Comments on AdS2 solutions of D=11 Supergravity(5)

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

Forthetwo-formY,wegetsimply

dY=0.(3.7)

NowwecanchoosethecoordinatesystemwhereK= ψandthemetricoftheinternalspaceis

ds2=e2φ(dψ+B)2+gijdyidyji,j=1,2,···,8.(3.8)

φ,Barerespectivelyascalarandavector eldde nedonthe8dimensionalspaceM8withcoordinatesyi.

Nowweclaimthat,whenthe9dimensionalinternalspaceiscompact,ηisachiralspinoronM,andasaresultφ=A.WewillneedtomakeuseofthefollowingidentitieswhichholdforanarbitraryDiracspinorηin9-dimensions.

(ηTη)2=(ηTγaη)2,

|ηTη| |ηTγaη|2=2(η γaη)2 2(η η)2,(3.9)(3.10)

whosederivationcanbefoundinRef.[6].WeconsiderthespinorbilinearηTη,whichiscomplex-valued.FromtheKillingequa-tionsweget

a(e AηTη)=aie 2AηTγaη.(3.11)

ForD=e AηTη,wethushave( aD)2= e 4A(ηTγaη)2= e 2AD2.Onecanalsoshowthat,fromtheKillingequationsEq.(2.9)andEq.(2.10),

2(e AηTη)=ai a(e 2AηTγaη)

= 2e 3AηTη,(3.12)

i.e. 2D+2e 2AD=0.Fromtheserelations,itiseasytoseethat 2(D 1)=0providedDisnotzero.Iftheinternalspaceiscompact,thisispossibleonlyifDisconstant,butitmeansD=0,sowehaveacontradiction.WethusconcludeηTη=ηTγaη=0,andfromEq.(3.10)ηischiralonM8andwehaveK2=e2φ=e2A.

Obviouslytheaboveargumentrequirestheinternalspaceshouldbecompact,andAshouldnotshowasingularbehavior.Butinthenextsectionwewillshowthat,fortheimportantclassofsolutionssuchasAdS4×SE7andthebubblinggeometryofRef.[7],eventhoughtheinternalmanifoldsarenotcompact,thesolutionscanberewritteninthemannerwewillconcludeinthissection.Weguessthatitmightbepossibletoimproveourproofforthechiralityofη,withoutassumingcompactinternalspace.

Nowthatηischiral,Yisaclosedtwo-forminM8,whichcanbeusedtode neanalmostcomplexstructure.Inordertoseewhetherthiscomplexstructureisintegrableornot,oneneedstochecktheexteriorderivativeofthecomplex-valued(4,0)-form ,de nedas

abcd=ηTγabcdη.(3.13)

Thechiralityofηagainrestricts tobeafour-formonM8,andasusualwithSU(n)-structures,J, satisfy

VolM8=e 4A16¯, ∧ J∧ =0.(3.14)