COURSES AND PLENARY TALKS REVISTA MEXICANA DE FÍSICA 49 SUPLEMEJ.""TO t, 14--18

B1ack holes and string theory

JUNIO 2003

Roben C. MyersDeparlmenl af I'hysics, McGil/ Universily, Manldal, Québec, Canada 1I3A 2n

Recibido el 22 de abril de 2001; aceptado c14 de mayo de 2001

This is a short summary of rny leclures given at the Fourth Mexican School on Gravitation ami Mathematical Physics. These lectures gavea bricf inlroduction lo black holes in string theory, in which r primarily focusscd on describing sorne of Ihe recent ca1culations of hlackhoJe entropy using Ihe statislical mechanics of J)-brane states. The following overview will al so provide lhe interested students with anintroduclion lo the relevant Iiterature.

KpYwnrds: Black holes, string theory, black hale enlropy

Este es un resumen breve de mi curso ofrecido en la Cuarta Escuela Mexicana de Gravitación y Física-Matemática. El curso conliene unacorta introducción a hoyos negros en teoría de cuerdas en la que me he concentrado principalmente a describir algunos de los cálculosrecientes de la entropía de hoyos negros, utilizando la mecánica estadística de estados de D-branas. El resumen presentado a continuacióntambién ofrece a los estudiantes interesados una introducción a la literatura relevante.

De.H:ripIores: I layas negros, leoría de cuerdas, entropía de hoyos negros

Pi\CS, ()4.70.Dy; 1l.25.Uv

1. Introduction

String thcory is a very broaú and extrernely rich arca of studyin lheoretical physics pursued by particle physicists, rnathe-malicians, ami rclativist<.;as well. \Vithin this communily ofstring Lheorisl<.;,there has long been a fascination with blackholes, ami studies of the latter have laken rnany differelllpoints of view, including:

l. Black holcs wilh string lhcory correclions 111.2. Black holes wilh quanlUm hair [2].

3. Two-dimensional black holes as WZW models 13J.4. Black holes in solvable rnodels of two-dimensionalgravily [41.

5. Black holes ami enwnglement entropy 15J.6. Blaek holes as low energy supergravity solulions [6].

7. Blaek holes as exaCl sigma model backgrounds [71.

8. Black holes as strings [81.

9. Black holes as D-branes 19-11].10. Black holes in Malrix lheory 1121.11. Black holes in lhe AdS/CFf correspondenee 113].12. Black holes as superconformal quantum mechan-

ies 114J.13. Black holes in brane world seenarios [151.

14. Blaek holes ami enhan,on physies [16].

The references ciled above are by no mean s complete. Con-sulting Paul Ginsparg's e-prinl archive [171, one finds thmin the pastten years, the high energy lheory (hep-th) seelionhas accumulaled in excess of 1600 papers about black holes.Above, 1havc only lisled a few reviews or salielll articles foreach lOpic 10 give lhe reader a briúgehead ¡nlo the relevanl

literalure. The interested students are encouraged lo explorethe associatcd refercnces ami citations of lhese papers withlhe Spires HEP database [18].

Clearly, 1 eould nol hope lo lel1 the full slory of blaekholes amI slring theory in two hour-Iong Iectures. Instcad 1only attempted lo introduce the students to the ninth itcm onlhe list abovc. That is, 1 describcd sorne of lhe recenl calcuvialions of black hole enlropy using lechniques involving D-branes. In particular, I focussed on lhe original calculations ofSlrominger amI Vafa [19J. These were the first ealeulalionsof any sort which successfully dcterrnined the Bckcnslein-Hawking cntropy with a statistical mechanical model in lerrnsof some underlying microphysical states. Therc are alreadyseveral extcnsivc reviews of the D-hrane description of blackhole rnicrophysics. In particular, I wOllld recommend thoscby Peel 1101 ami by Das ami Malhur [l1J. 1 would alsohighly recommend Juan Maldaeena's Ph.D. lhesis [91 as awellvwrittcn and pedagogical introduclion to this topic. \Vilhregards to furlher background referenccs, Clifford Johnson'sreview f20) of D-brane physics is very good. For a generalintroductioll to modem string theory, the standard referenceis now Polchinski's text [211. Inlerested sllldcnls may alsowish 10 look at a similar butlonger series of lectures OH blackholcs in slring theory, which I prescnted in Jcmsalcrn in theprcvious year [221.

