Hagedorn versus Hawking-Page transition in string theory

PHYSICAL REVIEW D 68, 066012 ~2003!

Hagedorn versus Hawking-Page transition in string theory

Alex Buchel and Leopoldo A. Pando ZayasMichigan Center for Theoretical Physics, Randall Laboratory of Physics, The University of Michigan, Ann Arbor,

Michigan 48109-1120, USA~Received 27 May 2003; published 29 September 2003!

We study the supergravity dual to the confinement or deconfinement phase transition for theN54 SU(N)super Yang-Mills theory onR3S3 with a chemical potential conjugate to a U(1),SO(6)R charge. Theappropriate supergravity system is a single-charge black hole inD55 N58 gauged supergravity. The appli-cation of the gauge-string theory holographic renormalization approach leads to new expressions for the blackhole Arnowitt-Deser-Misner mass and its generalized free energy. We comment on the relation of this phasetransition to the Hagedorn transition for strings in the maximally supersymmetric plane wave background withnull Ramond-Ramond five-form field strength.

DOI: 10.1103/PhysRevD.68.066012 PACS number~s!: 11.25.Tq, 04.65.1e, 04.70.Dy

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I. INTRODUCTION

The gauge-string theory correspondence@1,2# relatesN54 SU(N) supersymmetric Yang-Mills~SYM! theory totype IIB string theory on the AdS53S5 background. One ofthe most interesting aspects of this correspondence is thallows for a comparison between quantities that are not ptected by symmetries. One class of these non Bogomol’Prasad-Sommerfield~BPS! quantities naturally arises bconsidering each side of the correspondence at finite tperature. For AdS5 in global coordinates the background gometry is dual to the SYM theory inR3S3, which opens thepossibility of phase transitions. These phase transitionsamong the typical dynamical processes one expects toable to address within the correspondence. There are vaarguments supporting the existence of a confinideconfining phase transition for the gauge theory in the sN→` limit. The nature of the transition, however, is vedifferent at small and large ’t Hooft coupling; namely, fol@1 this confinement-deconfinement phase transitionbe identified@3# with the Hawking-Page~HP! phase transi-tion in an AdS background@4#. On the other hand, atl50, itwas shown in@5# that the density of gauge invariant statesN54 SYM theory on S3 exhibits Hagedorn behavior. Baseon these arguments, Polyakov@6# suggested an interpolatinformula for the density of states:d(D);eb(l)D, withb(l);const for smalll andb(l);l21/4 for largel.

A very peculiar behavior is expected for the free energythe gauge theory. Thel50 calculation@5# suggests generically a partition function near the Hagedorn transition of tform

Z~x!'1

~xH2x!j8~xH!,

where x5e21/T. From here we see that the free energydivergent: Z5e2bF,

F51

bln~xH2x!j8~xH!.

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On the other hand, at strong coupling we expect the fenergy to be finite and given by the appropriate interpretaof the free energy of the gravity solution. The natural cojecture is that for a generic value ofl the string theory par-tition function in AdS53S5 would have a Hagedorn temperature precisely equal to the Hawking-Page crititemperature. Clearly, to substantiate this claim one needknow the quantization of strings in AdS53S5 with back-ground Ramond-Ramond fields, which is currently nunderstood.1

While quantization of strings in the full AdS53S5 back-ground is not understood completely, string theory in a pticular Penrose-Gu¨ven limit is exactly soluble in the lightcone@8#. The precise dictionary between string theory quatities and a particular largeR charge sector ofN54 SYMtheory was formulated in@9#. Moreover, the statistical mechanics of these string theories has been studied in bothcanonical@10,11# and grand canonical@12,13# ensembles.2 Itwas found that, much as in the case of flat space, stringplane waves have a Hagedorn temperature. In the case ogrand canonical ensemble@where in addition to the temperature one introduces a chemical potential conjugate toU(1),SO(6)R charge# the free energy was shown to be finite near the Hagedorn temperature@13#: F;Ab2bH.This suggests the possibility of a phase transition. Howethe specific heat is negative and diverges,cV;(b2bH)23/2, obscuring the nature of the transition.

