Comments on AdS2 solutions of D=11 Supergravity(6)

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

UsingtheKillingequationsoneobtains

d(eA )= aie AK∧ .(3.15)

Forω≡eAe iaψ andusingK=e2A(dψ+B),wehave

dω= aiB∧ω,(3.16)

whichisan8dimensionalequationonM8.FromthegeneralresultofSU(n)-structuresandtheclassi cationoftorsionclasses,wearriveattheconclusionthatthecomplexstructuregivenbyYisintegrableanddB=RistheRicciformoftheK¨ahlermanifoldM8.ConsideringY2～(η η)2～e2Aandω2～e2A|ηTη|2～e4Awhenevaluatedusingthemetricg,itistherescaledmetricg¯ij=eAgijwhichisK¨ahler.

OnecanrephraseEq.(3.6)toget

F=F¯+3eAdA∧K,(3.17)

e3AR=F¯+Y,(3.18)

whereF¯isthetwo-form eldFrestrictedtoM8.NowifwecontractEq.(2.9)withη ,wehaveF¯ijYij= 6e 2A.WethushavetheexpressionforthescalarcurvatureRofM8whoseK¨ahlermetricisgivenasg¯,

R=2e 3A.(3.19)

Considerationofotherspinorbilinearsdoesnotgenerateindependentequations,butweneedtoimposetheBianchiidentityandtheequationofmotionforF,toguaranteethatthesupersymmetriccon gurationreallysatis esalltheequationsofmotion[10].ItturnsoutthatdF=0isaconsequenceofsupersymmetry,ascanbeeasilyseenfromEq.(3.6)andEq.(3.7).UsingtheequationofmotionforF, a(e 2AFab)=0,andEq.(3.4),onecanderivethefollowingequationforthescalarcurvatureofM8.

R 1