COURSES A:"D PLENARY TALKS REVISTA MEXICANA DE FÍSICA.J9 SUPLEMENTO J, 38--43 1U1\'102oo3

AdS/CFT correspondence and quantum induced dilatonic muIti-brane-worldsShin'ichi Nojiri

Depar/ment of Applied Physies, Na/ional Defenee Aeademy,lIashirimizu Yokosuka 239, JAI'AN

e-mail: nojiri@cc.nda.ac.jp

Sergei D. OdintsovInstituto de Fisica de la Universidad de Guanajuato,Apartado postal £-143,37150 Lean, G/o., Mexieo

Tomsk State Pedagogical Universily,634041 'Iomsk, RUSSIA

e-mail:odintsov@~rltg5.uglo.mx.odintsov@ilp.uni-Ieipzig.de

Recibido el 3 de mayo de 2(x) 1; aceptado el 25 de junio de 2(X) 1

d5 dilatonic gravity aclion wilh surface counterterms molivated by AdSICFr correspondence ami with con tribu tion s of brane quantum CI,"Tsis considercd around an AdS-like bulk. The elTective equations of motion are constructed. They admit two (outer and inner) or multi-bmnesolutions wherc brane CPTs may be different. The role ol' the quantum brane CFT consisls in inducing complicated bmne dilatonic gravity.For eX~XJnentialbulk pOlentials the number of AdS-like bulk spaces is found in analytical formo The corrcsponding Hat or curved (de Sitteror hyperboIic) dilatonic two branes are created, as a rule, thanks to quantum effects. The observable early Universe may corrcspond loinflationary bmne. The found dilalonic quanlum two brane-worlds usually contain naked singularities but in a couple of explicit examplesIhe curvature is finite and a horizon (corresponding to wormhole-like space) ¡¡ppcars.

Keywords: AdS/CFT corrcspondence, dilalonic fields

Se considera la acción de la teoría de la gravitación dilatónica d5, incluyendo conlratérminos de superficie motivados por la corrcspondenci¡¡AdS/CFT y con contribuciones de CFT's cuánticas de mcmbranas alrededor de un "bulk" de lipo AdS. Se construyen las ecuaciones efectivasde movimiento las cuales permiten la existencia de soluciones con dos membranas (interior y exterior) o multibranas donde las CFT's debrana<¡pueden ser diferentes. La función de la CI,"Tcuántica de branas consiste en inducir una compleja gravitación dilatónica de brJnas. Parael caso de potencialcs "bulk" exponenciales se encuentra de forma analítica el númcro corrcspondiente de espacios de tipo AdS. Por reglagenerol, debido a los efectos cuánticos se crean las 2-branas dilatónicas planas o curvadas (de Siller o hiperbólica). El Universo tempranoobserv¡¡ble puede COlTcs¡XJndcra una brana inflacionaria. Los mundos dilatónicos y cuánticos de branas que se encuentran por lo común,contienen la singularidad desnuda, pero en algunos ejemplos la curvatura cs finila y aparece un horizonte (correspondicnte a un espacio detipo agujcro de gusano).

Descriptores: Correspondencia AdS/CFT, campos dilatónicos

PAes: 11.25.Wx; !1.27.+d

1. Introduction

Recent booming activity in the study of brane-worlds iscaused by several reasons. First, gravily on 4d brane embed-ded in higher dimensional AdS-like Universe may be local-ized [1,2]. Second, lhe way lO resolve lhe mass hierarchyproblem appears [1]. Thinl, new ideas abou! lhe cosmolog-ical constant problem solution come to game [7,8]. A very¡ncomplete list of references mainly on the cosmological as-pecL' ofbrane-worlds is given in Refs. 3 and 4 (and referenceslherein), but it is growing every day.

The essential elemenl of brane-world models is lhe pres-cnce in lhe theory o[ two [ree pararnclcrs (bulk cosmologi-cal constant and brane tension, 01' brane cosrnological con-stant). Thc role of brane cosrnological constant is to fix lheposition of the brane in terms of tension (that is why branccosrnological conslant and brane tension are almost the sarnelhing). Being completely consistent amI malhemalically rea-sonable, such way of doing things may look nol complelely

satisfaclOry. Indeed, lhe physical origin (and prediclion) ofbrane tension in terms of sorne dynamical mechanism maybe required.