Following the relation between the Hagedorn transitiand Hawking-Page phase transition outlined above, itnatural to ask whether the Hagedorn physics of strings inPP wave background is related to the physics ofHawking-Page phase transition for the black holes in gloAdS5 that carry large U(1)J,SO(6)R R charge. The study ofthis connection is the main motivation of this paper.

In the next section we discuss the thermodynamicssingle-charge black holes in global AdS5 geometry. The ex-

1An interesting result which might help solve this problem wreported recently in@7#.

2Finite temperature string theory on various PP wave backgrouhas also been discussed in@14#.

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A. BUCHEL AND L. A. PANDO ZAYAS PHYSICAL REVIEW D 68, 066012 ~2003!

plicit solutions were constructed previously in@15# and someaspects of the thermodynamics were studied in@16#. Wepresent an alternative computation of the charged blackthermodynamic properties, which utilizes the holograprenormalization approach of@17#. As we explain, this leadsto a different expression for the Arnowitt-Deser-Misn~ADM ! mass than the one used in, say,@16#. In Sec. III wereview the relevant aspects of the thermodynamics of strin the PP background in the grand canonical ensemble@12#.Unfortunately, we find that the regime where black hohave a largeJ charge corresponds to very different valuesthe conjugate chemical potentialmJ , from the one used todefine the ‘‘corresponding’’ PP wave string grand canonipartition function.

II. SINGLE-CHARGE BLACK HOLESIN FIVE-DIMENSIONAL GAUGED SUPERGRAVITY

In this section we discuss the construction~and the ther-modynamics! of a single-charge black hole solution inD55 N58 gauged supergravity. The asymptotic backgrouis the global AdS5 , and the black hole would carryU(1),SO(6)R electric charge. This solution was originalfound in @15# as a special case of the STU model. The thmodynamic properties of these black holes were studiedviously in @16#. We present a new computation for the themodynamics, which gives different results for the ADmass~and appropriately the Euclidean action! than the oneused in@15,16#, and in many subsequent papers.

A. The black hole geometry

The black hole geometry can be obtained@15# as a solu-tion of the D55 N58 gauged supergravity. The relevaeffective five-dimensional action is

S551

4pG5E

M5

d5jA2gL

51

4pG5E

M5

d5jA2gF1

4R1

1

2g2V2

1

16H4/3F2

21

12H22~]H !2G , ~1!

whereg is the gauge coupling,R is the scalar curvature,Fmn

is a U~1! field-strength tensor, andV is theH scalar potential,

V52H2/314H21/3. ~2!

From Eq.~1! we find the following equations of motion:

!H5H21~]H !211

2H7/3F223g2H2

]V]H

,

05]m~A2gH4/3Fmn!,

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sf

l

d

-e--

Rmn51

2H4/3FmgFn

g11

3H22]mH]nH

2gmnF2

3g2V1

1

12H4/3F2G . ~3!

We take the following ansatz for the charged black hole mric:

ds5252e22A/3f dt21eA/3@ f 21dr21r 2~dS3!2#, ~4!

whereA,f are functions of the radial coordinater only, and(dS3)2 is the round metric on the unit radiusS3. Addition-ally, we takeH[H(r ), and the only nonvanishing component of the gauge potential,F5dA, At[At(r ). With thisansatz, using the equations of motion~3!, we can rewrite thegravitational Lagrangian in Eq.~1! as a total derivative,

A2gF1

4R1

1

2g2V2

1

16H4/3F22

1

12H22~]H !2G

52F 1

12A8 f r 31

1

2r 2~ f 21!G8, ~5!

where primes denote derivatives with respect tor.Omitting the computational details, the relevant tw

parameter family$m,r% of solutions of Eq.~3! is

eA5H, f 512m

r 2 1g2r 2H, H511q

r 2 , At5q

r 21q,

~6!

where we introduced

q5m sinh2 r, q5m sinhr coshr. ~7!

In what follows it will be important thatq are the physicalcharges~i.e., the conserved charges to which Gauss’s lapplies!. Note the useful relation

q25q~q1r 12 !~11g2r 1

2 !. ~8!

