Finiteness Properties of the N=4 Super-Yang--Mills

arXiv:hep-th/0605164v24Feb2008CERN-PH-TH/2006-090FinitenessPropertiesoftheN=4Super-Yang–MillsTheoryinSupersymmetricGaugeL.Baulieu∗,G.Bossard†,S.P.Sorella‡∗TheoryDivision,CERN,Switzerland§∗†LPTHE,CNRSandUniversit´esParisVI-ParisVII,Paris,France¶‡DepartamentodeF´ısicaTe´oricaInstitutodeF´ısica,UERJ,UniversidadedoEstadodoRiodeJaneiro,BrazilkAbstractWiththeintroductionofshadowfields,wedemonstratetherenormalizabilityoftheN=4super-Yang–Millstheoryincomponentformalism,independentlyofthechoiceofUVregularization.Remarkably,byusingtwistedrepresentations,onefindsthatthestructureofthetheoryanditsrenormalizationisdeterminedbyasubalgebraofsupersymmetrythatclosesoff-shell.Startingfromthissubalgebraofsymmetry,weprovesomefeaturesofthesuperconformalinvarianceofthetheory.Wegiveanewalgebraicproofofthecancellationoftheβfunctionandweshowtheultravioletfinitenessofthe1/2BPSoperatorsatallordersinperturbationtheory.Infact,usingtheshadowfieldasaMaurer–Cartanform,theinvariantpolynomialsinthescalarfieldsintracelesssymmetricrepresentationsoftheinternalR-symmetrygrouparesimplyrelatedtocharacteristicclasses.TheirUVfinitenessisaconsequenceoftheChern–Simonsformula.∗emailaddress:[e-mail]†emailaddress:[e-mail]‡emailaddress:[e-mail]§1211-Geneve23,Switzerland¶4placeJussieu,F-75252ParisCedex05,France.kRuaS˜aoFranciscoXavier524,20550-013Maracan˜aRiodeJaneiro,Brazil.1IntroductionInarecentpaper,newfieldshavebeenintroducedforsupersymmetricgaugetheories,whichwecalledshadowfields.ThesefieldsareelementsofBRSTdoublets.Theydeter-mineaBRSTlikeoperatorforthesupersymmetryinvariance,whichisfullycompatiblewiththegaugesymmetry[1].ShadowfieldsalsoallowfortheconstructionofnewclassesofgaugesthatinterpolatebetweentheusualFaddeev–Popovgaugesandnewones,whichareexplicitlysupersymmetric.ThesegaugesgiveanadditionalSlavnov–Tayloridentity,forcontrollingsupersymmetryatthequantumlevel.TheobservablesaredeterminedfromthecohomologyoftheBRSToperatorforgaugeinvariance,andtheirsupersymme-trycovariancecanbeestablishedfromthenewSlavnov–Tayloridentity.IthasbeenconjecturedformanyyearsthattheN=4super-Yang–Millstheoryisasuperconformaltheory.Itsβfunctionvanishesatallordersinperturbationtheory[2,3].ThesuperconformalinvarianceisabasicfeatureoftheAdS/CFTMaldacena’sconjecture[4].InthispaperweusethenewsupersymmetricgaugestodirectlyproveinperturbationtheorysomeoftheresultsthatareimpliedbytheconformalinvarianceoftheN=4super-Yang–Millstheory,withtheonlysymmetryhypothesisofsuperPoincar´esupersymmetry.In[1],therenormalizationofsupersymmetrictheories,usingshadowfields,wasde-tailedwhenthesupersymmetryalgebraclosesoff-shell,thatis,forcaseswhereauxiliaryfieldsexistforclosingthewholesupersymmetryalgebra.IntheN=4super-Yang–Millstheory,nosuchauxiliaryfieldsexist.Here,wewillbypassthispointbycombiningthemethodologyof[1]andtheexistenceofasupersymmetrysubalgebrawith9generators,whichislargeenoughtoconstraintheclassicalN=4super-Yang–Millsaction,andsmallenoughtobeclosedwithouttheuseofequationsofmotion.Thissubalgebrawasintroducedin[5,6],usingmethodsthatarespecifictotopologicalfieldtheoryandinvolvetwistedvariables.TheSlavnov–TayloridentitiesthatareallowedbytheintroductionoftheshadowfieldsenableustoprovetherenormalizabilityoftheN=4theory,withoutanyassump-tiononthechoiceoftheultravioletregularization.Inthispaper,weshowtheabsenceofapossibleanomalyfortheN=4theory.Algebraicmethodshavebeenusedtoshowthatinvariantpolynomialsdependingononeofthescalarfieldsofthesupersymmetrymulti-pletareprotectedfromrenormalization[7].WewillextendthisresulttothewholesetoflocalobservablesoftheN=4super-Yang–Millstheory,whichareinvariantpolynomialsinscalarfields,takingtheirvaluesinanygiventracelesssymmetricrepresentationofthe1R-symmetrygroup.Theseoperatorsarethe1/2BPSprimaryoperators.Wethenobtainthefinitenessofthewhole1/2BPSmultiplets,bysupersymmetrycovariance.Wewillgiveanewproofofthecancellationoftheβfunctionatallordersinperturbationtheory.Remarkably,alltheseresultsareobtainedbyusingsmallsectorsofthesupersymmetryalgebra.Thelatterexist,thankstothepossibilityoftwistingthesupersymmetryalge-bra,bycombiningtheR-symmetryandtheLorentzsymmetry.Inordertoprovethestabilityoftheactionunderrenormalization,itisinfactsufficienttouseasectorofthesupersymmetryalgebrawith6generators.And,toprovethatall1/2BPSprimaryoperatorsareprotectedoperators,oneusesasectorofthesupersymmetryalgebrawith5generators.AkeyfeatureofthisproofistheChern–Simonsformula,wheretheshadowfieldcanbeidentifiedasaMaurer–Cartanform.Itappearsthatthescalarobservablesoftopologicalfieldtheoriesdeterminethe1/2BPSprimaryoperatorsbycovarianceundertheR-symmetryofthesupersymmetrictheory.Infact,theirfinitenesspropertyiscloselyrelatedtotheexistenceofcharacteristicclasses.Wewillsupposeeverywherethatthegaugegroupissimple.2N=4super-Yang–Millstheoryinthetwistedvari-ables2.1FieldsandsymmetriesConsidertheN=4multiplet(A,λα,φi)inaflateuclideanspaceSpin(4)∼=SU(2)+×SU(2)−,whereα,iareindicesinthe4andthe6oftheinternalsymmetrySL(2,H),whichistheeuclideanversionoftheSU(4)R-symmetryinMinkowskispace.Thecompo-