AdS/CFT on the brane JiroSoda 0 1 0 2 n a J AbstractItiswidelyrecognizedthattheAdS/CFTcorrespondenceisausefultool 7 tostudystronglycoupledfieldtheories.Ontheotherhand,Randall-Sundrum(RS) braneworld models have been actively discussed as a novel cosmological frame- ] work. Interestingly, the geometrical set up of braneworlds is quite similar to that h t in the AdS/CFT correspondence.Hence, it is legitimate to seek a precise relation - p between these two different frameworks. In this lecture, I will explain how the e AdS/CFT correspondence is related to the RS braneworld models. There are two h differentversionsofRSbraneworlds,namely,thesingle-branemodelandthetwo- [ branemodel.Inthecaseofthesingle-branemodel,werevealtherelationbetween 1 thegeometricalandtheAdS/CFTcorrespondenceapproachusingthegradientex- v pansion method. It turns out that the high energy and the Weyl term corrections 1 foundin the geometricalapproachcorrespondto theCFT mattercorrectionfound 1 in the AdS/CFT correspondence approach. In the case of two-brane system, we 0 1 also show that the AdS/CFT correspondenceplays an important role in the sense . thatthelowenergyeffectivefieldtheorycanbedescribedbytheconformallycou- 1 0 pledscalar-tensortheorywheretheradionplaystheroleofthescalarfield.Wealso 0 discussdilatonicbraneworldmodelsfromthepointofviewoftheAdS/CFTcorre- 1 spondence. : v i X r 1 Introduction a Itisbelievedthatstringtheoryisacandidateoftheunifiedtheoryofeverything.Re- markably,string theory can be consistently formulatedonly in 10 dimensions[1]. Thisfactrequiresamechanismtofillthegapbetweenourrealworldandthehigher dimensions.Conventionally,theextradimensionsareconsideredtobecompactified JiroSoda DepartmentofPhysics,KyotoUniversity,Kyoto606-8501,Japan e-mail:[email protected] 1 2 JiroSoda to a smallcompactspace of the orderof the Planck scale. However,recentdevel- opmentsofsuperstringtheoryinventedanewidea,theso-calledbraneworldwhere matterresidesonthehypersurfaceinhigherdimensionalspacetime[2,3,4,5,6]( seealsoearlierindependentworks[7,8]).Thishypersurfaceiscalled(mem)brane. This idea originates from D-brane solutions in string theory. Interestingly, the D- brane solution also gives rise to the AdS/CFT correspondence which claims that classical gravity in an anti-de Sitter(AdS) spacetime is equivalent to a strongly coupled conformally invariant field theory (CFT). Since the origin is the same, braneworldsandtheAdS/CFTcorrespondencemayberelatedtoeachother.Inpar- ticular,Randall andSundrum(RS) braneworldmodels[9, 10] have a similar geo- metricalsetuptothatintheAdS/CFTcorrespondence.Hence,inthislecture,Iwill try to reveal relations between the AdS/CFT correspondence and RS braneworld models[11]. Themethodwewilluseisthegradientexpansionmethod.Physically,itisalow energyexpansionmethod.Historically,themethodhasbeenusedinthecosmolog- icalcontext[12,13,14,15].Inparticular,itisknowntobeusefulforanalyzingthe evolutionofcosmologicalperturbationsduringinflation.SincetheAdSspacetime canberegardedastheinflatinguniverseinthespatialdirection,thegradientexpan- sionmethodisalsoexpectedtobeusefulintheAdSspacetime.First,byutilizingthe gradientexpansionmethod,weapproximatelysolveEinsteinequationsinthebulk. Then,thejunctionconditionsatthebranegivetheeffectiveequationsofmotionon thebrane.Thus,wecanunderstandthelowenergyphysicsinthebraneworld.The similar butslightlydifferentmethodisalso usedin theAdS/CFT correspondence. WewillidentifyaconcreterelationbetweenthegeometricalandtheAdS/CFTcor- respondenceapproachby detailed comparison.The