The ideology may be differenl, in lhe spiril of Refs. 5 and6. One considers the addition of surface countertenns to theoriginal action on AdS-like space. These tenns are responsi-hlc for rnaking lhe variational procedurc to be well-defined(in Gibbons-Hawking spirit) and for elimination of thc lead-ing divergences of the action. Brane tension is not consid-ered as free parameter anyrnore but it is fixed by the condi-lion of finiteness of spacetime when brane goes lO infinity.Of course, lcaving the theory in such form would rule outthe possibility of the cxistcncc of consislent branc-world so-lutions. Fortunately, other parameters contribute to brane tcn-sion. I[ Orle considers thal there is a quanlum CFT living onIhc brane (which is more close to lhe spiril of AdS/CFT corre-spondence [9 Il lhen such a CFT produces con formal anomaly(or anomaly induced effective action). This contributes tothe brane tcnsion. As a result the dynamical rnechanisrn lo

ADS/CFrCORRESPONDE1\'CE AND QUANTUM INDUCED DILATONIC MUI;r¡-BRANE-WORLDS 39

get brane-world wilh fla! or eurved (de Siller or Ami-de Sil-ter) brane appears. The curvature of such 4d Universe is ex-pressed in tenns of sorne dimensional para meter l which usu-ally appea" in AdS/CFT sel-up and of eonlent of <¡uanlumbrane malter. In other words, brane-world is the consequenceof!he fael (verified experimenlally by everybody Iife) of thepresenceofmatteron the brane! Forexample, sign ofconfor-mal anomaly terms for usual malter is such that in the one-brane case !he de Sitler (ever expanding, inflationary) Uni-verse is a preferable solution of the brane equation.{a)

The scenario of Refs. 5 ami 6 may be eXlended to in-clude lhe presence of dilalon(s), as il was done in ReL lO, orlhe formulation of <¡uanlum cosmology in Wheeler-De Wittfonn [llJ. Then whole scenario looks even more related withAdS/CFT correspondence as dilalOnic gravily nattlrally fol-lows as !he bosonic seclor of d5 gauged supergravity. More-over, the extra prize-in form of dynamical determinationof 4d boundary value of dilatan-appears. In ReL 10 lhequantum dilatonic one brane Universe has been presentedwilh possibilily lo get inflationary or hyperbolic or flat braneswith dynamical determination of brane dilaton. An interest-ing question is rclatcd with the generaliz<1.tion of such sce-nario in dilatonic gravity for the multi-brane casco This willbe lhe purpose of the presem work.

In the next section we present the general action ofd5 dilatonic gravity with surface counterterms and quantumbrane CFf contribution. This action is convenient for thedescription of brane-worlds where lhe bulk is an AdS-likespacetime. There could be one or two mal or curved) branesin !he theory. As it was already mentioned, the brane tcnsionis fixed in our approach, inslead of it lhe effective brane ten-sion is induced by <¡uanlum effecls. In Seco 3, lhe explicilanalytical solution of bulk e<¡uation for a number of expo-nemial bulk pOlentials is presenled. There is lhe possibilityto have lwo (inner and oUler) branes associaled with each ofthe aboye bulk solutions. It is interesting that quantum crc-ated branes can be flal, or de Siller (inflalionary) or hyper-bolic. The role of quantum brane malter corrections in get-

S = 5ElI + 5Gn + 25, + W,

ting such branes is eXlremely imporlant. Nevertheless, thereare few particular cases where such branes appear at the c1a~-sicallcvel, i.e. without quantum corrections. We also brief1ydescribe how to get generalizations of the aboye solutionsfor quantum dilatonic multi-brane-worlds with more than twobranes. A brief summary of results is given in the final sec-lion where also the result of studying thc character of sin-gularities for proposed two-brane solutions is presented. Inmost cases, as usually occurs in AdS dilatonic gravity, the so-lutions contain naked singularities. However, in a couplc ofcal:les the scalar curvature is finite and there is horizon. Thecorresponding 4d branes may be inlerpreled as wormholes.

