PLENARY'Ii\LKS REVISTA MEXICANA DE FfslCA 49 SUPLEMEIoo"fO 2, 61~>6

B1ack brane entropy from branc-antibranc systems

Dlf H. DanielssonInstitutionenJór Teoretisk Fysik

Box 803, SE-751 08 Uppsala. Swedene-mail: ulf@teorJys.lm.se

Alberto GüijosaDepartamento de Ffsica de Altas Energ{as

Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de MéxicoApartado postal 70-543, 045IO México, D.F.. Mexico

e-mail: alberto@nuc/ecu.unam.mx

Martín KruczcnskiDepartment o/ Physics, Universily oJ7'oronto

60 Sto George st .• Toronto ON, M5S IA7 Canadae-mail: martink@physics.utoronto.ca

Recibido el 25 de abril de 2002; aceptado el 27 de agosto de 2002

SEPTIEMBRE 2003

In the context of string theory, il is ¡x>ssiblc to cxplain the microscopic origin of the cntropy of ccrtain black holes in leems of D-brancsystems. "lo date. most of the cases studicd in detail refee to extremal (supcrsyrnmctric) or ncae-extremal black holes. In this work we propasea microscopic modc1 for ccrtain black branes (extended versions of black holes) which would apply lo cases arbitrarily far from extremality,inc1uding the Schwan_"child ca..;e.Thc modcl is based on a system ofD-brancs and anti-D.branes, and is able to reproduce several propertiesof the corresponding supergravily solution. In particular, the microscopic entropy agrecs with supcrgravity, except for a factor of 2p/p+l,wherc p is the dimension of Lhebrane.

Keywords: Thcory {lf fundamental suings; quantum aspccts of black holes

En el contexto de teoría de cuerdas es posible explicar el origen microscópico de la entropía de ciertos agujeros negros en términos desistemas de D-branas. llasta ahora, la mayoría de los casos estudiados en detalle se refieren a los hoyos negros extremos (supcrsimétricos) ocasi extremos. En este trabajo se propone un modelo microscópico para cierta'i branas negra'i (versiones extendidas de agujero negro) que seaplica a ca'iOSno extremos, incluyendo Schwarzschild. El modelo se basa en un sistema de D-branas y anti-D-branas y permite reproducirvaria..;propiedades de la solución de supergravedad correspondiente. En particular, la entropía microscópica concuerda con supcrgravedad,excepto por un factor de 2v/p+ I donde p es la dimensión de la brana.

Descriplores: Teoría de cuerdas fundamentales; aspectos cuánticos de agujeros negros

PACS: Il.25.-w; 04.70.Dy; 11.27.+d; Il.lOWx

1. Strings, D-branes, and entropy

String thcory replaces poinl partides with strings, one-dimensional objeets whose tension TF ;: 1/2,,1; defines adimensionful parameter 1" known as the string lengtll. Besi-des moving as a whole, a doscd string can oscillate in dif-ferent ways, and upon quantization tIlese intemal modes gi-ve rise LOa perturba tive speetrum eonsisting of an infinitetower of states with masses m2 = 4N/l;, N = 0,1, ...At tIle bollom of tIle tower tIlere are mass!ess states, eo-rresponding to a graviton h.v, an antisymmetrie tensor ficldBlJ.lJ' a scalar cp (the dilatan) whose vacuum expectation va-lue determines lhe string eoupling constant g, = exp(

62 ULF H. DANIEI..SSON, ALBER'TO GÜUOSA, AND MAR"liN KRUC¡;ENSKI

la assigns!he blaek lhree-brane an entropy

SBIl = 25/'3-3/'v'rrQVI/4¿;M3/' + ... , (1)

where the dOl•.•denote coereetions which are higher order in¿;M.