differenceshows up when we consider two brane systems. Indeed, we do not have the conventional AdS/CFT correspondenceforthetwo-branesystem.Instead,wehaveaconformallycoupled radiononthebranewhichreflectstheconformalsymmetryofthetheory.Thisob- servationisusefulforunderstandingwhybraneinflationsuffersfromtheetaprob- lem.Itisapparentthatthegradientexpansionmethodcanbeapplicabletovarious braneworldmodels.Moreover,aswewillsee,thegradientexpansionmethodpro- videsaunifiedviewofbranworldsandausefultooltomakecosmologicalpredic- tions. Theorganizationofthislectureisasfollows:Insection2,weintroduceRSmod- els and derive the effective Friedman equation on the brane. Here, two important corrections,i.e., the dark radiationand the high energycorrections,are identified. This sets our starting point. In section 3, we review two different views from the brane,namely,theAdS/CFTcorrespondenceandthegeometricalholography.These twoframeworksgiveusacomplementarypictureofthebraneworlds.Insection4, we present key questions to make our concerns manifest. In section 5, we review the gradientexpansionmethod.In sections 6 and 7, we apply the gradientexpan- sionmethodtothesingle-branemodelandtothetwo-branemodel,respectively.We obtaintheeffectivetheoryforbothcases.Insection8,wegiveanswerstothekey questions.ThiscompletesexplanationofrelationsbetweentheAdS/CFTcorrespon- dence and RS braneworldmodels. In section 9, we extend our analysis to models AdS/CFTonthebrane 3 withabulkscalarfield.Sincethepresenceofbulkfieldswouldbreaktheconformal invariance,itisinterestingtoconsiderdilatonicbraneworldsinconjunctionwiththe AdS/CFTcorrespondence.Thefinalsectionisdevotedtotheconclusion. 2 Braneworlds inAdS spacetime Inthissection,wewillintroduceRSbraneworldmodels.Wewillderivetheeffective Friedmann equation and identify the effects of extra-dimensions. In this lecture, wewillconcentrateoncosmologyalthoughwecanapplytheresultstoblackhole physics. 2.1 RS Models Randall and Sundrum proposed a simple model where a four-dimensional brane withthetensions isembeddedinthefive-dimensionalasymptoticallyanti-deSitter (AdS) bulk with a curvature scale ℓ. This single-brane model is described by the action[10] 1 12 S = d5x√ g R+ s d4x√ h+ d4x√ hL , (1) 2k 2 − ℓ2 − − − matter Z (cid:18) (cid:19) Z Z whereRandk 2arethescalarcurvatureandgravitationalconstantinfive-dimensions, respectively.WeimposeZ symmetryonthisspacetime,withthebraneatthefixed 2 point.ThematterLmatterisconfinedtothebrane.Throughoutthislecture,hmn rep- resentstheinducedmetriconthebrane.Remarkably,theinternaldimensionisnon- compactinthismodel.Hence,wedonothaveto careaboutthestabilityproblem. The basic equations consist of the equations of motion in the bulk and junction conditionsatthebranepositionduetothepresenceofthebrane.Alternatively,the basicequationscanberegardedasthe5-dimensionalEinsteinequationswithsingu- larsources.Letusrecallthe4-dimensionalcomponentsof5-dimensionalEinstein tensorcanbeexpressedby (5) (4) Gmn = Gmn +Lnm Kmn gmn K + , (2) − ··· whereLnm denotestheLiederivativealon(cid:2)gtheunitno(cid:3)rmalvectortothebrane,nm . Here,wedefinedtheextrinsiccurvatureby Kmn = d mr nm nr (cid:209) r nn . (3) − − ByintegratingEinsteinequationsal(cid:0)ongthenorm(cid:1)altothebrane,weobtainthejump oftheextrinsiccurvature(Kmn+ gmn K+) (Kmn− gmn K−)fromthelefthandside − − − 4 JiroSoda andthetotalenergymomentumtensoronthebranefromtherighthandsidedueto thedeltafunctionsources.Thus,takingintoaccounttheZ2symmetryKmn Kmn+ = ≡ Kmn− ,weobtainthejunctionconditions − k 2 Km n d nm K = sd nm +Tm n . (4) − atthebrane 2 − (cid:12) Here,Tmn represe(cid:2)ntstheenerg(cid:3)y(cid:12)(cid:12)-momentumtens(cid:0)orofthematte(cid:1)r. Originally,theyproposedthetwo-branemodelasapossiblesolutionofthehier- archyproblem[9].Theactionreads 1 12 S = d5x√ g R+ 2k 2 − ℓ2 Z (cid:18) (cid:19) (cid:229) s d4x h + (cid:229) d4x√ hLi , (5) − i − i − matter i= , Z i= , Z ⊕⊖ p ⊕⊖ where and representthepositiveandthenegativetensionbranes,respectively. ⊕ ⊖ In principle, one can consider multiple-branes although they are not discussed in thislecture. 2.2 Cosmology The homogeneous cosmology of the single-brane model is easy to analyze [16]. BecauseoftheBirkofftheoremduetothesymmetryonthebrane,itissufficientto considerAdSblackholespacetime: dr2 ds2= h(r)dt2+ +r2 dc 2+f2(c ) dq 2+sin2q df 2 , (6) − h(r) k (cid:2) (cid:0) (cid:1)(cid:3) where sinc fork=1 f = c fork=0 (7) k sinhc fork= 1  − and  M r2 h(r)=k + . (8) −r2 ℓ2 Note that M is the mass of the black hole, k is the curvatureof the horizon and ℓ isthe AdScurvatureradius.Supposethebraneismovingin thisspacetimewith a trajectoryt=t(t ),r=a(t ),wheret isapropertimeofthebrane(seeFig.1).The inducedmetriconthebranebecomes AdS/CFTonthebrane 5 ds2 = dt 2+a2(t ) dc 2+f2(c ) dq 2+sin2q df 2 . (9) ind − k ThisisnothingbuttheFriedman-R(cid:2)obertson-Wa(cid:0)lkerspacetimewh(cid:1)e(cid:3)reaisthescale factor.Themotionofthebranecannotbearbitrary.Itisconstrainedbythejunction condition: k 2s k 2 1 c c K c = T c T , (10) − 6 − 2 −3 (cid:20) (cid:21) where Kc c , Tc c , T are a cc component of the extrinsic curvature, a cc com- ponentand the trace part of the energymomentum tensor of matter on the brane, m respectively.Fromthenormalizationconditionn nm =1oftheunitnormalvector Fig.1 TheMinkowskibranerepresentedbythedottedlineisastaticbraneinthePoincarecoor- dinatesystemoftheAdSspacetime.While,thecosmologicalbranerepresentedbythethickline ismovinginthebulk.Themotionofthebraneinducestheexpansionofthebraneuniverse.The Cauchy horizon ofAdSspacetime corresponds tothebig-bang singularity. In thecase of AdS- Schwarzschildspacetime,thehorizonshouldbethepasthorizonoftheblackhole.Thebigbang islocatedbeyoundthehorizon. nm =( a˙,t˙),weobtain − 1 a˙2 t˙= + . (11) sh(a) h2(a) Here,thedotisaderivativewithrespecttothepropertimet .Now,onecancalculate Kcc as 6 JiroSoda 1 ¶ Kcc =−(cid:209) c nc =−¶ c nc +G ccr =−2nr¶c gcc . (12) Hence,fromEqs.(10),(11)and(12),wehaveanequation 1 k 2 h(a)+a˙2= (s +r ) . (13) a 6 q Thus,weget k M a˙2 k 4 +ℓ2+ = (s +r )2 , (14) a2 −a4 a2 36 or k 4s 2 1 k 4s k M k 4 H2= + r + + r 2. (15) 36 −ℓ2 18 −a2 a4 36 Bysettingk 2s ℓ=6,wefinallyderivedtheeffectiveFriedmannequationas[17,18, 19,20,21] k k 2 M k 4 H2= + r + + r 2, (16) −a2 3ℓ a4 36 where H =a˙/a is the Hubbleparameter.The Newton’sconstantcan be identified as 8p G =k 2/ℓ. The curvatureof the horizon k correspondsspatial curvature of N theuniverse.TheblackholemassMisreferredtoasthedarkradiation[22]which is not real radiation fluid but a reflection of the bulk geometry. This effect exists eveninthelowenergyregime.Thelasttermrepresentsthehighenergyeffectofthe braneworld[23]. Asto the two-branemodel,the same effectiveFriedmannequation(16) canbe expectedoneachbranebecausetheaboveequation(16)hasbeendeducedwithout referringtothebulkequationsofmotion. Giventhiscosmologicalbackground,itisnaturaltoinvestigatecosmologicalper- turbationinthebraneworld[24].Inthecaseofthesingle-branemodel,itisshown thatthegravityinMinkowskibraneislocalizedonthebraneinspiteofthenoncom- pact extra dimension. Consequently, it turned out that