Comments on AdS2 solutions of D=11 Supergravity

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

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arXiv:hep-th/0607093v2 15 Jul 2006NakwooKimandJong-DaeParkDepartmentofPhysicsandResearchInstituteofBasicScience,KyungHeeUniversity,Seoul130-701,KoreaE-mail:,Abstract:Westudythesupersymmetricsolutionsof11-dimensionalsupergravitywithafactorofAdS2madeofM2-branes.Suchsolutionscanprovidegravitydualsofsuperconfor-malquantummechanics,orthroughdoubleWickrotation,thegenericbubblinggeometryofM-theorywhichare1/16-BPS.Weshowthat,whentheinternalmanifoldiscompact,itshouldtaketheformofawarpedU(1)- brationoveran8-dimensionalK¨ahlerspace.Keywords:

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

1.Introduction

Itisoneofthemostintriguingissuesinstringtheorytoprove/disprovetheso-calledMaldacenaconjecturebetweenanti-de-Sittergravityandconformal eldtheories

[1].Alotofremarkableagreementshavebeenencounteredsofar,soitwouldbemoreappropriatetosayonewouldliketodeterminetheregionofvalidityaspreciseaspossible.

ThisinevitablyleadsustothestudyoflesssupersymmetricsolutionsofString/M-theory.Thespectrumofsupersymmetricsolutionsisrichenoughtoprovideexampleswithrealisticfeaturessuchascon nementandasymptoticfreedom(see[2]forexample),andyetthankstothepowerofsupersymmetrywecanperformexplicitchecksusingprotectedquantities.

InthispaperwewillemployatechniquewhichisfoundtobeverypowerfulandatthesametimeilluminatinginthesearchfornewsupersymmetricbackgroundsofString/M-theory.BasedontheexistenceofKillingspinors,onecanconstructvariousdi erentialformsasspinorbilinearsanddeterminethelocalformofthemetricandthegauge eldsexploitingthedi erentialandalgebraicrelationsbetweenthedi erentialformswhichcanbederivedusingtheKillingspinorequations.AsampleofworkswhichmakeuseofthistechniquecanbefoundinRef.[3].

Inthispaper,asasequeltothepreviousone[4],weanalyzetheconsequencesofsu-persymmetryin11-dimensionalsupergravity,combinedwiththeansatzofAdS2factor,i.e.SO(2,1)isometry.Inparticular,forsimplicity,weconsiderpureM2-branecon gurations.AconvenientwayofseeingthemisasM2-braneswrappedontwo-cyclesinaCalabi-Yau5-fold,andthereadersarereferredto[5]forstudiesalongalternativeavenuesonsimilarcon gurations.Inthepresentwork,itwillbeshownthat,the9-dimensionalinternalmani-foldshouldtaketheformofawarpedU(1)- brationonaK¨ahlerbase.Wealsowritedownthe8-dimensionalnonlinearpartialdi erentialequationforthecurvaturetensor,whichisrequiredifthesupersymmetriccon gurationistosolvetheequationsofmotion.

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

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)

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)

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

UsingtheKillingequationsoneobtains

d(eA )= aie AK∧ .(3.15)

Forω≡eAe iaψ andusingK=e2A(dψ+B),wehave

dω= aiB∧ω,(3.16)

whichisan8dimensionalequationonM8.FromthegeneralresultofSU(n)-structuresandtheclassi cationoftorsionclasses,wearriveattheconclusionthatthecomplexstructuregivenbyYisintegrableanddB=RistheRicciformoftheK¨ahlermanifoldM8.ConsideringY2～(η η)2～e2Aandω2～e2A|ηTη|2～e4Awhenevaluatedusingthemetricg,itistherescaledmetricg¯ij=eAgijwhichisK¨ahler.

OnecanrephraseEq.(3.6)toget

F=F¯+3eAdA∧K,(3.17)

e3AR=F¯+Y,(3.18)

whereF¯isthetwo-form eldFrestrictedtoM8.NowifwecontractEq.(2.9)withη ,wehaveF¯ijYij= 6e 2A.WethushavetheexpressionforthescalarcurvatureRofM8whoseK¨ahlermetricisgivenasg¯,

R=2e 3A.(3.19)

Considerationofotherspinorbilinearsdoesnotgenerateindependentequations,butweneedtoimposetheBianchiidentityandtheequationofmotionforF,toguaranteethatthesupersymmetriccon gurationreallysatis esalltheequationsofmotion[10].ItturnsoutthatdF=0isaconsequenceofsupersymmetry,ascanbeeasilyseenfromEq.(3.6)andEq.(3.7).UsingtheequationofmotionforF, a(e 2AFab)=0,andEq.(3.4),onecanderivethefollowingequationforthescalarcurvatureofM8.