B. The thermodynamics

Given the explicit single-charge black solution~6!, it isstraightforward to extract its thermodynamics. The oublack hole horizon is atr 1 , the largest non-negative zero othe functionf,

f ~r 1!50. ~9!

The inverse Hawking temperatureb[1/TH , and theBekenstein-Hawking entropySBH are given by

b51

TH52p

~r 12 1q!1/2

11g2q12g2r 12 ,

SBH5Ahorizon

4G55

p2

2G5r 1

2 ~r 12 1q!1/2. ~10!

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HAGEDORN VERSUS HAWKING-PAGE TRANSITION IN . . . PHYSICAL REVIEW D68, 066012 ~2003!

The computation of the generalized free energyV;(1/b)I E ~determined from the properly regularized Eucliean gravitational actionI E) and the ADM massM is slightlymore subtle and we discuss this in some detail. In@15,16#,the ADM mass is taken to be

M ~m,q!;3

2m1q. ~11!

In particular, Eq.~11! vanishes form5q50, which impliesthat the mass of global AdS5 is assumed to be zero. But thlatter statement contradicts the gauge and string theoryrespondence@1,2#: it was shown in@17# that MAdS5

.0;moreover, its precise value exactly coincides with the~posi-tive! dual gauge theory Casimir energy. It is instructivesee what goes wrong with the prescription for computingADM mass used in@15#. In @15#, following the proposal of@18#, the ADM mass of the geometry~4! was defined as thefollowing surface integral at radial infinity:

M521

8pG5E

S3N~K2K0!, ~12!

whereN is the norm of the timelike Killing vector andK isthe extrinsic curvature, dependent on the black hole pareters $m,q%; finally, K0 is taken to be the correspondinextrinsic curvature of the global AdS5 geometry. Literallyfollowing this prescription for the single-charge black holeinterest here does not give~11!; rather, we findM;g2qr2

→`. The technical reason for this is that the 5D gauge fieof the black hole solution modify the first subleading~as r→`) correction off in Eq. ~6!, which cannot be subtracteby comparing with the ‘‘uncharged’’ global AdS5 geometry.Now, the finite mass term results from the second subleadterm in f, and thus the subtraction~12! necessarily gives adiverging result.3

In the rest of this section we present a modified presction for computingI E and M that, first of all, gives finiteresults for the above quantities. Additionally, in the limitvanishing charge and nonextremality parameter we recothe global AdS mass of@17#. Our prescription relies on theMaldacena proposal for the existence of dual~local! four-dimensional quantum field theory for the black hole geoetry ~4!. In a sense, this is a simple application of the renmalization ideas of@17#. We begin by summarizing ouresults:

I E5bp

G5S 2

1

8m2

1

12q2g21

3

32g221

1

4r 1

2 D ,

M5p

G5S 3

8m1

1

4q2

1

12q2g21

3

32g22D . ~13!

3The application of the procedure of@18# to the recently studiedblack hole solution@19# would also give a diverging result for thADM mass. Again, the problem appears to be due to the additiomatter fields compared to the extremal geometry.

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Notice that, using Eqs.~10! and ~13!, we find the expectedthermodynamics relation4

I E5b~M2m qq!2SBH , ~14!

wherem q is the chemical potential conjugate to the physicblack hole chargeq, related to the gauge potentialAt @Eq.~6!# at the horizon,r 5r 1 ,

m q5vol~S3!

8pG5At~r !U

r 5r 1

5p

4G5

q

r 12 1q

. ~15!

To obtain Eq.~14! we used Eq.~8!. In the limit of vanishingcharge, we find from Eq.~13!

M ua5053p

32g2G51

3pm

8G5, ~16!

which with the identificationm[r 02, g[1/, is precisely the

result obtained in@17# for the Schwarzschild black hole inglobal AdS5 .

