Comments on AdS2 solutions of D=11 Supergravity(10)

We study the supersymmetric solutions of 11-dimensional supergravity with a factor of $AdS_2$ made of M2-branes. Such solutions can provide gravity duals of superconformal quantum mechanics, or through double Wick rotation, the generic bubbling geometry of

aspresentedin[4].Therelevant6-dimensionalK¨ahlermanifoldtakesaformofS2-bundleovera4-dimensionalK¨ahler-Einsteinmanifold.Itwillbeveryinterestingtointroducesuchaconcreteansatz,andsolveEq.(3.20).Weexpect,asisthecasewithYp,q,themetricoftheK¨ahlerbasemightbeingeneralnotcomplete,buttheentire9-dimensionalinternalmanifoldcanbemadecomplete.Weplantopresentthenewsolutionsandtheglobalanalysisinafuturepublication.

Anotherinterestingdirectionistointerpretoursolutionsasgeneralizedbubblingge-ometry.ForthecaseofAdS5×S5solutionsinIIBsupergravity,the1/2-BPSbubblingsolutionsgivenin[7]canbedescribedintermsofadistributionfunctioninthephasespaceof1-dimensionalfreefermions.ForM-theory,determiningthesolutionsisinsteadreducedtosolvingaTodaequationin3-dimensions,andwebelievethedynamicsof1/2-BPSop-eratorsmustbeencodedtherein.Thesupergravitysolutionsdualtolesssupersymmetricgiantgravitonoperatorsareconsideredfor1/8-BPSinRef.[4]andfor1/4-BPSinRef.[11].Theybothconcludethatthe10dimensionalsolutionisbasedonaK¨ahlerspacewhichis6and4-dimensional,respectively.Naturallyoneexpectstheyoriginatefromthesymplecticstructureoftheeigenvaluedynamics.Likewise,Eq.(3.20)canbeinterpretedastheequa-tiongoverningthedynamicsofgenericsupersymmetricoperatorsofM-branesconformal eldtheory.Oneimportantfeaturewehavebeenignoringinthispaperistheglobalprop-ertyandtheboundaryconditionsofthesolutions.Sinceouranalysisisgeneralenoughtoencompassthe1/2-BPS uctuationsoflesssupersymmetricconformal eldtheoriesonM-branes,aswellaslesssupersymmetric uctuationsofmaximalconformal eldtheoriesonM-branes,wewill rstneedto xtheboundaryconditionaccordingtotheconformal eldtheoryweareinterestedin,andthensolveEq.(3.20)to ndgravitydualstoBPSoperators.

Acknowledgments

WearegratefultoHo-UngYeeandSang-HeonYifordiscussions.TheresearchofNakwooKimissupportedbytheScienceResearchCenterProgramoftheKoreaScienceandEn-gineeringFoundation(KOSEF)throughtheCenterforQuantumSpacetime(CQUeST)ofSogangUniversitywithgrantnumberR11-2005-021,andbytheBasicResearchProgramofKOSEFwithgrantNo.R01-2004-000-10651-0.NakwooKimandJong-DaeParkarebothsupportedbytheKoreaResearchFoundationGrantKRF-2003-070-C00011.References

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