2. SUllIllIury

In lhe early seventies, Bekenstein [231 made the bold eonjee-ture thal black holes carry an intrinsic entropy givcn by thesurface area of the horizon mcasured in Planck units rnultivplied by a dirncnsionless nurnber of arder one. In part, thisconjecture was mOlivalcd by Hawking's arca theorcrn [241

BLACK HOLES AND STRlNG TIIEORY 15

whieh had shown lhat, like entropy, lhe horizon arca of ablack holc can never dccrcasc in general rclativity.

Thc nexl crucial insight carne from Hawking while invcs-tigaling quantum fields in a blaek hole spaeetime [25]. Hefound mal cxlcmal observen; dClcct lhe cmission of thennalradiation from a blaek hole with a lemperalure proportional10 itl'i surface gravily.(a) /'\.:

In fact, such a loss of information violates unitary time evo-IUlion, one of lhe basie tenets of qu:tntum mceh:tnies, the lhe-ory whieh lead lo lhe blaek hole evaporation in !he firsl place.This paradox h:ts profound implieations as il w:ts originallysuggested lO indieate !hal quantum mcch:tnies and generalrelativily simply can not be combined in a consistent rnanner.II was long feh !hat a resolution of ei!her of !hese puzzleswould yicld sorne insighl inlo the nalure of quanlum gravily.This ís the esscntial source of the fascination which stringtheorisls and particle physieisls have for blaek holes.

Reeentiy, progress inlo these queslions has becn madewilh new insights from slring theory. This progress is a spin-off fmm lhe research into string dualities [28J and lhe re-alizalion of lhe important role of extended objcels beyondjusI slrings [29J. In particular, a elass of extended objcelsknown as Diriehlel branes or D-braJles [20] have provenvery valuable fmm a ealeulational sl1ndpoinl. These objeelshave a simple description in lhe framework of perturbative orweakly-inleraeling slrings, and yet lhey exhibit rieh dynam-ies, ineluding a wide variety of eomplicated bound slates.

In lhe low eJlergy or long wavelength limil, string lheoryis aeeurately deseribed by Einstein gravily eoupled to vari-ous kinds of mallero In my lectures, 1 focussed on whal isknown :ts !he l'ype I1b superslring !heory. In this case, onehas a ten~dimcnsional supergravity theory where the mat.ter fields inelude IwO sealars (lhe dilaton :tnd !he axion),lwo two.fonn potenlials, a fOllr-form potemial and variousfermions. As we saw in Don Marolf's lcelures [30], variouskinds of extended objeets can carry eharges under lhe formfields. The fundamenlal strings of lhe theory aet :ts !he elee-lrie SOUfees for onc of the two-fonn potentials, known as thcNS (Neveu-Sehwarz) two-form. The olher potential, the RR(Ramond-Ramond) two-form, ha\) e¡eetrie sources known asDl-branes, and magnetic SOllrces knowll as D5-branes (i.e.,lhese are D-brancs cxtended in one and five spatial dimen-sions, respeelively). From lhe full quanlum string lheory, weknow there is an analog of Dirac charge quanlízalion for or-dinary clcetrie charges and magnelic monopoles in four di-mensions 131), whieh requires lha! the RR two-fonn ehargescome in discrelc unill) {29]. Hcncc if a system carries a eertainRR eharge, one can use lhe eharge to eountlhe lotal numbcrof constituent D-branes that must have bcen used in m;seffi-bling lhe syslem,

In lhe leelures, I focussed on a particular farnily of blaekhole solulions in the l'ype I1b supergravity theory. From alen-dimensional poinl of view, lhese solutions describe blaekfive-branes earrying lhree distinetlypes of eharge, ineludingbOlh eleetrie and magnetie eharge wilh respccllO lhe RR lwo-formo Henee we can say lhat the blaek brane was formed bybringing logether some number of D l~branes aod D5-hranes,NI ami N •. Furlhermore, the deta;ls of lhe solution allow uslo infcr that all of the D5-branes wcre arrangcd in parallcl ona eornmon live-dimensional hypersurface and that, similarly,the O I-branes wcre parallel 00 a common line in lhis sur-face. In ordcr lhat thc resulting black brane has a finite mass,

(3)

(1)h"knT= -.2"e

which givcs a precise rclatioo confirrning Bekcnstcin 's carlicrconjccturc.