2. Dilatonic gravity action with brane quantumcorrections

Let us present the initial action for di latonic AdS gravity un-der consideration. The melric of (Euclidean) AdS has lhe fol-lowing form:

1

ds2 = dz2 + L 9(4);jdx'dxj, 9(4)'j = e2A(')g'j. (1)i,j=l

Here gij is the metric of the Einstein manifold, which is de-fined by Tij = kgij, where Tij is the Ricci tensor constructcdwith gij and k is a constant. One can consider two copies oflhe regions given by z < Zo and glue lwo regions pUlling abrane at z = Zo. More generally, one can consider two copiesof regions io < z < Zo and glue the regions putting twobranes at z = io and z = Zo. Hereafter we call the brane atz = io as "inner" brane and that at z = Zo as "outer" brane.

Let us firsl consider lhe case wi!h only one brane atz = Zo and start with Euclidean signature action S whichis the sum of the Einstein-Hilbcrt action SEII with kineticlerm and potenlial V(rjJ) = 12/12 + "'(rjJ) for dilaton rjJ,lhe Gibbons-Hawking surface lenn 5011, lhe surface counlerlerm S, and the lrace anomaly induced action W:(b)

(2)

1 J 5 [ 1 "12 ]5E11 = 16"C d x .,)9(5) R(5) - '2 \7"<1>\7 rjJ+ y¡ + "'(rjJ) ,

5GB = S:C J d"x .,)9(1) \7"n",

1 J' [6 1 ]S, = -16"Cl d x .,)9(1) T + ;¡"'(rjJ)

W = bJ d'x.jgFA+b' J d'x.jg{A [2Ef+R"vV"Vv - ~iiEí2+ ~(V"ii)V"] A+ (8- ~Eíii) A}

-112 {b"+~(b+b')} J d"x.jg[R-6EíA-6(V"A)(V"A)f

J ~ [2 - -- 2-21---]+C d"XVgArjJ Eí +2R"v\7"\7v-;¡REí +3(\7"R)\7" rjJ.

Rev. Mex. Fú. 49 SI (2003) 38--43

(3)

(4)

(5)

(6)

40 SHIN'ICHl NOJ1RI AND SERGEI D. ODlNTSOV

3. Dilatonic ql1antum brane-worlds

Lct liS considcr lhe solution of ficld equations foc lhe lwO-brancs model. FirSl of a1l, ooe defines a new eoordinale z by

(11)

(14)

( 13)

(12)

ql4

p, = 'fJ3 .

( 10)

1a = - -, PI i' ",v'u ,P,

!_3qlYo - 4'

1a="'J3'

2 3PI -+ 2'

3f(y) = 2e,

-, - ~kyP,(b) In case of k> O, e, == e2/p~ should be positive and lhereis an ouler hrane solution, at le"sl if F2 (c2/2k) 2: -8b',whcrc

Case '2(a) hulk solution

amI y:f:. is givcll by

yJ == 3~l(1'"JI _ 4~;)(e) Solulion for k = O

</>(y) = PI In (P2y)12

<I>(</»=-p+elexp(a</»+e2exp(2a</» (9)

Case 1(a) hulk solution

6kplP;e, = 3 _ 2p; ,

f(y) = 3 - 2p;4ky

(b) \Vhen k i' O ami P; < 2, there is an ouler brane solu-tion if F, (y+) 2: -8b', and there is an inner brane solution ifFI (y_) ::::81;-'Here F, is defined by

4

= f(y)dy2 + y L iiij(xk)dx'dxi (8)i,j=l

Here [¡ij is lhe melrie of lhe 4 dimensional Einslein manifoldasin(l).

Herc we only summarize lhe oblained resulls (for moredelails, see ReL 21). Oenerally the obtaincd bulk Solulionshave the fonn:

z = J dy J f (y) ami solves y wilh respeel lo Z. Then lhewarp faelOr is e2A(z,k) = y(z). Here one assumes lhe mClrieof 5 dimensional spacetime as foIlows:

d ' - d .d vs - g(5)~1J X X

(7)b" = O.

b' = _ 12N' + 12N - 124(471')2

One can wrile Ihe corrcsponding cxprcssion [oc dilatoo cou-plcd spinor maller 114J whieh also has non-trivial (slighilydifferent in form) dilatonie eontribution 10 CA than in ease ofholographie conformal anomaiy 113J for N = 4 super Yang-Milis theory.

