Comments on AdS2 solutions of D=11 Supergravity(3)

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

Thispaperisorganisedasfollows.InSection2,wesetuptheproblemandderivethe9-dimensionalKillingequationsfromthe11-dimensionalone.InSection3,weconsiderthespinorbilinearsandtheirderivativesto xthelocalformofthemetric.InSection4weillustratethatthewell-knownsolutionswhichhaveAdS2factorscanbeindeedcastintheformaswehavepresentedinthispaper.InSection5weconcludewithcommentsanddiscussionsonfurtherworks.

2.Ansatz

InthisarticleweconsidersupersymmetricsolutionsofD=11supergravity,whoseLa-grangiandensityinthebosonicsectorisgivenasfollows,

L=R 1 1

6C∧G∧G,(2.1)

whereCisthe3-formpotentialandG=dC.

Byde nition,supersymmetricbackgroundsallownontrivialsolutionstotheKillingspinorequationwhichisobtainedbysettingthesupersymmetrytransformationtozero.ForD=11supergravity,wethushave,forpurelybosonicbackgrounds,

δψM= M +1