R 1

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 +

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

Onecanalsocheckthat,asrequiredbyEq.(4.4),

d =2i

4)∧ .(4.8)

AndEq.(3.20)isindeedsatis ed.

Thenextexampleisthe1/2-BPSbubblinggeometryofM-theorygiantgravitonsobtainedin[7].Thisclassofsolutionsdescribegeneric1/2-BPSoperatorsofM2-braneorM5-braneconformal eldtheories.Fromsuperalgebraarguments,suchcon gurationsshouldpossessSO(3)×SO(6)symmetry,whicharerealizedasaS2×S5factorinthemetric.S2canbetreatedasaWick-rotatedAdS2to tintoourresult.AnothercommentinorderisthatnowthecanonicalKillingvectormadeoutoftheKillingspinorbecomestime-like,sooursolutionscandescribemagnetic,orM5-branes,aswellaselectricorM2-branecon gurations.Thesolutionsaresummarizedasfollows.

ds2

11= 4e2λ(1+y2e 6λ)(dt+Vidxi)24λ

+e

y(1 y yD)

Vi=1

2 3[d( yD)+( yD)2dy]

F=dBt∧(dt+V)+BtdV+dB

Bt= 4y3e 6λ

dB =2 3[(y 2

yD+y( yD)2 yD)dy+y i yDdxi]

=2 3[y2( y1

3)2+ds2

KE4]

4e2λ(1+y2e 6λ)(dt+V)2

e 4λ

+

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

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

aspresentedin[4].Therelevant6-dimensionalK¨ahlermanifoldtakesaformofS2-bundleovera4-dimensionalK¨ahler-Einsteinmanifold.Itwillbeveryinterestingtointroducesuchaconcreteansatz,andsolveEq.(3.20).Weexpect,asisthecasewithYp,q,themetricoftheK¨ahlerbasemightbeingeneralnotcomplete,buttheentire9-dimensionalinternalmanifoldcanbemadecomplete.Weplantopresentthenewsolutionsandtheglobalanalysisinafuturepublication.

Anotherinterestingdirectionistointerpretoursolutionsasgeneralizedbubblingge-ometry.ForthecaseofAdS5×S5solutionsinIIBsupergravity,the1/2-BPSbubblingsolutionsgivenin[7]canbedescribedintermsofadistributionfunctioninthephasespaceof1-dimensionalfreefermions.ForM-theory,determiningthesolutionsisinsteadreducedtosolvingaTodaequationin3-dimensions,andwebelievethedynamicsof1/2-BPSop-eratorsmustbeencodedtherein.Thesupergravitysolutionsdualtolesssupersymmetricgiantgravitonoperatorsareconsideredfor1/8-BPSinRef.[4]andfor1/4-BPSinRef.[11].Theybothconcludethatthe10dimensionalsolutionisbasedonaK¨ahlerspacewhichis6and4-dimensional,respectively.Naturallyoneexpectstheyoriginatefromthesymplecticstructureoftheeigenvaluedynamics.Likewise,Eq.(3.20)canbeinterpretedastheequa-tiongoverningthedynamicsofgenericsupersymmetricoperatorsofM-branesconformal eldtheory.Oneimportantfeaturewehavebeenignoringinthispaperistheglobalprop-ertyandtheboundaryconditionsofthesolutions.Sinceouranalysisisgeneralenoughtoencompassthe1/2-BPS uctuationsoflesssupersymmetricconformal eldtheoriesonM-branes,aswellaslesssupersymmetric uctuationsofmaximalconformal eldtheoriesonM-branes,wewill rstneedto xtheboundaryconditionaccordingtotheconformal eldtheoryweareinterestedin,andthensolveEq.(3.20)to ndgravitydualstoBPSoperators.

Acknowledgments

WearegratefultoHo-UngYeeandSang-HeonYifordiscussions.TheresearchofNakwooKimissupportedbytheScienceResearchCenterProgramoftheKoreaScienceandEn-gineeringFoundation(KOSEF)throughtheCenterforQuantumSpacetime(CQUeST)ofSogangUniversitywithgrantnumberR11-2005-021,andbytheBasicResearchProgramofKOSEFwithgrantNo.R01-2004-000-10651-0.NakwooKimandJong-DaeParkarebothsupportedbytheKoreaResearchFoundationGrantKRF-2003-070-C00011.References

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