\Vc can also considcr lhe case whcrc thcrc are lwObrancsat z = Zo and z = zo, adding the aetion eorresponding 10 lhebrane at z = Zo 10 the aClion in (2).

Hcrc !he quantitics in the 5 dimensional bulk spacctirnc arespccified by the suffiees (5) and !hose in !he boundary 4 di-mensional spaeelime are spceified by (4). The factor 2 infrom of St in (2) is coming from !he fael !hat we have lwobulk regions which are connceled wi!h eaeh o!her by thebrane. In (4), n. is lhe unil vcelor normal 10 the boundary.In (4), (5) and (6), one chooses !he 4 dimensional bound-ary metric as 9(4).v = e2Aii.v. \Ve should distinguish Aand ii.v wi!h A(z) and ii'j in (1). The metric ii'j is givenby ii.vdx.dxv == 12 (da2 + dOl). \Ve also specify the quan-lities given by ii.v by using -. G (e) and F (i) are the Oauss-Bonnet invariant and the square of the \Veyllensor.

In !he cffeet;ve aClion (6) indueed by brane quantum mat-ter, with N sealar, N,/, spinor, N, veetor fields, N, (= O nr1) gravilons ami NIID highcr dcrivativc confonnal scalars, b,b' ami bU are

b = N + 6N'/2 + 12N, + 6UN2 - 8NHD

120(471')2

b' = _ N + UN,/, + 62N, + 14UN2 - 28NllD360(471')2

Usually, bU may be changed by the finite renormalizalionof a local countcrtcnn in lhe gravitational cffcctive aClion.As il was the ease in ReL 10, the lerm proportional to{bu + ¡(b + b')} in (6), and therefore bU, does nol eonlributeto the equations of motion. Note lhal CFf maller indueed ef-fective aClion may be eonsidered as brane dilatonie gravily.

For lypieal examples motivatcd by AdS/CFf correspon-dence 191one has: a) N = 4 SU(N) SYM theory:

e N'-1b = -b' = '4 = 4(471')2,

b) N = 2 Sp(N) theory:

b = 12N2 + 18N - 224(471')2 '

Re\~Mex. Fís. 49 S I (2003) 3X -43

ADS/CITCORRESPOND)iNCE Ai'D QUANTUM I~DUCEDDILATO:\'IC \1UITl.BRANE-WORI.DS

F ( ) = _3_ (YO )2e2 - 4kyo _ Y5 klyo _ le,),Yo - IG"G 2 3 21 + 4 24'

41

( 15)

(e) In case of k < O, F,(y) has atlcast one mimimum if e, < k'l' (..j3fi - 1) or e, > O. If lhe value of F,(y) allhe

maximum is larger lhan -8b', lhere is an ouler brane solution. If e, > Oand F,(O) < 8il or e, < O ami F2 (e,/2k) < 8b',there can be an ¡nner brane solutioll.(d) In case of k = O, if e, > O, lhe solution is given by

Case 3(a) bulk solution

Yo = l,¡c:; ( ~:1: _1) .2 V3 .j3 ( 16)

1e, = 3kpl, a = :1: .j3 ,

¡(y) = 21y'P28JY (elY + 7kjP2jj)

.j3PI = '1'2'

( 17)

(b) When el '" el / y'P2 < O, k > Oamllhere can be outer brane solution if F3 (49k' /(1) > -8b', where

3 [ YoF3(yo) '" -- --IG"G 2yo

( 18)

(e) When el > Oami k < O, lhere always exisls onler brane solution if F3 (49k2/e;) > -8b'.

Frorn the above results in case 1 3. we find that therevery uften appear Lwo(inner ami outer) branes solution as inlhe firsl modcl by Randall ami Sundrum 111. Moreover, thebranes may be eurved as de Siller or hyperbolie spaee whiehgives lhe way ror an ever expanding inflationary Univcrse.Such Solulions ofLen can exist even if Lhereis no any quan-tum effecl. i.e .. b' = O.