Howcvcr, thcsc rcvclatioTls about lhe thcrmal nalurc ofblaek holes lead lo two relaled puzzles. The above diseus-sion describes blaek hole entropy within lhe framework oflhcnnodynamics. whcrc it is associatcd with lhe energy in asyslem whieh is unavailable to lIo work, e.g., in Eq. (2), T 8Sindieates lhe hem loss in some proeess. For ordinary lhermalsystems, stmislieal meehanies provides a eomplementary in-lerprelalion of enlropy by taking into aeeounllhe mieroseopiedegrecs of frccdom of the syslem. In this eonlexl, entropy hasquile a different signilicanee. II is a measure of!he laek of de-tailed informmion about lhe mierophysieal stale of a syslem.However, in lhe case of blaek holes, it remained a loogstand-ing problem 10 find a statistieal meehanieal derivation of theentropy.

An even more dramatie puzzle is the blaek hole informa-tion loss paradox. Classical general rclativity says that what-ever falls inlO a blaek hole eannol afterwards be observedfrom the oUlsille, In principie lhough, we eould diseover whal[eH in by entering lhe hlack hole ourselves. However if quan-tum processcs cause the black hole to radiate away ilS energythermally so tha! eventually lhe blaek hole disappears, lhenlhe information ahaUl whal has fallen in is complctcly 10s1.

For a Sehwar1$ehild blaek hole, " = e' /(4G M) and so onefinds !hal llawking's result typieally eorresponds lO an in-crcdibly small lcmpcraturc: T '" 10-7 K foc a solar massblaek hole.

Prcviously, cxtcnsivc sludics of solutions of Einstein'sequations had eulminaled in the formulation of four laws ofblaek hole meehanies [27] . llawking's diseovery of a blaekhole lemperalure w:ts the key 10 realizing lhal lhese previ-ous resullS were the laws of lhermodynamies applied 10 blaekhales. For cxamplc, thcrc is a corrcspondcncc bctwecn lhefirstlaw iryeaeh of lhese frameworks:

é'" , 2-aA = e 8M +----+ T8S = 8U, (2)8"G

Here, Mé' is nalllrally idenlified with!he blaek hole's inler-nal energy, U. Henee given Hawking's relation (1) belweenthe surfaee gravity and lhe lemperature, the eorrespondeneebetwccn lhese lwo relations is eompleted by identifying [25)

3 AS = k"e _hG 4'

Rev. Mex. Fís. 49 SI (2£XJ)) 14-18

16 ROBERT C. \.1YERS

chargc ami horizon area, we imagine tha1 lhe dircclions inlhe ahove hypcrsurfacc are wrappcd on c¡feles lo form a fivc~dimensional 1001s. Thc third chargc is a momcntum alongIhe circIc common 10 lhe D )-hrancs ami D5-brancs. A stall-

da ni rcsult of KK (Kaluza-Klcin) lhcory Ís 111alsuch un in-ternal momcntum must also he quantizcd 132), ami so weuse N p lO denote lhe Tlumocr of lTlomcntum quama carricdhy Ihe solutioTl. From lhe point of view o[ lhe Cf[CClivc fivc-dimensional thcory, lhe Pcnrosc dingram for thcsc SOIUliollsis similar lo that of lhe Rcissncr-Nordstrom solution in fourdimclIsioTls 126J. Thc solution prcscTltcd is distinguishcd hylhe fae! thar thcrc is a supersymmctric Iimil in which lhe hori-zon arca rcmains finite ami the black brane becornes extremal(i.e., the surface gravity ami hence Ihe Hawking lemperaturevanish). Identifying black hole Solulions wilh these proper-ties is nOlltrivial as can he seen by the facI Ihat ir any 01'¡he ¡hree charges is set 10 zero, lhe horizon is replaced bya llulI singularily in lhe supersymmetric limil. Evaluating Ihehlack hole entropy according to the Bekenstein-Hawking for-Illula (3) yields

(4)

Note tha! the right hand side is apure number that does noldepend, e.g., on Ihe details nf Ihe compactificalion to f1vedi.mcnsiolls.

This result (4) for lhe classical supergravity solution re-líes onlwo inputs from the underlying Type I1b string llleory.Thc I¡rst was the charge qtlantiz.alion conditions alluded 10

ahove, and lhe second was a formula for Newton's constmIl intell dimensiolls: lÚ1rG = (21r)7 g/-ff> 8. Here Newtoll's con-stanl is cxprcssed in lenns of [wo parameters arising in thepcrturhalive string lheory: g¿¡, the string cOllpling COIIstaT1l,adimcllsionlcss parameler which descrihes the strength with\vhich the fundamental (closed) slrings interacl, and f¿¡, Ihestring scale which can he rcgarded as the typical siz.e of afundamcntal string.

