9902131Introduction to the Maldacena Conjecture on AdSCFTJen

关于反德西特空间上共形场论的基础文献

NBI-HE-99-05

IntroductiontotheMaldacenaConjectureonAdS/CFT

arXiv:hep-th/9902131 v2 19 Feb 1999

JensLyngPetersen

TheNielsBohrInstitute,Blegdamsvej17,

DK-2100Copenhagen ,Denmark.

ABSTRACT

Theselecturesdonotatallprovideageneralreviewofthisrapidlygrowing eld.Insteadaratherdetailedaccountispresentedofanumberofthemostelementaryaspects.

关于反德西特空间上共形场论的基础文献

1Introduction

TheMaldacenaconjecture[1]isaconjectureconcerningstringtheoryorMtheoryoncertainbackgroundsoftheformAdSd×MD d.HereAdSdisanantideSitterspaceofspace-timedimensiond,andMD disacertaincompacti cationspaceofdimensionD dwithD=10forstringtheoryandD=11forMtheory.Inaddition,thebackgroundisspeci edbyastatementaboutthe uxofacertain eldstrengthdi erentialform.Theconjectureassertsthatthequantumstring-orM-theoryonthisbackgroundismathematicallyequivalent-ordualasthewordgoes-toanordinarybutconformallyinvariantquantum eldtheoryinaspace-timeofdimensiond 1,whichinfacthastheinterpretationof“theboundary”ofAdSd.Thisseemstoputtheformulationofstring/M-theoryonanovelandratherunexpectedfooting.Alsotherelationbetweenquantumandclassicaltheoryisilluminatedinasurprisingwaybytheconjecture.SeveraldetailsinMaldacena’soriginalformulationwereleftunspeci ed.MostofthoseweresubsequentlygivenapreciseformulationbyindependentworksofGubser,KlebanovandPolyakov[2]andbyWitten[3].Aprioriitmightseemverystrangethatquantumtheoriesindi erentspace-timedimensionscouldbeequivalent.Thispossibilityisrelatedtothefactthatthetheoryinthelargerdimensionis(amongotherthings)aquantumtheoryofgravity.Forsuchtheoriestheconceptofholographyhasbeenintroducedasagenericproperty,andtheMaldacenaconjectureisanexampleoftherealizationofthat(fordiscussion,seeforexample[4]).

Inthemeantimealargenumberofcheckshavebeenperformedwhichweshallnotattempttoreviewinthesenotes(forsomerecentreviewswithmanyadditionalreferences,seeforexample[5,6,7,8]).Supposingtheconjectureistrue,itremainssomewhatunclearwhatthemostsigni cantconsequencewillbe.Ontheonehandtheconjectureallowsonetoobtainnonperturbativeinformationonordinary,butmostlyconformallyinvariantquantum eldtheories,especiallyatlargeN(ofagaugegroupU(N)),fromclassicalstring/M-theoryorevenclassicalsupergravity.Thisisaremarkableunexpecteddevelopment,andtheonethathasmostlybeenpursueduntilnow.Ontheotherhanditisconceivablethattheconjecturewillplayanimportantroleintheeventualnon-perturbativeformulationofM-theory,forwhichthematrixmodelofBFSS[9]wasa rstproposal.Inasomewhatdi erentlineofdevelopment,Witten[10]showedhowtoapparentlyovercometheoriginalrestrictiontoconformallyinvariant(andmostlysupersymmetric)quantum eldtheories,providinginfactanentirelynewframeworkforstudyinglargeN“ordinary”QCDandsimilartheories.Arathernewideaabouthowtoachievethesameendinperhapsamoree cientwayhasrecentlyappeared[11].Thatapproach,howeverwillnotbecoveredhereatall(seealso[5]).Inanycase,theAdS/CFTdevelopmentattractsanenormousinterest.

Intheselecturesweshallattemptaveryelementaryintroductiontoasomewhatrestrictednumberofbasicaspects.Insect.2webeginbyreviewingpropertiesofantideSitterspaces,theirisometries,thefactthattheymaybeassociatedwitha“boundary”andthefactthattheisometrygroupofantideSitterspacebecomestheconformalgroupontheboundary.ItfollowsthatifaquantumtheoryonantideSitterspaceisdualtoanotherquantumtheoryontheboundary,thenthatsecondtheorymustnecessarilybeconformal.

Insect.3weexpandonthediscussionin[1]andprovideashortreviewofclassicalsuper-gravitysolutionsinthepresenceofbranes.Bothsocalledextremal(BPS)andnon-extremal

关于反德西特空间上共形场论的基础文献

solutionswillbeconsideredforlaterreference.Thissubjecthasalreadybeenreviewedonnu-merousoccasions(seeforexample[12,13,14,15]).WedescribehowthesocallednearhorizonapproximationinsomecasesleadtogeometriesoftheformAdSd×SD d.Thisfacthasbeenknownforseveralyearsbytheexperts,butitsfullsigni cancewasonlyrealizedbyMaldacena.Insec.4wefollowratherclosely[3]anddescribeindetailseveralinstructivealbeitrathertrivialexamplesofhowthedualitybetweenthebulktheoryandtheboundarytheoryworksinthecaseoffreetheories.Animportantobjectwhichhasageneralsigni canceisthegeneralizedpropagatordescribingpropagationofcertainmodesfromaspace-timepointinthebulkofantideSitterspacetoa“point”ontheboundary.Thispropagatorwasthekeyobjectinthediscussionsin[2,3]andwillbeconstructedinafewofthesimplestcases.AtthesametimetheMaldacenaconjecturewillbemademoreprecise.

Insect.5wefollow[10]anddescribehowcertain nitetemperaturescenariosmaybeusedtoprovideamechanismforbreakingconformalinvarianceandsupersymmetry,andtherebyobtainaframeworkforstudyinglargeNQCD.

2ElementarypropertiesofantideSitterspaces

1

WebeginbyconsideringtheEinstein-Hilbertactionwithacosmologicalterm.

S= s

|g|(R+Λ)

(1)

Weconsider( rst)Minkowskimetricwiths= 1,andwetakeittobe“mostlyplus”.WeshallalsoconsiderEuclideansignature,s=+1.Noticethatthesignoftheaction ipsifwegofroma“mostlyplus”toa“mostlyminus”metric.AntideSitterspace(AdS)aswellasdeSitterspacearesolutionsoftheemptyspaceEinsteinequation:

Rµν

1

2

R=

Λgµν D

2 D

gµν(2)

SothesespaceshavethepropertythattheRiccitensorisproportionaltothemetrictensor:TheyareEinsteinspaces.Weshallbeinterestedinvariousexamplesofsuchspaces,inparticularinoneswithmaximalsymmetry,forwhichinadditionwehave

Rµνρσ=

R

关于反德西特空间上共形场论的基础文献

2.1AdSn+1byembedding

Itisusefultoconsideran(n+1)-dimAdSn+1asasubmanifoldofapseudo-Euclidean(n+2)-dimensionalembeddingspacewithcoordinates(ya)=(y0,y1,...,yn,yn+1)andmetric

ηab=diag(+, , ,..., ,+)

with“lengthsquared”

y2≡(y0)2+(yn+1)2

n i=1

(yi)2

preservedbythe“Lorentz-like”groupSO(2,n)(with“twotimes”)actingas

ya→y′=Λabyb,

a

Λab∈SO(2,n)

(4)

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