LeL us make few remarks on the forrn of rnelric. If uneeonsiders lhe metrie in lhe form (1), lhe warp faelor e>A(z)

docsnot bchavea~an exponenlial fUTlctiollof z bul asa pO\\'erof z. This would rcquirc thal we nccd a regioll (of compleLespaeelime) where lhe polenlial aod the dilalon beeome almoslconstanl. h results in difficulties when one lries lo explain Lhehierarehy using lhis model.

Hence. wc presented a number of dilatonic (inf1alionary,Oal or hypcrbolie) two brane-world Universes which are ere-aled by quamum effeels of brane mallero Somelimes, suehUniverses may be realized due lo a speeifie choice ofthe dila-tonie potenlial even allhe elassieal level.

In some papers (forexamplc in Ref. 18), the solution wilhmany braneswas proposed. In sllch model, Lhereare two AdSspaceswith differeTlLradii or diffcrenL values of lhe cosmo-logical constants. They are glllcd by a brane. whose tcnsionis given by lhe differenee of lhe inverse of the radii. In lhe~olution. the value of dA/dz in lhe melric of the fonn (1)jumps allhe brane, whieh lells the value of ¡(y) in lhe metriechoice in (8) jumps on the brane sinee J ¡(y) = dz/dy =1/(2y(dA/dz )J. Imagine une ineludes lhe quanlllm effeels on

the brane. Then one can. in general, gllle two AdS-like spaceswith same valuesof the coslllological constan l. Lct liSassumethat thcre is a brane at y = Yo and there are lwo AdS-likcspaces in y > Yo ami y < Yo glllCd by lhe hrane. Qne nowdenotes lhe qualllily in lhe AdS-like spaee in y> Yo (y < Yo)by lhc suffix + (-). Then wc can oftcn generalil.e lhe lwobrane-world lO the Illulti-brane case.

4, Discussion

In summary, wc presentcd lhe generalil',•..nion of lhe quantllmdilatonie brane-world 1101 where lhe I1rane is nal, spheri-cal (de Sitter) or hyperbolic ami iL is induced hy quanturneffects of CFT living on lhe brane. In lhis gencralizationonc may have two brane-worlds or even rnulti.brane-worldswhich provcs lhe gencral character of sccnarios suggcstcd inRef. 5 ami 6 where instcad of arbitrary brane tension addedby hand, Ihe effeclivc branc tension is prodllced by houndaryquantuITI fields. What is more interesting is lhaLlhe bulk so-lutions llave analylical form, at Icast, for the spccific choiceof bulk potcntial under cOTlsideratioTl.

In classical dilatonic gravity lhe variety of brane-worldsolutions has been presellted in Ref. 16 where also lhe ques-tion of singularities hasoeendisclIsscd.The f1ne-tllllcdcxalTl-pIe of a bulk potcntial where one gcts a bulk solution whichis nOl singular has been presented. In Ollr solutions, lhe CUf-

vaturc singularily appcars al y = o.

He\'. Mex. Fís. 49 SI (2003) 38--43

42 SHIN'ICHl NOJlRI AND SERGEI D. ODlNTSOV

In ea,e 1, when y ~ O and lhe eoordinales besides yare fixed, lhe infinilesimally small dislanee ds is given byds = V!dy ~ dy/(q,fij), whieh tells lhat the distanee be-twcen the branc and the singularity is finitc. Then in cases ofk = O and k < O, the singularity is naked when we Wiekre-rotale spaeetime 10 lhe Lorenlzian signature. When k > O,lhe singularity is nO!exaelly naked afler lhe Wiek re-rotationsincc the horizon is given by y = O, i.e. the horizon coincideswith the curvature singularity.

