Comments on AdS2 solutions of D=11 Supergravity(2)

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

1.Introduction

Itisoneofthemostintriguingissuesinstringtheorytoprove/disprovetheso-calledMaldacenaconjecturebetweenanti-de-Sittergravityandconformal eldtheories

[1].Alotofremarkableagreementshavebeenencounteredsofar,soitwouldbemoreappropriatetosayonewouldliketodeterminetheregionofvalidityaspreciseaspossible.

ThisinevitablyleadsustothestudyoflesssupersymmetricsolutionsofString/M-theory.Thespectrumofsupersymmetricsolutionsisrichenoughtoprovideexampleswithrealisticfeaturessuchascon nementandasymptoticfreedom(see[2]forexample),andyetthankstothepowerofsupersymmetrywecanperformexplicitchecksusingprotectedquantities.

InthispaperwewillemployatechniquewhichisfoundtobeverypowerfulandatthesametimeilluminatinginthesearchfornewsupersymmetricbackgroundsofString/M-theory.BasedontheexistenceofKillingspinors,onecanconstructvariousdi erentialformsasspinorbilinearsanddeterminethelocalformofthemetricandthegauge eldsexploitingthedi erentialandalgebraicrelationsbetweenthedi erentialformswhichcanbederivedusingtheKillingspinorequations.AsampleofworkswhichmakeuseofthistechniquecanbefoundinRef.[3].

Inthispaper,asasequeltothepreviousone[4],weanalyzetheconsequencesofsu-persymmetryin11-dimensionalsupergravity,combinedwiththeansatzofAdS2factor,i.e.SO(2,1)isometry.Inparticular,forsimplicity,weconsiderpureM2-branecon gurations.AconvenientwayofseeingthemisasM2-braneswrappedontwo-cyclesinaCalabi-Yau5-fold,andthereadersarereferredto[5]forstudiesalongalternativeavenuesonsimilarcon gurations.Inthepresentwork,itwillbeshownthat,the9-dimensionalinternalmani-foldshouldtaketheformofawarpedU(1)- brationonaK¨ahlerbase.Wealsowritedownthe8-dimensionalnonlinearpartialdi erentialequationforthecurvaturetensor,whichisrequiredifthesupersymmetriccon gurationistosolvetheequationsofmotion.