As mcnlioned previollsly, the D-branes can also be anal.ysed using the lechniques of perturhative string Iheory. Fromthis point of view, one is considering a parlicular bound st:lteof NI O I-branes and Ss 05-hrancs. The N p uniL~of mo-Illentum are earried by fundamenlal strillgs eonnccting theDI -branes am! DS-branes. 11should be evidenllhal lhere are01' the on1cr 01' NI N5 diffcrcnl spceics of strings that canserve in lhis role, ami further lhal Ihis mOITIenluITImay bepartitiolled mnongst the variolls string exeilalions in manydiffcrent ways. i.e., an individual string may carry anywherebetween I and Np units of mornenlum. Therefore the super-symmctric ground state of this bOIlTllISIate has a large dcgcn.eracy. 'D. Given (his degencraey, Orle can assign a stntisticalmeehanieal entropy lOIhe syslem aecording lo 5= 10g'D. Aprecise cvalllalion of the dcgeneracy lhen exactly reproduceslhe enlropy given in Eq. (4). Henee Ibis ealeulalion yields astriking agrecment betweeTllhe Hawking-Bekenstein cntropyami the slalislical elltropy of the D-brane microstates.

Now al this POill(, Ihe attenlive sludem musl have heenasking: what does a calculation of the dcgencracy 01' a O.brane bound state have lo do wilh a calculation 01' theHawking-Bekenstein entmpy of a black hole Solulion? Therelalion belween lhese calculations is Lhatlhe D-hrane ooundslate in perturbative string lheory ami the blaek hole in su.pergravity are actually complcmentary descriptions of lhesame syslcm, valid in diffcrellt regimes 01' lhe cOllpling. Oc-spile lheir simple deseription in pcrturbative string theory,O-hranes are nonperturhalive objects. This can be sccn fmmthe tension (energy density) for a single O.branc which is in.verscly proportional 10 Ihe slring coupling, T '" l/g¿¡. How-ever, Ihe gravilational footprinl may slill be small as (his in.vol ves eoupling the O-br:me lo gravity with Newton's con-slant. \Vith lhe resulL quoted aboye, one finds, foc examplc,ror a colleetion 01' 05.br:mes: r~ = GNsT '" Núg¿¡f¿¡2.

This radius r9 can be regarded as lhe length scale over whichthe Cllfvatures ami other fields are strong. Hence in a regimev.'here lVJj9s « 1, r9 is ¡ess lhan the string scale f¿¡ amI so is adistance that we can'l expect to resolve cffcctively in the pcr.turhative string lheory. In this regime, lhe D-branes are effee-lively deseribed by a perturbalive pielure where Ihe D-branesare rcprcscnled by a source in nal empty space whieh eou-pies \\/Cukly to the fundamental strings. On the other hand, in(he regime Ns9¿¡ » 1, ry is mueh larger than lhe string scalee¿¡ ami a non linear supergravity solution providing a hack-ground for lhe propagalion of lhe slrings is a bettcr deserip-lion of the physics. Note that lhis "slrong coupling" regimemay still have gf> « 1. Then the fundamental strings coupleweakly to each other but interact s(rongly with lhe col1ec(ion01' O-brnnes.

Therefore lbe perlurbalive S1ring pielure ami lhe blaekhole sollltion providc eomplementary pictures of the sameDI-OS hound statc. Olle poinl of view is that when y¿¡ (amihence Newton's cOllstant) is increascd aboye the weak cou-pling regime, the syslem undergoes gravilational collapseforming a black hole. The tlnal step in thc argumeTll is lhatIhe system under considcration is supersymmetric 1331. Su-pcrsymmelry plays :m esseTllial role hefe in that it guaran-Ices Ihal Ihe number of ground states is indepcndent of lhestrength of the slring coupling, i.e., the ground slate degener-acy is a kinematic quantity rather than a dynamical quantity.Therefore one can expect thal lhe calclllations in bOlh regimeswill yield lhe same resulL.

