On Uniqueness of supersymmetric Black holes in AdS

arXiv:0712.0132v2[hep-th]12Dec2007December2007OnUniquenessofsupersymmetricBlackholesinAdS(5)PedroJ.Silva1InstitutdeCi`enciesdel’Espai(IEEC-CSIC)andInstitutdeF´ısicad’AltesEnergies(IFAE),UAB,E-08193Bellaterra(Barcelona),Spain.ABSTRACTWestudythepossibilityofhavingBlackholeofsphericalandringhorizontopologywithfiveindependentchargesintheU(1)3-modelof5Dgaugesupergravity.Tostudythesepossibilitiesweconsidernotonlytheknownresultobtainedbylocalsupersymmetryanalysisbutincludetheinputcomingfromnon-localpropertiesofthesolutions,liketheattractormechanism,theentropyfunctionofSen,theEuclideanformulationandgeneralpropertiesoftheuplifttotendimension.Forthesphericalcase,wefoundthatthereisnoroomformoregeneralBlackholesthantheonesalreadydescribeinhep-th/0601156.Ontheotherhand,ifasolutionofringhorizontopologyexists,weconcludethatitmustbelabeledbythreeindependentparametersonly,sinceithastosatisfytwoindependentconstraintsthatweexplicitlyfindintermsofitschemicalpotentials.Attheendofthearticle,basedonallthelocalandnon-localinformation,weputforwardaconjectureontheconstraintsthatcharacterizegeneralBlackholesdualtoN=4SYM.1IntroductionStringtheoryhasprovidedthefirstmicroscopicderivationoftheentropyforsomeextremalBlackholes(Bh)indiversedimensions(seeforexample[1,2]).Allthecalculationsarebasedontheidentificationofagivensetofmicroscopicstatesforminganensemble(labelbytheBhcharges)withtheBhgeometry.InfourdimensionswehaveBhuniquenesstheoremswhichguarantythattoagivensetofasymptoticchargestherecorrespondsonlyoneBhsolution.Thisisingeneralnottrueforhigherdimensionwheresuchtheoremsnolongerexist.InparticularinfivedimensionswehaveBlackrings(Br)andsphericalBhsolutionsasanexplicitexampleofnon-uniqueness.Consideringtheabove,itseemsimportanttounderstandbetterwhatkindofBhexistinstringtheorycompactifications,ifwewanttounderstandtowhichmicroscopicensemblestheseBharerelated.Unfortunately,thisisaverydifficultproblemtosolveingeneralthatneverthelesscanbesignificantlysimplifiedifweincludesupersymmetryasaninput.1E-mailaddress:[e-mail]1Infact,supersymmetricBhinminimalfivedimensionalsupergravityanditsextensiontotheU(1)nvectormultipleswereclassifiedin[3,4].Here,localanalysisofsupersymmetrywasusedtoclassifyallthenearhorizongeometries(NHG)tolatermatchthemwithasymptoticdatathatdefinestheBhsolutions.ThemainresultintheminimalcasesistheuniquenessofthecorrespondingsupersymmetricBMPVBh[5]andthesupersymmetricBrsolutionsof[6].Inthisarticle,weconsidertheveryimportantframeworkoftendimensionaltypeIIBsupergravitywithAdS5⊗S5asymptoticconditions,relatedbydualitytoU(N)N=4superYang-Millsinfourdimensions.ThistheorycanbeconsistentlytruncatedandcompactifiedontheS5togiveSO(6)gaugesupergravityinfivedimensions,thatfurthermorecanbetrun-catedtotheU(1)3-modelthatalsoadmitsafinaltruncationtominimalgaugesupergravity.ThesedifferenttruncationstellusthatwecanhaveBhsolutionsatfourdifferentlevels.1.Bhin10DtypeIIBsupergravity.2.BhinSO(6)5Dgaugesupergravity.3.BhinU(1)35Dgaugesupergravity.4.Bhinminimal5Dgaugesupergravity.AllsupersymmetricBhshouldbedescribeinthedualCFTbysupersymmetricensemblesofstateslabelbythequantumchargescorrespondingtotheSO(4)angularmomentaJi,i=(1,2)andtheSO(6)R-chargesQI,I=(1,2,3).Atpresent,thereonlyexistexplicitsupersymmetricBhsolutionsinthelasttwotruncations(3)and(4)[7,8,9],butisnotclearifthesesolutionsarethemostgeneralBhorjustaparticularfamily.Thekeypointisthatthesesolutionscomewithaconstraintamongtheconservedchargesthatisnotapparentinthedualtheory[10].RegardingtheclassificationofBhsolutionsinthisframework,onlytruncation(3)and(4)havebeenstudiedsofar[11,12,13,14,15],whereunfortunatelylocalanalysisofsu-persymmetrywastoocomplicatedtobefullysolved,inbothcases.Morespecifically,onlytheNHGwithtwoU(1)symmetriesareclassifiedandthenecessaryconnectionwiththecorrespondingasymptoticregionwasnotachieved,leavingtheclassificationunsettle.Inthisworkweproposeacomplementaryapproachtohelpontheclassificationofsuper-symmetricBhsolutions,bypassingtheproblemoftheasymptoticformofthemetric.WebaseourconstructiononglobalpropertiesofsupersymmetricBhliketheattractormech-anism[16],itsEuclideanformulation,thermodynamics[17,18]andgeneralpropertiesoftheupliftingofgaugesupergravitiesto10D[19].Weexplicitlyconsideronlythelasttwotruncations,butthesenewideasshouldhelpalsointheothermoregeneralcasesandinfactwediscussthispossibilityandsomeofitsimplicationinthelastsection.OurmainresultsarethatintheU(1)3-model,therearenootherBPSBhwithS3horizonthantheoneswealreadyknow,andthatifthereisaBPSBrinAdSitishighlyconstraintwithonly3independentparameters.Evenmore,fortheseBr,thereseemstobeanobstructiontotaketheun-gaugelimit(sendingtheAdSradiustoinfinity),asthechemicalpotentialsconjugatedtothedipolemomentabecomeunbounded.Moregenerally,wefoundthatallBPSBhintheU(1)3-modelpresentthesamesingleconstraintamongits2chemicalpotentials.Finally,andatamorespeculativelevel,weproposeasaconjecture,thatallBPSBhdualtoN=4SYMshowthissameconstraint,thatisthesamerelationthatappearsinthedefinitionoftheSYMIndexof[10].Theplanofthisworkisthefollowing,wesummarizethemainresults