the conventionallinearized Einsteinequationapproximatelyholdsatscaleslargecomparedwiththecurvature scale ℓ. It should be stressed that this result can be attained by imposing the out- going boundary conditions. It turns out that this is also true in the cosmological background[25]. Inthecaseofthetwo-branemodel,GarrigaandTanakaanalyzedlinearizedgrav- ityandhaveshownthatthegravityonthebranebehavesastheBrans-Dicketheory atlowenergy[26].Thus,theconventionallinearizedEinsteinequationsdonothold evenonscaleslargecomparedwiththecurvaturescaleℓinthebulk.Charmousiset al.haveclearlyidentifiedtheBrans-Dickefieldastheradionmode[27]. AdS/CFTonthebrane 7 In the end, we would like to know how nonlinear gravity in the braneworld is deviated fromthe conventionalEinstein gravity.A partialanswer will be givenin thefollowingsections. 3 View from thebrane Intheprevioussection,wehaveconsideredanisotropicandhomogeneousuniverse and seen that the effective Friedmann equation on the brane can be regarded as the conventional Friedmann equation with two kind of corrections, i.e., the dark radiationandhigh-energycorrections.Here,wereviewtwodifferentapproachesto extendtheaboveresulttomoregeneralcases. 3.1 AdS/CFT Correspondence LetusstartwiththeAdS/CFTcorrespondence[28,29,30].Aftersolvingtheequa- tionsofmotioninthebulkwiththeboundaryvaluefixedandsubstitutingthesolu- tion g intothe 5-dimensionalEinstein-HilbertactionS , we obtain the effective cl 5d action for the boundary field h=g . The statement of the AdS/CFT cor- cl boundary | respondenceis that the resultanteffectiveaction can be equatedwith the partition functionalofsomeconformallyinvariantfieldtheory(CFT),namely exp[iS [g ]] <exp i hO > , (17) 5d cl CFT ≈ (cid:20) Z (cid:21) whereOisthefieldinCFT.Intherighthandside,hshouldbeinterpretedasasource field.ThisactionmustbedefinedattheAdSinfinitywheretheconformalsymmetry exists as the asymptotic symmetry. Hence, there exist infrared divergenceswhich mustbesubtractedbycounterterms.Thus,thecorrectformulabecomes exp[iS [g ]+iS ]=<exp i hO > , (18) 5d cl ct CFT (cid:20) Z (cid:21) whereweaddedthecounterterms S =S S [R2terms], (19) ct brane 4d − − whereS andS arethebraneactionandthe4-dimensionalEinstein-Hilbertac- brane 4d tion,respectively.Here,thehighercurvatureterms[R2terms]shouldbeunderstood symbolically. Inthecaseofthebraneworld,thebraneactsasthecutoff.Therefore,thereisno divergencesintheaboveexpressions.Inotherwords,nocountertermisnecessary. We can regardthe aboverelation as the definition of the “cut off” CFT. Thus, we 8 JiroSoda canfreelyrearrangethetermsasfollows S +S =S +S +[R2terms], (20) 5d brane 4d CFT wherewehavedefined expiS <exp i hO > . (21) CFT CFT ≡ (cid:20) Z (cid:21) This tells us that the brane models can be described as the conventionalEinstein theory with the cutoff CFT and higher order curvature terms [31, 32, 33, 34]. In termsoftheequationsofmotion,theAdS/CFTcorrespondencereads (4) k 2 Gmn = Tmn +TmnCFT +[R2terms], (22) ℓ (cid:0) (cid:1) where the R2 terms represent the higher order curvature terms and TmnCFT denotes theenergy-momentumtensorofthecutoffversionofconformalfieldtheory.When weapplythisresulttocosmology,weseeCFTcorrespondstothedarkradiationin the braneworld and the higher curvature terms can be reduced to the high-energy corrections. 