Comments on AdS2 solutions of D=11 Supergravity(7)

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

isawell-knownexample,whichisgivenasthenear-horizonlimitofthreeM2-branesintersectingoverapoint.Assuch,thepreservedsupersymmetryisinfact1/4forthissolution.

WenextconsiderAdS4×SE7solutionswhereSE7isa7dimensionalSasaki-Einsteinmanifold.Theycanbeconsideredasthenear-horizonlimitofM2-branesolutionswhenputonasingularityofCalabi-Yau4-folds.Assuch,ingeneralthesesolutionsare1/8-BPS.

Itiswellknownthatincanonicalform,anySasaki-EinsteinmanifoldscanbewrittenasaU(1)- brationoveraK¨ahler-Einsteinmanifold.ForSE7,usingthestandardconvention,

σds2=(dα+

=

=4112ds2AdS4+dsSE7

coshρ 4coshρ4cosh2ρdφ+2+ds2

KE6)+ 4dα +σ/4 2+ds2KE6sinh2ρ4 2 ,(4.2)

whereweputα→α +φ/2.

ComparedtothegeneralformofthesolutiongiveninEq.(3.21),evidentlyweexpectthatthe8-dimensionalmetric 2coshρsinh2ρ22ds=(4.3)+dsKE6+44

shouldbeK¨ahler,andtheRicci-formisgivenas dα +σ/4R=d2

,(4.5)cosh3ρ

and nally,Eq.(3.20)shouldbesatis ed.

Inordertocheckthis,itismoste cienttoconstructthealmostK¨ahlerformJandthe(4,0)-form ,andcomputetheirexteriorderivatives.AreasonableguessforJis

coshρ1+J=4

8 σcoshρdρ+2isinhρdα +