The original calculations of the five-dimensional hlackhole [191 were quiekly ex lended lo spinning blaek hnles 1341.four-dimensional blaek holes 1351 ano also near.exlremalblaek holes [36J. In Ihe laller case, where lhe lempcralure isslightly grealer than zero, one caTldevelop a O-branc modclfoc the proccss of Hawking evaporation {37). This micro.scopie model [38) even eaplures Ihe grey body faelors whiehmodify Ihe (hennal speclrum foc the particles radiated lOasymptotic infinity 1391. Por the near.extremal calclllations,one has 10Sl supersymmelry and so one musl modify lheargurnents which relate the results in the weak and strong

ReJ!.Mex. ns. 49S1 C(X)3) 14-18

BLACK HOLES AND STRII'G THEORY 17eoupling regimes [40]. Further, the robustness of these D-brane models is related 10 the AdS/CFT eorrespondenee [13].which comes into play in dcscribing the ncar.nocizon rcgionoflhe five-dimensional blaek holes.

Thcse resu!LI) rcprcscJllcd a majar brcakthrough in our UJl-derstanding of blaek hole mierophysies, as a statislieal me-ehaniea! interpretalion of blaek hole entropy had e1uded lhe-orelieal physieists for over 20 years following the diseoveryof Hawking radiation. While the paradox of infonnaliotl lossin black hole cvaporation rcmains unrcsolvcd, D.brancs sccmto provide a robust model of alleast eenain evaporating blaekhoIes and so the tools lo rcsalve this perplcxing paradox sccrnlo be at hand. In any evem, the present remarkable ealcula-lions alrcady provide further sanetion for slring theory as thethcory lo rcconcilc quantutn mechanics ami general rclativity.

(al The surfaee gravity may be lhought of as lhe redshiftedaccelcration of a fiducial obscrvcr moving just olltsidclhe horizon [26].

1. e.G. Callan, Re. Myers, and M.J. Perry, Nuel. I'hys.11311 (1989) 673; R Myers, Nucl. Phys. 11289 (1987)701.

2. S. Coleman, 1. Preskill, and F. Wilczek, Nuel. I'hys.11378 (1992) 175 [hep-th/9201059].

3. E. Witten , "On blaek holes in string lheory", [hep-Ih/91 I 10521 in Slrings and Symmelries, edited by N.Berkovitz el al .• (World Seielllifie, Singapore, 1992);H. Verlinde, "Blaek Holes And Strings In 1\vo Di-mensions". in String The01Y and Quantllm Gravity '91,edited by 1.Harvey el al .• (World Seienlifie, Singapore,1992).

4. 1.A. Harvey and A. Strominger, "Quanlum aspeets ofblaek holes", [hep-Ih/9209055] in Recenl DÍI~clions inParricle 7heory: From Superstrings and Black /-loles inIhe Slandard Model (lilSI-92), edited by lA Harveyand 1. Polehinski, (World Seientifie, Singapore, 1993);S.B. Giddings, ''Toy models for blaek hole evapo-ration", [hep-th/92091 13] in Slring Qllanlllm Grav-ily and Physics al Ihe Planek Energy, edited by N.Sanehez, (World Seicntifie, Singapore, 1993),

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Acknowledgments

We would like 10 eongratulale lhe organizers of the FounhMexiean Sehoo1 on Gravitation and Mathematieal Physiesfor arranging a very sueeessful gathering. 1 would also likelO lhank lhem for giving me the opponunily lo leeture attheir sehool, as well as enjoying lhe plca"nt surroundingsof Huatuleo. This work was supported in parl by NSERC ofCanada and Fonds FCAR du Québee. I would to lhank the In-slitule forTheoretieal Physies at UCSB for hospitality duringthe latter stages of writing lhese notes. Researeh at lhe ITPwas supported in pan by the U.S. National Seienee Founda-lion under Gram No. PHY99-07949. Finally I would like lolhank Neil Constable, Frederie Leblond and David Winters[oc carcfully rcading an carlicr draft of thcsc notes.

holkar, 1.P. Gauntlctt, 1.A. Harvcy, and D. Waldram,NlleI. l'hys. 11474 (1996) 85 [hep-Ih/951 1053].

9. 1.M. Maldaeena, Ph.D. Thesis. [hep-Ih/9607235].

10. A.W. Peet, 7i1Slleelures on blaek holes in slring Ihe-ory, [hep-th/0008241].

11. S.R. Das and S.D. Malhur, Ann. Re\'. NlleI. Parl. Sei.50 (2000) 153 [gr-qe/0105063].

12. T. Banks, 7i1Sl leelures on malrix Iheory, [hep-th/991 1068].

13. O. Aharony el al., Phys. Repl. 323 (2000) 183 [hep-th/9905111 ].

14. R Britto-Paeumio, 1. Miehelson, A. Slrominger, andA. Volovich, Leclures on ,mperconformai quantwn me-chanies and mr"li-black hole mod,,¡¡ spaees, [hep-th/9911066 ].