On me microscopic side, the system is understood to Oca Slaek of N 03-branes plus a gas of open slrings. The mas-slessopen string molles should make the dominant cOTltribu-tion to the cnlropy, anl1 so we can identify me cxcess masswith the energy of a gas of mal)slessparticles (gluons, sea-lars, and superparlners): ¿;M = nb(".2 /16)N2VT', wilh Tthe temperalurc 01'lhe gas and nb me number ofbosonic (ma-lrix) degrees of freedom (lhe numeriea] eocffieienl !henlakestheir fennionic partners into account). Thc entropy of the sys-tcm is entirely due to the gas,

Sg = nb ".2 N2VT' = n:/4: v'rrQV1/'¿;M3/4. (2)12 3

The funelional dependence of Sg and SUIl on ¿;M amI Qis identieal; lbe nUlllerieal eoeffieiel1l agrees if nb = 6 [3).This number sltould be eonlrasled willt lhe g,N « 1 resu1l,nb = 8, arising from the two transversc polarizatiolls of thegauge field Aa amI lhe six sealars

BLACK BRANE ENTROPY FROM BRANE-AN'l1BRANE SYSTEMS 63The laehyonie mass of 1> expresses an instability of lhe

system. lhe meaning of which bccomes c]ear upon examina-lion of the taehyon potential [12),

The pOlenliai (4) is of lhe 'Mexiean hal' lype: il has a ma-ximum V = 27.3al 1> = O (lbe unstable 'open slring' vaeuum)and a minimum V = O at i1>1= V" /2 (Ihe 'c1osed s!ring'vacllum). From lhe cTlcrgy differcncc betwcen the lwo vacuawe deduce lhat condensation of the tachyon fmm the fonner10 the lalter corresponds lo lhe annihilation of lhe brane amilhe antibranc, which in particular implies me disappcarance01'all open slring degrees of freedom [13].

If wc now consider this same system al a finite tempcra-ture, standard lhermal field lheory rea,oning tells us whal loexpeel: a smalllemperalure should lead lo an effeetive pOlen-lial in which lhe location of the minimum ha~shiftcd awayfrom l1>i = V,,/2. The physieal rea,on for lhis is thal mo-ving towanls 1> = O can be lhermodynarnieally favorable: ilcosL' energy, bUl it also reduces lhe ma'5 of the taehyon andtherefore inercases the entropy of lbe laehyon gas. The 0pli-mal configuration is the ooe that minimizes the free cncrgyof lhe systelll, amllhis will vary with the temperalure.

LClllS now detennine whclhcr and under what conditionslhe oren string vactlutn, t = 4> = 0, can be a minimum of theeffeetive pOlenlial. When lhis happens, lbe 'taehyon' willnolongcr be tachyonic, and so (al sub.string-scale tempcraturcs)lhe massless fields will make the most importanl conlributionto lhe free energy. If one slarL, at 1> = O, lhen sliding downthe laehyon pOlential (4) lowers lhe energy of the system, bulil also givcs mass to lhe relalive gaugc ficlds, and so decre-ases the entropy of lhe gas. Wc are intereslcd in eSlablishingwhich ()f lhese effccts dominates.

Our system consisl"; of N brane-antibranc pairs, a tach-yon condensate, ami a gluon (+ lransversc scalars + super-parlners) gas. Ils free energy al temperature T = {3-1 is gi-ven by

which wc have cxpressed in tenns of me gravitational cou-pling" '= VCN/S". If one tries to inlerprel this as the en-lrOpy of a gas of particles with energy E = M some well-known problems arisc, First, sincc me power of E is largerthan one, a simple lhermodynamical calculalion shows lhatlhe spccific hc..'11is negativc. Second, by cxlensivily lhe ex-poneTll of V is forced lo be ncgative, which implies lhat mepressure p = T(DS/DV)M is negative. Finally, when lheblaek brane evaporales eompletely through Hawking radia-!ion, aH of il~excilulions disappcar, meaning mal Ihe fieldlheory whieh describes it mieroseopieally should have a va-cuum wilh 110dcgrees of frccdom (01hcr than c10sed strings).