Let us compute the renormalized~in the sense of@17#!Euclidean gravitational actionI E of Eq. ~1!. First, we regu-larize Eq. ~1! by introducing the boundary]M5 at fixed~large! r with the unit orthonormal spacelike vectornm}d r

m

S5r 5

1

4pG5E

r 1

r

drE]M5

d4jAgELE

521

4pG5E

r 1

r

drE]M5

d4jA2gL

51

4pG5E

r 1

r

drF 1

12A8 f r 31

1

2r 2~ f 21!G8E

]M5

d4j

5bp

2G5F 1

12A8 f r 31

1

2r 2~ f 21!GU

r 1

r

, ~17!

where the subscriptE represents that all the quantities arebe computed in the Euclidean signature. We used Eq.~5! toobtain the second identity in Eq.~17!. As usual, to have awell-defined variational problem in the presence of a bouary requires the inclusion of the Gibbons-HawkingSGH term

SGH521

8pG5E

]M5

d4jAhE¹mnm

5bp

2G5F2

1

12A8 f r 32

1

4r 3f 82

3

2r 2f G , ~18!

wherehmn is the induced metric on]M5 ,

hmn[gmn2nmnn . ~19!

al 4Black holes with ‘‘hair’’ for which a similar relation can beproved are discussed in@20,19#.

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Finally, as in@17#, we supplement the combined regularizaction (S5

r 1SGH) by the appropriate boundary countertermconstructed from the local5 metric invariants on the boundar]M5 :

Scounter51

4pG5E

]M5

d4j~a1AhE1a2R4AhE!

5bp

2G5@a1eA/6f 1/2r 316a2r f 1/2e2A/6#, ~20!

whereR4[R4(hE) is the Ricci scalar constructed fromhmn ,anda i are coefficients that can depend only on the curvatof the asymptotic AdS5 geometry.6 The counterterm parametersa i are fixed in such a way that therenormalizedEu-clidean actionI E is finite,

I E[ limr→`

~S5r 1SGH1Scounter!, uI Eu,`. ~21!

Using the explicit solution~6! we find the answer~13! for I Eand

a153

2g, a25

1

8g21. ~22!

We now proceed to the computation of the ADM massthe background~1!. Following @17#, we define

M5ESd3jAsNSe, ~23!

where S[S3 is a spacelike hypersurface in]M5 with atimelike unit normalum, NS is the norm of the timelikeKilling vector in Eq.~4!, s is the determinant of the inducemetric onS, ande is the proper energy density,

e5umunTmn . ~24!

The quasilocal stress tensorTmn for our background is ob-tained from the variation of the full action

Stot5S5r 1SGH1Scounter, ~25!

with respect to the boundary metricdhmn ,

Tmn52

A2h

dStot

dhmn. ~26!

Explicit computation yields

5The locality condition is very important and it follows from thlocality of the dual gauge theory. In our case this dual gauge theis the N54 SYM theory in the deconfined phase with a givchemical potential conjugate to theR charge. An example where thlocality condition does not hold will be discussed elsewhere@21#.

6For this reason the values ofa i must be the same as the corrsponding parameters in@17#.

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r

Tmn51

8pG5F2Qmn1Qhm22a1hmn

14a2S R4mn2

1

2R4hmnD G , ~27!

where

Qmn51

2~¹mnn1¹nnm!, Q5TrQmn. ~28!

It is straightforward to verify that witha i as in Eq.~22! themass as defined in Eq.~23! is finite, and is given by Eq.~13!.

C. The phase diagram of a single-charge black hole

To make the connection with the dualN54 SU(N) SYMtheory onR3S3 we recall@2#

1

g3G55

2N2

p, g5,21. ~29!

We would like to identify the thermodynamic characteristiof the single-charge black hole computed in the previosection $I E ,M ,SBH ;TH ,m q% with the appropriate gaugetheory quantities$V,E,S,T,mJ%,

$I E ,M ,SBH ;TH ,m q%↔$V,E,S;T,mJ%, ~30!

where the thermodynamic potentialV is related to the Helm-holtz free energyF in the standard way,

V5F2mJJ5E2TS2mJJ. ~31!

The identification we are after must preserve the relation~14!and satisfy the first law of thermodynamics for the gracanonical ensemble with$T,mJ% as independent variables:

dV52S dT2J dmJ . ~32!

