AdS/CFT on the brane

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by  Jiro Soda

AdS/CFT on the brane
JiroSoda
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J AbstractItiswidelyrecognizedthattheAdS/CFTcorrespondenceisausefultool

7 tostudystronglycoupledfieldtheories.Ontheotherhand,Randall-Sundrum(RS)
braneworld models have been actively discussed as a novel cosmological frame-

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work. Interestingly, the geometrical set up of braneworlds is quite similar to that
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t in the AdS/CFT correspondence.Hence, it is legitimate to seek a precise relation
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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
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differentversionsofRSbraneworlds,namely,thesingle-branemodelandthetwo-
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branemodel.Inthecaseofthesingle-branemodel,werevealtherelationbetween

1 thegeometricalandtheAdS/CFTcorrespondenceapproachusingthegradientex-
v pansion method. It turns out that the high energy and the Weyl term corrections
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foundin the geometricalapproachcorrespondto theCFT mattercorrectionfound
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in the AdS/CFT correspondence approach. In the case of two-brane system, we
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1 also show that the AdS/CFT correspondenceplays an important role in the sense
. thatthelowenergyeffectivefieldtheorycanbedescribedbytheconformallycou-
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0 pledscalar-tensortheorywheretheradionplaystheroleofthescalarfield.Wealso
0 discussdilatonicbraneworldmodelsfromthepointofviewoftheAdS/CFTcorre-
1 spondence.
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r 1 Introduction
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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]
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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
− − −
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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
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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