As we will sec, lhese same properties are in facl posses-sed by D-branc-8nli- D-branc systcms, which have bccn mu-eh sludied of lale [7[. An anti-Op-brane (Op-brane) is simplya Dp-branc wilh rcvcrsed oricntatinn-il has the same massM = TpV as lhe O-brane but opposite eharge, Q = -1. Asystem wilh N 03-braoes and the same numberof03-branesis therefore a nalural candidale lo describe the neulral blacklhree-hrane [SI. We will slmly lhis system in lbe nexl seelion.

Other string-lheorctic attcmplS lo givc a microscopic des-eription of Sehwansehild blaek holes have been made pre-viously, most nntably via the stringlblack hole corresponden-ce [9J, and in lhe Matrix lheory [10J eonlexl [11].

2. Branc-Antibranc systcms at finitetemperature

The exeitalions of a 03-03 pair are deseribed by open stringswhose endpoinls are anehored on lhe branes. The 3-3 and 3-3" strings give rise lo the usual massless gauge fields and sca-lars {Aa,;}, {A",;¡;;} (plus superpartners) on the bmneand lbe anlibrane. The 3-3 and 3-3 slrings, on lhe olher hand,yield a eomplex sea lar field 1> with negalive mass-squared(i.e., a lachyon), ami addilional masslcss fennions. Since iloriginales fmm slrings running between lhe brane ami lheamibrane, the lhe laehyon is eharged under lhe relative U(l)(i.e .. A; '= A. - A.), bul is neutral under lhe overall U(l)

+ - -(A. = A. + A.).

V(1)) = 2T3exp[-2[t(1)WJ,

with t relaled to 1> lhrough an error funelion,

11>1= /f¡Erf(ltl).

(4)

(5)

, 4" 1=' [1 - e-.,jz'+l3'm,]F(1),{3) = 2T3Tre-21tl + -( )3eN2{3-4 dxx21n --~=== ,21r o 1 + e-Jx2+IPm2

(6)

Iwhich is positivc for large enough lemperature. Disregardinglhe Tlumerical constants, we thus leam that for

the open slring vncuum is a minimum of the free energy(equivalcntly, a maxirnum of the entropy for fixcd lotalenergy). amllhe orane-antibrane pairs do not annihilate,

where t is an N x N matrix, m - 11>1 is lbe mass given lo lhegluons by the Higgs effeet, ami the numerical constant e = 8for lhe relevant 8 hosonic +8 fennionic degrees of freedom.Slarting al 1> = O and letting a single diagonallaehyon con-dense by an 31ll0unt ó4J gives maloiS to N out ofthe N'l spcciesof parlidcs in Lilegas, ami so changcs (6) by

T> 1- .,j y.NI.

(8)

Rev, Mex. Fú", 49 S2 (2003) (i I (í(i

64 ULF H. DANIELSSON, ALBERTO GÜUOSA, ANO MARTlN KRUCZENSKl

Given lhis sel of assumptions, !he energy and entropy ofour mieroseopie system follow a~

15151 1~8FT =n;2'3-'5-'7l"jKV-'MFT, (12)

whcrc wc have cxprcsscd thc D3-brane tcosion in tenns ofthe gravilalional eoupling constan!, T3 = .¡rr /1 due lo a thermal expeelalion value for !he relativegauge fields, (A - A -}T ~ NT2, eorresponding to ama"m" ~ vg;NT.

lt is important lo note lhal !he regime (8) lies in !hephysieally aeeessible sub-string-seale region only for non-perturbative values of !he eoupling, y,N > 1, where wewould expeel lhe system lo have a dual supergravily deserip-lion. The system under consideration bccorncs then a viableeandidate for!he desired mieroseopie deseription of a neutralblaek lhree-brane.