We propose to identify

T[TH ,

S[SBH ,

mJ[m q . ~33!

Given Eqs.~33!, ~14!, the first law~32! uniquely determines7

V[THI E1g2p

12G5q2,

ry7Strictly speaking, there is a single overall constant, independ

of temperature and the chemical potential, in the definition ofE andV. This constant must be set to zero in order to reproduce the refor the agreement of gauge theory Casimir energy and the Amass@17# at $T50,mJ50%.

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he


E[M1g2p

12G5q2. ~34!

We do not have an independent way of justifying the idenfication ~34!, apart from the argument presented above. Nthat, for vanishing physical chargeq, q vanishes as well, andwe get the standard identificationbF[I E .

In what follows we set the five-dimensional gauge copling ~or equivalently the AdS5 scale! g51. Thus, from Eq.~29!, all the gauge theory thermodynamic quantities woscale proportionally toN2, as appropriate for the deconfinephase. On the other hand, in the confined phase we exthese thermodynamic potentials to scale proportionallyN0, effectively zero in the largeN limit. For fixed tempera-ture T[TH and chemical potentialmJ , the physical gaugetheory phase~in the largeN limit 8! has the chemical potentiaVphys

1

N2 Vphys~T,mJ!5minH 1

N2 V~T,mJ!,0J . ~35!

We expect the confinement or deconfinement phase tration to occur at$T(mJ),mJ% such that

V„T~mJ!,mJ…50. ~36!

We found that the best way to parametrize the thermodynics is to use the analogue of the ‘‘unphysical’’ chargeq in theblack hole case.9 The summary of the thermodynamics isfollows.

The critical curves for the generalized free energyV as afunction of ~T,q! are presented in Fig. 1. The lower lin@Tlower(q)# corresponds to the vanishing of the outer blahole horizon. For values of~T,q! below it, bothr 1 andV areimaginary. The upper line@Tupper(q)# corresponds to theconfinement or deconfinement phase transition as in~36!: for values of~T,q! in the strip between the two lineV.0. For ~T,q! above the phase transition curve,V,0. As-ymptotically, asq→1`, we have

Tlower51

2pq1/21

1

2pq21/2,

Tupper5Tlower19

16pq23/21O~q25/2!.

~37!

At the phase transition temperatureTcritical[Tupper, thechemical potentialmJ /N2 and the chargeJ are shown inFigs. 2 and 3, respectively. Asymptotically, asq→1` wefind

mJ

N2 51

21

3

16q211O~q22! ~38!

8There is no phase transition, but rather a crossover for finiteN.9Note that the identification~33! for the chemical potential implies

that J[q.

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si-

-

q.

and

J5q13

81O~q21!. ~39!

For completeness, we also present the asymptoticq→1` behavior of Sc[S(Tcritical ,q), Ec[E(Tcritical ,q), r hor[r 1(Tcritical ,q):

1

N2 Sc53p

4q21/21

3p

4q23/21O~q25/2!,

1

N2 Ec51

2q1

3

41O~q21!,

FIG. 1. The critical curves for the thermodynamic potentV(T,q), where qP@0,1`), TP@0,1`). Below the lower lineIm VÞ0. In the strip between the two linesV.0, and above the topline V,0. The upper line corresponds to the critical line for tconfinement or deconfinement phase transition@Eq. ~36!#.

FIG. 2. The chemical potentialmJ conjugate to the chargeJ as afunction of q, at the critical temperatureT5Tcritical .

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r hor5)

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)

4q23/21O~q25/2!. ~40!

III. PLANE WAVE BACKGROUNDAT FINITE TEMPERATURE

One of the most interesting aspects of the plane wbackground is that, being a Penrose-Gu¨ven limit of AdS53S5, it can be used as a specific example@9# of the gaugeand string theory correspondence, on one side of which this an exactly solvable theory@8#.

In this section we review the thermodynamic propertiesthis background and their implications for the field theoThe finite temperature partition function of string theorythe plane wave background with Ramond-Ramond~RR! nullfive-form field was obtained in@10,12,13,11#. One of themost salient features that has been established is thetence of a Hagedorn temperature for strings in this baground.

