A non-renormalization theorem for conformal anomal

arXiv:hep-th/9906030v230Aug1999SPIN-1999/11THES-TP99/04hep-th/9906030Anon-renormalizationtheoremforconformalanomaliesAnastasiosPetkou†1andKostasSkenderis⋆2†DepartmentofTheoreticalPhysics,AristotleUniversityofThessaloniki,Thessaloniki54006,Greece⋆SpinozaInstitute,UniversityofUtrecht,Leuvenlaan4,3584CEUtrecht,TheNetherlandsAbstract:Weprovideanon-renormalizationtheoremforthecoefficientsoftheconformalanomalyassociatedwithoperatorswithvanishinganomalousdimensions.Suchoperatorsincludecon-servedcurrentsandchiraloperatorsinsuperconformalfieldtheories.Weillustratethetheorembycomputingtheconformalanomalyof2-pointfunctionsbothbyacomputationintheconfor-malfieldtheoryandviatheadS/CFTcorrespondence.Ourresultsimplythat2-and3-pointfunctionsofchiralprimaryoperatorsinN=4SU(N)SYMwillnotrenormalizeprovidedthata“generalizedAdler-Bardeentheorem”holds.Wefurthershowthatrecentargumentsconnectingthenon-renormalizabilityoftheabovementionedcorrelationfunctionstoabonusU(1)YsymmetryareincompleteduetopossibleU(1)Yviolatingcontactterms.Thetreelevelcontributiontothecontacttermsmaybesettozerobyconsideringappropriatelynormalizedoperators.Non-renormalizabilityoftheabovementionedcorrelationfunctions,however,willfollowonlyifthesecontacttermssaturatebyfreefields.1email:[e-mail]2email:[e-mail]11IntroductionSupersymmetryhasbeenaninstrumentaltoolinprobingthestrongcouplingregimeofstringandquantumfieldtheories.Inmanycases,informationwhichisregardedasdynamicalinnon-supersymmetrictheoriesfollowsbysymmetryconsiderationsinsupersymmetricones.Certainsectorsofthetheoryaresoseverelyconstraintthatnoquantumcorrectionsareallowedatall.Afamousexampleisthenon-renormalizationtheoremof[1]whichstatesthatthesuperpotentialdoesnotreceiveanyperturbativecorrections.Anothersymmetrywhich,whenpresent,severelyconstrainsthestructureoffieldtheoriesisconformalsymmetry.Conformalinvariancehasbeenquitesuccessfullyexploitedintwodimensionswheretheconformalgroupisinfinitedimensional.Inmorethantwodimensionstheconformalgroupisfinitedimensional,neverthelessitispowerfulenoughtodetermine2-and3-pointfunctionsuptoconstants[2–4].Theaimofthepresentpaperistodiscussnon-renormalizationtheoremsthatfollowfromthecombinationofsupersymmetryandconformalinvariance.Inparticular,wewillstudythetraceanomalyinthepresenceofexternalsourcesforcompositeoperators.Wewillshowthatthetraceoftheenergy-momentumtensoracquirescontributionsproportionaltothesourcesandthatcertainratiosofthecoefficientsofthesetermstotheoverallnormalizationsof2-and3-pointfunctionsdonotrenormalize.Recentstudiesoftheconstraintsimposedbysuperconformalinvariancecanbefoundin[5–12].Oneofthemotivationsforthisinvestigationcomesfromtherecentlyproposedduality[13–15]betweenstringorM-theorycompactificationsonadSd+1spacetimesandd-dimensionalsuper-conformalfieldtheories(adS/CFTduality).Inparticular,IIBstringtheoryonadS5×S5withNunitsofF5fluxisbelievedtobedualtoN=4,d=4SU(N)SYMtheory.The’tHooftcouplingconstant,λ=g2YMN,isrelatedtotheadScurvatureandNisrelatedtothestringcouplingconstant,suchthatthestrongcouplingregimeofthe’tHooftlargeNlimitofN=4,SU(N)SYMtheory(i.e.N→∞withλfixedbutlarge),hasaweaklycoupleddescriptionintermsofanti-deSittersupergravity.Anessentialtestforanystrong-weakcouplingdualityin-volvesquantitiesthatdonotrenormalizeasonegoesfromweaktostrongcoupling.Computingsuchquantitiesinbothweakandstrongcouplingregimesandfindingagreementonlyprovidesanon-trivialconsistencycheck;apossibledisagreementbetweenthetwocomputationswoulddefinitelydemonstratethefalsificationofthedualityconjecture.Oneoftheaimsofthisinves-tigationistoprovidesuchnon-renormalizationtheorems,andsubsequentlytesttheadS/CFTduality.Turningthingsaround,onecanalsouseanunexpectedagreementbetweenquantitiescom-putedinthetwodualtheoriesasanindicationofapossiblenon-renormalizationtheorem.Forinstance,usingtheadS/CFTcorrespondencethegravitationaltraceanomalywascomputedin[16]andexactagreementwasfoundwiththeweaklycoupled(free)fieldtheorycalculation,indicatingthatthereisanon-renormalizationtheorematwork.Indeed,sucharenormaliza-2tiontheoremhasbeendiscussedin[17–19].In[14,15]aprescriptionwasgivenforcalculatingboundaryGreen’sfunctionsfromanti-deSittersupergravity.Withthisprescriptionalltwo-andthree-pointfunctionsofchiralprimaryoperatorswerecalculatedinthelargeNlimitin[20].Surprisingly,theresultwasfoundtocoincidewiththeweakcoupling(freefieldtheory)computation.Thisledtheauthorsof[20]toconjecturethatthe3-pointfunctionsofallchi-ralprimaryoperatorsforlargeNareindependentofλandtospeculatethattheymightbeindependentofgYMevenforfiniteN.Subsequentfieldtheorycalculations[21,22]supportedtheconjectureinitsstrongestform.In[23–25]thenon-renormalizationofthe2-and3-pointfunctionwasclaimedtofollowfromabonusU(1)YsymmetrythatN=4SYMshouldhaveinanappropriatelimit.However,aswewilldiscuss,contacttermcontributionstocorrelationfunctionsrendertheargumentof[23,24]incomplete.Onlyifthecontacttermssaturatebyfreefieldsnon-renormalizationwouldfollow.Theresultsofthepresentpaperimplythat(properlynormalized)2-and3-pointfunctionso