3.2 GeometricalHolography Here,letusreviewthegeometricalapproach[35].IntheGaussiannormalcoordi- natesystem: ds2=dy2+gmn (y,xm )dxm dxn , (23) (5) wecanwritethe5-dimensionalEinsteintensor Gmn intermsofthe4-dimensional (4) Einsteintensor Gmn andtheextrinsiccurvatureas (5) (4) l Gmn = Gmn +Kmn ,y gmn K,y KKmn +2Kml K n − − 1 + gmn K2+Ka b Kb a 2 6 (cid:16) (cid:17) = gmn , (24) ℓ2 wherewehaveintroducedtheextrinsiccurvature 1 Kmn = gmn ,y, (25) −2 AdS/CFTonthebrane 9 andthelastequalitycomesfromthe5-dimensionalEinsteinequations.Ontheother hand,theWeyltensorinthebulkcanbeexpressedas 3 l ab Cym yn =Kmn ,y−gmn K,y+Km Kln +gmn K Kab −ℓ2gmn . (26) Now,onecaneliminateKmn ,y gmn K,y from(24)using(26)andobtain − (4) l Gmn = Cym yn +KKmn Kml K n − − 1 3 gmn K2 Ka b Kb a + gmn . (27) −2 − ℓ2 (cid:16) (cid:17) TakingintoaccounttheZ symmetry,wealsoobtainthejunctionconditions 2 k 2 Km n d nm K = sd nm +Tm n . (28) − y=0 2 − (cid:12) (cid:2) (cid:3)(cid:12) (cid:0) (cid:1) Here,Tmn representstheenergy-m(cid:12)omentumtensorofthematter.EvaluatingEq.(27) atthebraneandsubstitutingthejunctionconditionintoit,wehavethe“effective” equationsofmotion (4) k 2 Gmn = Tmn +k 4p mn Emn (29) ℓ − wherewehavedefinedthequadraticoftheenergymomentumtensor 1 1 1 1 p mn = Tm l Tln + TTmn + gmn Tab Tab T2 (30) −4 12 8 −3 (cid:18) (cid:19) andtheprojectionofWeyltensorCym yn ontothebrane Emn =Cym yn y=0 . | Here,weassumedtherelation 6 k 2s = (31) ℓ sothattheeffectivecosmologicalconstantvanishes. Because of the traceless property of Emn , when we consider an isotropic and homogeneousuniverse,itiseasytoshowthatthisgivesthedarkradiationcompo- nent(cid:181) 1/a4. Theexistenceofthehigh-energycorrections(cid:181) r 2 isapparentinthis approach. Thegeometricalapproachisusefultoclassifypossiblecorrectionstotheconven- tionalEinsteinequations.Onedefectofthisapproachisthefactthattheprojected WeyltensorEmn cannotbedeterminedwithoutsolvingtheequationsinthebulk. 10 JiroSoda 4 Does AdS/CFT playany roleinbraneworld? Tomakeourconcernsexplicit,wegiveasequenceofquestions.Wetreatthesingle- branemodelandtwo-branemodel,separately. 4.1 Single-branemodel IstheEinsteintheoryrecoveredeveninthenon-linearregime? Inthecaseofthelineartheory,itisknownthattheconventionalEinsteintheoryis recoveredatlowenergy.Ontheotherhand,thecosmologicalconsiderationsuggests the deviation from the conventional Friedmann equation even in the low energy regime.Thisisduetothe darkradiationterm.Therefore,we needto clarifywhen theconventionalEinsteintheorycanberecoveredonthebrane. HowdoestheAdS/CFTcomeintothebraneworld? It was argued that the cutoff CFT comes into the braneworld. However, no one knows what is the cutoff CFT. It is a vague concept at least from the point of view of the classical gravity. Moreover, it should be noted that the AdS/CFT correspondenceisaspecificconjecture.Indeed,originally,Maldacenaconjectured that the supergravity on AdS S5 is dual to the four-dimensional N = 4 su- 5 × per Yang-Mills theory [28]. Nevertheless, the AdS/CFT correspondenceseems to be related to the brane world model as has been demonstrated by several peo- ple [32, 33, 34, 36, 37, 38, 39, 40, 41]. Hence, it is important to reveal the role oftheAdS/CFTcorrespondencestartingfromthe5-dimensionalgeneralrelativity. HowaretheAdS/CFTandgeometricalapproachrelated? Thegeometricalapproachgivestheeffectiveequationsofmotion(29) (4) k 2 Gmn = Tmn +k 4p mn Emn . ℓ − Ontheotherhand,theAdS/CFTcorrespondenceyieldstheothereffectiveequations ofmotion(22) (4) k 2 Gmn = Tmn +TmnCFT +[R2terms]. ℓ Anapparentdifferenceisremarka(cid:0)ble. (cid:1) It is an interesting issue to clarify how these two descriptions are related. Shi- romizuandIdatriedtounderstandtheAdS/CFTcorrespondencefromthegeomet- ricalpointofview[42].Theyarguedthatp mm correspondstothetraceanomalyof