In case 2, lhe siluation is nO! ehanged for k = O, k > Oand k < O wi!h C2 > O from lhal in case 1 and the dis-tanee belween lhe brane and lhe singularily is finite sineeds ~ (dy/,fij)J3/2c2 when y is small. When k < OwilhC2 < 0, however, the singularity is nOl naked sincc thcrc is akind ofhorizon al y = c2/2k, where 1/f(y) = O.We shouIdnO!e lhe sealar eurvalure R(5) is finile. This lells lhal y is nota proper eoordinale when y ~ c2/2k. If a new eoordinale '1 isinlrodueed: '12 == 2 (y - c2/2k), !he metrie in (8) is rewritlenas follows

2 3 2 (C2 1]2) ~,. k i .ds = - 4kdr¡ + 2k +"2 .~ g¡j(x )dx dx'. (19)

1,]=1

The radius of !he 4d manifold wilh negative k, whose met-rie is given by {j¡j, ha, a minimum c,f2k al '1 = O, whieheorresponds lOy = c2/2k. The radius inereases when 1'11 in-creases. Therefore lhe spaeetime can be regarded as a kind ofworrnhole, where lwo universes eorresponding lo '1 > Oand'1 < O, respeetively, are joined al '1 = O.

In case 3, lhe singularity is naked (lhe singularily is nolexaeUy naked when k > O as in case 1) in general and !he

(a) A similar mechanism for anomaly driven inl1ation in the usual4d world has been invenled by Starobinsky (15] and genera1-izcd for dilaton fields in Ref. 17.

(b) For the introduclion to anomaly induced effectivc action incurvcd space-time (wilh torsion), see Seco 5.5 in Rcf. 12. Thisanomaly induccd aetion is due to brane CFT living on lheboundary of dilatonic AdS-like space.

1. L. Randa)] and R. Sundrum, Phys. Re\!. Lel!. 83 (1999) 3370,hep-th/990522l.

2. L. Randall and R. Sundrum, Phys. Rev. Lel!. 83 (1999) 4690,hep-thJ9906064.

3. A. Chamblin and 11.5. Roall, Nuel. I'ltys. B 562 (1999) 133,hep-lhJ9903225;N. Kaloper, I'ltys. Rel'. 060 (1999) 123506,hep-th/9905210; A. Lukas, B. Ovrul, and D. Waldrarn, Phys.Rev. 061 (2000) 064003, hep-lhJ9902071;T. Nihei, I'ltys. Leu.B 465 (1999) 81. hcp-lhJ9905487; 1I. Kim and 11.Kim, I'ltys.Rev. [) 61 (2000) 064003, hcp-th/9909053; D. Chung and K.Frcese, Phys. Rev. [) 61 (2000) 023511; J. Garriga and M.Sasak.i, hep-thl9912118; K. Koyama and J. Soda, hcp-th/9912-118; J. Kim and B. Kyac, Phys. Lelt. B 486 (2000) 165, hep-thlOO05139; R. Maartens, D. Wands, B. Bassctt, and T. Heard,hep-phJ9912464;S. Mukohyama. hep-thJOO07239;L. Mendesand A. Mazurndar, gr-qc/OO09017.

distance between ¡he brane and the horizon is finite cxceptthe k > O and Cl < O case sine e lhere is a horizon al,fij = -7 k / Cl where the sealar eurvature is finile.

The priee for having analytieal bulk result, (exaeUy solv-able buIk pOlential) is !he presenee of (naked) singularity.One can, of eourse, presenl the fine-luned examples of bulkpotential as in Refs, 10 and 16 where !he problem of singu-larily does not appear. Moreover, bulk quantum effcct, maysignifieally modify classieal bulk eonfigurations [6,19,20]whieh presumably may help in !he resolution of lhe (naked)singularily probIem. However, in sueh situation there are noanalytieal bulk solutions in dilatonie gravity.

Thcre are various ways lo extend me resulte; oC pecsentwork. Fin~t oC aH, one can construct multi-branc dilatonic 80-

lutions within ¡he current scenario Coc anathce cialises oC bulkpotential. Howcver, this requires the application of numeri-cal melhods. Seeond, il would be inleresling lo describe thedelaiIs of brane-world anomaly driven innation (with non-trivial dilaton) at late times when it should deeay to stan-dard FRW cosmology. Third, within a similar sccnario onecan eonsider dilalonie brane-world blaek holes whieh are eur-rcntly under investigation.

Acknowledgments

SDO would like to lhank lhe organizers of the Fourth Mex-iean Sehool: Membranes 2000 especially O. Obregon and 1.Socorro for !he kind possibility to present an invited lalk atthe Sehoo!.

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