3. Micro5COpic rnodel

Given lhe eonsidertions of the previous seelions, lhe modellhat we eonsider [8] is !he low-energy lheory on lhe world-volume of a system of N D3-branes and N anti-D3-branes.Sinee lhe number of uneondensed branes and antibranes canvary, the actual values are ehosen lo maximize lhe lotal en-tropy of lhe syslem for fixed eharge and ma".

The lemperature is assumcd (and later eonfirmed) lO sa-tisfy 1/ vg;N « T « 1 in string unit~. As seen in (8), !hefirsl inequality is required for stabilily: !he seeond one allowsus lo ignore the ma"ive open slring modes. For !he aboyelemperature range to exisl we musl have y,N » 1, so weare nceessarily in the strong-eoupling regime. We will takey, « 1 to suppress closed string loops. Under lhese eondi-tions, when a brane-antibrane pair annihilales lhe energy goeslo !he gas of open strings on the remaining branes and anti-branes, ralher lhan being emittcd as closed slrings (Hawkingradiation), sinee!he latter process is disfavorcd for small y,.

Sinee we are trying 10 formulale a mieroseopic model fora supergravity solution, !he restrietion lOstrong eoupling wa~of eourse expected.1n lhe absenee of a weakly-eoupled regi-me, and knowing thal !he lheory is nol supersymmetrie, !hebest one can do is lO use plausibilily argument~ lOdeterminelhe entropy. From lhe strueture of lhe brane-antibrane world-volume !heory one can show !ha!, if O(N2) ficlds are exeilcdon !he branes and anlibranes, !hen !he laehyons and !he fer-mions aequire ama" of order vg;NT, whieh is in fael whalwe obtained (for!he taehyon) in See. 2. [fsueh is!he ea~e, !hefaet !hal (for y,N » 1) !he lemperalure is mueh lower lhanlheir mass means lha! lhese fields are nO!exeiled, and lhe !he-ory on lhe branes dceouples from the !heory on !he antibra-ncs. Wc then havc two copies oE (3 + l)-dirncnsional N = 4U(N) SYM. (The supersymmetries preserved by eaeh eopyare different, so !he overall syslem is nol supersymmelrie.)As we have seeo io See. 1, at strong-eoupling!he AdS/CFTeorrespondeoee [4] prediet~ !hal the eotropy of eaeh eopy islhe same as thal of a frce ficId lheory wi!h 6N2 bosons aodthe same number of fermions [3].

Rev. Mex. n•.4952 (2003) 61-66

BLACK BRANE ENTROPY FROM BRANE-ANTIBRAI'E SYSTEMS 65

As discussed before, il is imeresting lO consider notonly me mass bul also lhe omer componenLs of lhe energy-morncnlum tensor, which luro out to be

where Eg = 3,,2N2VT4/8 (E¡; = 3,,2Ñ2Vt4/8) is meenergy of me gas on me branes (antibranes). Notice lhal, sin-ce me meories are decouplOO, the lwO gases could a priorihave differenllemperalures, l' # t.

Lel us now consider me supergravily expressions, follo-wing a procOOure similar lOReL 15. The energy, emropy andeharge of a non-extremal mree-brane are given by

MFO' = (N + Ñjr,V + Eg + E¡;, (16)

SFT = 2~3-¡"ívt (EgVN + E¡;/N), (17)QFT = N - Ñ, (18)

TIA. = -(N + Ñjr, + 3~ (Eg + E¡;), (19)

