Comments on AdS2 solutions of D=11 Supergravity(4)

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

Wenowneedtointroducethebasisforthegammamatriceswhichisconvenientforthedimensionaldecompositionwehavechosentoconsider.Withtangentspaceindices,theyare

Γµ=γµ 1,

Γa=γ3 γa,µ=0,1,a=2,···,10,(2.6)(2.7)

whereγµ’sarethe2dimensionalgammamatrices,whileγa’sdenote9dimensionalones.Forsimplicitywetakethebasiswhereallγµandγaarereal,andofcourseΓM’sarealsorealmatrices.

AstheansatzfortheKillingspinor ,weassumeits2dimensionalpartsatis es

¯µε=a

γµγ3ε,butin2dimensionsthisjustcorrespondstotheparity

inversionsoforde nitenesswechooseEq.(2.8).

We

canthentakethe11dimensionalspinor =ε η+c.c.,whereηisa9dimensionalspinor.

Combiningtheingredientsgivenabove,itisstraightforwardtoderivethe9dimensionalKillingspinorequations,whichcanbepresentedasfollows.2

1aie Aη+/ Aη

bcb 2A(2.10)γaFbc 4Fabγη=0.e24

Inthefollowingwewillanalyzehowtheaboveequationsrestrictthelocalformofthe9-dimensionalinternalmetric.

3.Spinorbilinearsandtheconsequencesofsupersymmetry

Wenowconsiderthevariousspinorbilinearsmadeoutofη,andexploittheKillingspinorequationsto ndthelocalformofthemetric.Wecanconsiderthereal-valueddi erentialformssuchas

f=η η,

K=η γaηdxa,

aiY=(3.1)(3.2)

3

= aAη η,e 2AFacη γcη(3.4)(3.5)

implyingthatonecansetη η=eA.Weproceedinthesamewayand ndthat aKb+ bKa=0,i.e.Kde nesaKillingvectorintheinternalspace,and

d(eAK)=F+Y.(3.6)