15. S. Dimopoulos and G. Landsberg, Blaek fIoles alIhe LIle, [hep-ph/0106295]; S.B. Giddings and S.Thomas,lligh energy eollidas as blaek hole/aelories:7he end o/shorl dislance physics, [hep-ph/Ol06219].

16. C.v. lohnson and RC. Myers, "l7re enham;on, blackItoles, and Ihe second law, [hep-thI0105159]; N.R.Constable, 7he enll'Opy o/ 4D black holes and Iheenhanr;on, [hep-th/0106038].

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University Press, Uniled Kingdom, 1998).22. Re. Myers, Blaek [[oles and Slring 7heory, lee-

lures at the 17th Jcmsa1em Winter Sehool in The-orelieal Physies: "Slring Theory al lhe Tum of

Rev. Mex. F[~i.49 SI (2003) 14-18

18 ROBERT C. MYERS

lhe Millennium", at the Hebrew University ofJerusalem, [srael, Deeember 28, 1999-January 6,2000[hu p://hug. phYS.huj i.ae.i l/willlersehooV].

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24. S.W. Hawking, Commlln. Math. l'hys. 25 (1972) 152:S.w. Hawking ami O.R.F. Ellis, 1'he Large Seale Struc-tllre oJ Spacetime, (Cambridge University Press, Cam-bridge, 1973).

25. S.W. Hawking, Commlm. Math. l'hys. 43 (1975) [99.26. R.M. Wald, General Relativity (University of Chieago

Press, Chieago, 1984).27. J.M. Bardeen, B. Carler, and S.W. Hawking, Com,mm.

Math. l'hys. 31 (1973) 161; see also Ref. 26.28. C. Vafa, "LeelUres on strings and dualilies", [hep-

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29. J. Polchinski, l'hys. Rev. I-el/. 75 (1995) 4724 [hep-Ih/95 100171.

30. D. Marolf, in these proceedings [gr-qeIOI051 15].31. Sec, for example, Seclion 6.12 of J.D. Jaekson, Clas-

sieal Electrodynamies, 2nd ed., (John Wiley and Sons,New York, USA, 1975).

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gapore, 1984): M.J. Duff, B.E. Nilsson, and e.N. Pope,Phys. Rept. 130 (1986) 1.

33. 1. Bagger and J. Wess, SIIpersymmelry and SIIpergral'-ity, (Prineeton University Press, PrineelOn, N.J., 1992).

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35. J.M. Maldaeena and A. Strominger, l'hys. Rev, Lel/.77 (1996) 428 [hep-thJ9603060J: e.v. Johnson, R.R.Khuri, and R.e. Myers, l'hys. Lel/. B 378 (1996) 78[hep-lhl9603061j; N.R. Conslable, e.v. Johnson, andR.e. Myers, J /l El' 0009 (2000) 039 [hep-thlOOO8226].

36. C.O. Callan and J. M. Maldaeena, NlleI_ l'hys. 8472(1996) 591 [hep-lhl96020431: O.T. Horowitz and A.Strominger, l'hys. Rev. Lel/. 77 (1996) 2368 [hep-Ih/96020511: OT_ Horowilz, J.M. Maldaeena, andA. Slrominger, l'hys. Lel/. ¡¡ 383 (1996) 151 [hep-Ih/9603109J; re. Breekenridge et al., l'hys. Lel/. ¡¡381 (1996) 423 [hep-lhI9603078]; O.T. Horowitz, D.A.Lowe, ami J.M. Maldaeena, l'hys. Rev. I-el/. 77 (1996)430 [hep-lh/9603195].

37. S.R. Das aml SD. Mathur, NlleI. l'hys. 1\478 (1996)561 [hep-th/96061851; NlleI. l'hys. 11482 (1996) 153[hep-lh/9607149].

38. J. Maldaeena and A. Strominger, l'hys. Rev. f) 55(1997) 861 [hep-th/96090261.

39. See, for example, N.O. Birrell amI P.e.w. Davies,Quantllmjields in cllrved space (Cambridge UnivcrsilyPress, Oreat Brilain, 1982).

40. J. Maldaeena, l'hys. Rev. f) 55 (1997) 7645 [hep-th/9611125].

Nev. Mex. Fis. 49 SI (2003) 14-18