Before moving on 10 me eharged case, let us discuss aninleresting issue lhal appears alrcady here. Sinee we repro-duce lhe blaek brane enlropy, il is clear mallhe speeific heatof our syslem is negative. To undersland why, lel us con si-der how Hawking evaporation proceeds in mis model, andcheck mal lhe lemperalure inereases when energy is radia-ledoWhen a closed string is emiued, energy is laken from meopen string gas, so a priori me lemperalure should deerea-se. However, we have found mal, in equilibrium, lhe energyin lhe gas is 3/2 me lension of lhe D-D pairs. This mcansthal, when me gas has 10Slenough energy so as lo maleh 3/2lhe tension of N - 1 pairs, il will be entropieally favorable[oc one paie lO annihilate. giving energy 10 the gas and ¡nc(ca-sing iLSlemperalure_ This is repealed again and again, effeeti-vcly incrca')ing the tcmpcraturc 00 average al) the ma'iS of thesyslem decreases. The process will conlinue until g,N ~ 1,wherc me ga'i tcmpcralure bccomes oí arder ane in st.ringunils, and me model (16)-{18), based only on lhe masslessopen slring modes. cea ses lo be valid. Allhis poinl we wouldexpecl all of lhe available energy lO go inlo a highly-excilOOlong string, so me brane-amibrane model makes contacl Wilhlhe stringlblack hole correspondence [9].

lt is straightforward lo generalize lhe modello lhe case ofa eharged black brane, where me numbers of D3-branes andanli-D3-branes are no longer equal, N # Ñ:

(27)

(25)

2 1 1-- = -+-=,TFT l' T

where TFT is me lemperalure defined as T'~l~= (8S/8M)Q(which is a faclor of 2'/4 smaller lhan me Hawking lempe-ralure,TII = l/"ro cosh a). 11would be inleresting lOtry lodctCCl these lWD distinct tcmpcraturcs dircctly in the super-gravily side, as was done for lhe DIID5 syslem in ReL 16.

As we emphasizOO before, lherc is nO!hing lo prevem usfrom poslulaling mal me lemperatures of lhe lwo gases aredifferenl, bUl il is nO!clear lo us why supcrgravily secms 10require mal me ir energies (or equivalemly, lhe pressures), belhe same. The equality of me lwo energies implies mal, asone approachcs lhe extremallimil M = Qr,V (i.e., a -t 00wilh ro ex e-a/2), me lcmperalure of lhe gas on me anli-branes grows wimoul bounds. Sincc me modcl is based onmassless open string modes, il is expeclOO lo be val id onlyif bom T and tare subslanlially lower Ulan lhe string scale.We expecl lhal as t -t 1;-1 , lhe energy available lo lhe gason me anlibranes gocs lOa highly excilOOlong string, whosecontribution lO lhe entropy is however negligible comparOOlo me gas on lhe branes.

As reviewOO in Sec. 1, in lhe near-extremal region it isknown mallhe black brane entropy can be preeisely reprodu-cOOusing a syslem wilhoul anlibranes [31, so as one approa-ches eXlremalily one would inluitively expccl a transition lothis c1ass of states. It is easy lO sce that lhe entropies of thcsclwo microscopic descriptions cross when M '" 6QT,V, whi-eh indccd suggest~a transition, with the brane-antibrane sys-lem being lhe preferred one further away from extremalily.

SSG = 2~3-~"ívt E¡ (VN + /N). (26)From (17), we see mallhis is lhe correel entropy for a gas

of particles on me N branes plus anolher gas on me Ñ anli-branes, bom wilh me same energy E. However, since the lotalenergy availablc for me gases is E, in lhe field meory modclwe have lO assign an energy E /2 lo each of mem, resultingin a mismalch in me entropy which is exactly me same asfound for lhe neutral case: SSG = 2'/4 SIT' Under me con-dition mal lhe encrgies in bolh gases are lhe same, one cancheek mal me expressions (24) are me ones lhal maximizelhe entropy for fixed charge and mass.

The facllhallhe energy densities of me lwo gases are lhesame implies mallheir lcmperalures T, tare differen!. Theyare relaled mrough

in tcnns of which the cntropy can be wriucn as

Comparing (20), (22) and (23) wim (16), (18) and (19)uniquely delermines

,,~ -,,~ 4 2N = _r4e2a N = -r e- a (24)

2~ O 1 2"" O 1

and identifies lhe energy E == Eg + E¡; of lwo lhe gases as3,,'

E=2K,2Vr~1

(22)

(20)

(21)

(23)

,,' ( 3)MSG = ,,2 rJV cosh 2a +:1 '2,,4

SSG = -2 rgV cosha,"•'" 4QSG = -ro sinh 2a."

