketobeabletocalculatethescalingbehaviorofthequantumfieldsinaconformaltheoryperturbedbysomerelevantfield,undertheactionofrenormalizationgroupflow.Theinfraredlimitofthistheoryshouldcorrespondtosome(possiblytrivial)conformalfieldtheory.Whilethisquestioncanbeaddressedperturbativelyinsomecases,forinstancetheminimalmodelsnearc=1[3],wegenerallyrequiresomenonperturbativetechniquestoanswertheabovequestion.Inthecaseoftwo-dimensionalfieldtheorieswithN=2supersymmetrysomeoftherequisitetechniqueshaverecentlybeendeveloped[9–13],whichallowforthenonperturbativecalculationofaclassofcorrelationfunctionsbothonandoffthecriticalpoint.ThisclassofcorrelationfunctionsisknownasthetopologicalsectorofN=2fieldtheories,andiscloselyrelatedtothecorrelationfunctionsofphysicalobservablesintopologicalquantumfieldtheories[14].Thetopologicalsectoriscomposedoftheexpectationvaluesofchiralfieldsevaluatedbetweenthesetofsupersymmetricgroundstates.Theequivalencebetweentwo-dimensionalσ-modelsonCalabi-YauspacesandcertainN=2superconformalmodels,firstobservedbyGepner[15],iswell-known,andN=2Landau-GinsburgeffectiveLagrangiansprovideexplicitrealizationsofthiscorrespondence[16–19].ThisequivalencehasledtotheapplicationofgeometricalmethodsinthecharacterizationofN=2superconformalfieldtheories[10,20,21].Usingthequasi-topologicalnatureofN=2supersymmetry,S.CecottiandC.Vafahaverecentlyuncovered1thegeneralizationofthesegeometricalaspectstoarbitraryN=2quantumfieldtheories[12,13].Inthispaperweshallapplytheirresultstothenonperturbativecalculationofthetopologicalsectorofatheorywhichinterpolatesbetweentwoconformalfieldtheoriesalongtherenormalizationgrouptrajectoryconnectingthem.InSection2wedefineanddiscussthepropertiesofthetopologicalsectorofN=2supersymmetricquantumfieldtheories,andtherelationbetweenthissectorandtopologicalquantumfieldtheoriesbasedontwistedN=2models.ThebasicgeometricalframeworkneededtosolveforthetopologicalsectorcorrelationfunctionsisreviewedinSection3.Section4describeshow,byusingtheN=2non-renormalizationtheorem,thesecorrelationfunctionsmaybecalculatednonperturbativelyalongarenormalizationgrouptrajectory.Wealsodiscusstwoquantitiesintroducedin[13]whichservetocharacterizethetheorybothattheconformalpoint,andoffofcriticality,knownastheRamondchargematrixandthealgebraicQ-matrix.Bothareeasilycomputedinthetopologicalsector.Asaconcreteapplicationofthisframework,weanalyzetherenormalizationgroupflowbetweentwoconformalfieldtheories,theA3N=2minimalmodelperturbedinsuchawaythatthetheoryflowstotheA2modelintheinfraredlimit.ThecomputationisdiscussedinSection5,andwecomparetheresultsofthenonperturbativenumericalcalculationofthecorrelationfunctionsoftheinterpolatingtheorywiththeexpectedasymptoticbehaviorinSection6.OurconclusionsarepresentedinSection7.2.TopologicalSectorofN=2QuantumFieldTheoriesWeconsidertwo-dimensionalquantumfieldtheorieswithanN=2supersymmetry.Weassumethetopologyofthetwo-dimensionalspacetobeacylinder,orequivalently,aspherewithtwopunctures.ThesuperchargesQ+,Q−,andtheirHermitianconjugates¯Q−,¯Q+,obeythealgebra:nQ+,Q−o=−∂,¯Q+,¯Q− =−¯∂,(2.1)withallother(anti-)commutationrelationsvanishing.Weimposeperiodicboundaryconditionsonthefermionicoperators,sothattheWittenindex,Tr(−1)F,whereFistheoperatorwhichcoun