Comments on AdS2 solutions of D=11 Supergravity(9)

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

Ifwede neα=α +t,thenthemetriccanberewrittenas

ds2

11=ye

+

42 4λe6λ22 4λ d 2 4ye[dt y(1+y2e 6λ)(dα +σ3 (1+y2e 6λ)V)]24ye 6λ

4 y(eD)dx1∧dx2+1

3 V)+yJKE4,(4.13)

whereJKE4istheK¨ahlerformofKE4.OnecaneasilyshowthatJisclosed,andforthe(4,0)-formgivenas

σD/2 yDdy+2i(dα + =4ye

y2(dα +σ

y2V∧ .(4.15)

5.Discussions

InthisworkwehavestudiedsupersymmetricM2-branecon gurationswhichhaveafactorofAdS2.Theycanbeinterpretedasthenear-horizonlimitofM2-branes,whoseworldvol-umeiswrappedona2-cycleinaCalabi-Yau5-fold.Ingeneral,theyarethus1/16-BPS.Itturnsoutthattheinternal9-dimensionalmanifoldshouldtakeaformofU(1)- brationwhosebasemanifoldisgivenbyaK¨ahlermanifold.ThereisarestrictionontheK¨ahlerbaseimposedbysupersymmetry:wehavefoundaLaplace-likeequationforthescalarcurvatureandRiccitensor,asgiveninEq.(3.20).Theresultpresentedinthisarticleisamusinglyverysimilartotheresultof[4],wherepureD3-branecon gurationswithAdS3arestudied.Inthatcase,theinternalmanifoldis7-dimensional,whichagaintakestheformofwarpedU(1)- brationona6-dimensionalK¨ahlermanifold.TheK¨ahlerbasecannotbearbitrary,ithastosatisfyanonlinearpartialdi erentialequationforthecurvature.

Itiscertainlyofgreatinterestto ndnewAdSsolutions,bydirectlytryingtosolveEq.(3.20).Infact,recentlyanewclassofAdS3solutionsinIIBsupergravityhasbeenpresented[8].Theauthors rststudiedAdS3solutionsof11-dimensionalsupergravity,andthen,throughT-dualityoperations,obtainedAdS3solutionsofIIBsupergravity,whereonlythe ve-form uxesareturnedon.ThisisofcourseverysimilartothediscoveryofthecelebratedSasaki-EinsteinsolutionsYp,qaspartofAdS5solutionsinIIBsupergravity

[9].Inparticular,itwasarguedthatthenewsolutionscanbeindeedwrittenintheform