.. [", 4 ( 1)] ..T~~= ,..,2TO -cosh2Q+ 2 811,Re\'. Me.x. Fis. 4952 (2003) 61-66

66 ULJ-:H. DANIELSSON, ALBERTO GÜlJOSA, A:":D MAKIÍN KRUC.lENSKI

However, il is hard lOsee how one could reLain!he agreemenlwith supergravity in the cntice paramctcr spacc: Lhe branc-anlibrane model gives lhe exact dependence of lhe black bra-ne entropy on M and Q, but gives a numerical value whieh isalways a factor of23/4 loo small, whereas the model ofReL 3is in accord with supergravity in the near.cxt.remal region, butdcvialcs significantly [rom it already al the prcsurncd lransi-tion poiot.

Another importanl propcrly of black branes is thal theyare generically unstable [17J. If lhe black lhrec-brane, forinslancc, livcs on a vcry large lOrus. then ii has lowcr cn-lropy than a ten-dimensional black hole, so lhe laller is lheprcfcrrcd configuration. It can be ShOWIl (8}that in oue mi-croscopic rnodel it is also convcnicnt lo reduce the sizc of thesyslem, bUlonly unli] it is of lhe order oflhe inverse lempera-lUre. The radius lhat maximizes the enlropy is found 10yicldan cntropy-energy rclalion for lhe syslem which is preciselylhat of a black hole in ten dimensions, as expecled from lhcsupcrgravity side.

Before closing, we should nOle lhat cnlirely analogousmodels [8] can be formulaled for !hc black two-brane andfivc-branc of eleven dimensional supcrgravity, whcrc lhe mi-croscopic descriplion is bascd upon M2-M2 and M5-M5 sys-lems in M-lheory. Using again the input from AdS/CFT, lheresults are in eomplele parallel wilh lhose of the D3-D3 case.

4. Conclusions

We have shown !hat a D-D syslem is sLable if lhe lemperaturesalisfies T > 1/,¡¡¡;NI, (which in lum requires g,N > 1).

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Building upon !his observalion and previous work [151, wchave formulalcd an cxplici¡ microscopie model for lhe blacklhrec-brane often-dimensionallype llB string lheory. Combi-ned wi!h inpul from lhe AdSICFT corresponden ce, !he modelyields an entropy which reproduces lhe supergravity result,up lo a puzzling faclor of 23/4. Since lhe AdSICFT corres-pondencc uses supcrgravity (in lhe ncar.extrernal region), theresult rnight appear to be rncrcly a consistency check, butthefacl is lhal we are describing a very differem regime (inclu-ding the Schwarzschild case), and a diffcrcnt situation, sincelhe number of branes can change. As wc have seen, our fieldthcory rnodel possesses several appealing [eatures which arein c10se corresponden ce with the propertics of black brancsand black holes in supergravily. Beyond the immediale taskof qucslioning its assumptions and attempting to resolve lhenumerical discrepancy in the entropy, wc belicve thal thercare several inleresting a'pects of lhe model which meril furl-her sludy.

Acknowledgments

AG would like lo thank lhe organizers of !he IV MexicanWorkshop on Gravilalion and Malhemalical Physics for theinvilation lo presenl this work [81. UD is a Royal SwedishAcademy of Sciences Research Fellow supported by a granlfrom lhe Knul and Alice Wallenberg Foundalion and by lheSwedish Natural Science Research Council (NFR). The workaf AG is supported by a repalriation granl from lhe McxicanNalional Council of Science and Technology (CONACyT).

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Re\~ Mex. Frs. 49 S2 (2003) 61-66