To clarify the relation between the string theory and fietheory quantities it is convenient to follow the Penrose limsuggested by Tseytlin and presented in@22#; namely, we startwith

x15t, x25R2~ t2c!, ~41!

whereR is the AdS radius andc is a coordinate parametrizing the great circle of S5. The Penrose-Gu¨ven limit along thisnull geodesic results in the standard maximally supsymmetric IIB plane wave background with null RR fivform flux and the following relation between string angauge quantities:

2p1

m5E2J, 2ma8p25

J

Al. ~42!

In the field theory interpretationE is the energy of states inR3S3 andJ is the U~1! R charge of the corresponding stat

FIG. 3. The U(1),SO(6)R chargeJ as a function ofq, at thecritical temperatureT5Tcritical .

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r-

Let us, following @12,13#, define a slightly more generapartition function of the form

Z~a,b;m!5TrHe2bp22ap1. ~43!

One of the virtues of this partition function is that it makexplicit the interpretation in the grand canonical ensembleintroducing a chemical potential. The partition function ofideal gas of type IIB strings in the maximally supersymmric plane wave background can be written in terms ofsingle-string partition function for bosonicZ1

B and fermionicZ1

F modes:

ln Z~a,b;m!5(r 51

`1

r@Z1

B~ar,br;m!2~21!rZ1F~ar,br;m!#.

~44!

The single-string partition function can be written as

Z1~a,b;m!5TrHe2bp22ap1. ~45!

Most of our conclusions will rely on the single-string aproximation. Using the expression for the light-cone Hamtonian obtained in@8#:

HLC51

a8p2Fm(

i 51

8

~a0i†a0

i 1S0i†S0

i !1 (n51

`

An21m2

3S (i 51

8

~ani†an

i 1a0i†a0

i !1~Sni†S0

i 1S0i†S0

i !D G , ~46!

wherem5ma8p2 , and identifyingt25a/2pa8p2 , we find

Z1~a,b;m!5aVL

4p2a8E

0

` dt2

t22 E

21/2

1/2

dt1e2ab/2pa8t2zLC~0,0!

3~t,ma/2pt2!zLC~0,1/2!~t,ma/2pt2!, ~47!

wherezLC(0,0) andzLC

(0,1/2) are, roughly, the partition functions omassive two-dimensional bosons and fermions, respectivand we refer the reader to@10# for the precise notation andfurther details. Our main concern will be with the Hagedotemperature, although other thermo-dynamic quantities sas the free energy can also be calculated@10,12,13,11#. Wedefine the Hagedorn temperature as the value above wthe partition function starts diverging:

b

16ma85g0S am

2p D2g1/2S am

2p D , ~48!

where the right hand side shows the difference betweenCasimir energies for bosons and fermions@10#. The aboveexpression can be made explicit in terms of integralsBessel functions@10,12,13,11#.

Let us now turn to the meaning of the expression forHagedorn temperature in terms of the field theory variabTaking into account Eqs.~42! and ~45!, the temperature andchemical potential of the gauge theory are

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bYM51

TYM5

am

2, mJ5

b

2ma8Al2

am

2. ~49!

We will use b85b/Al, or correspondingly introduce thchemical potential conjugate toJ85J/Al. Note that the lat-ter quantity remains finite in the Berenstein-MaldaceNastase~BMN! limit. Thus, the equation for the Hagedortemperature as a function of the chemical potential in gatheory quantities takes the final form

1

16S 1

TH1mJ8D5 (

p51

`1

p@12~21!p#K1S 2p

THD . ~50!

The dependence of the chemical potential on the Hagedtemperature is presented in Fig. 4. Notice that for high Hadorn temperatures the relation between the chemical potial and temperature simplifies to

mJ8'2p2TH . ~51!

IV. HAGEDORN vs HAWKING-PAGE TRANSITION

The main motivation of this paper is to compare tHagedorn behavior of strings in the maximally supersymetric plane wave background and the phase transitionlarge charge black holes in the global AdS5 background. Toreiterate, the hope is that these large charge black holesholographically dual to the largeR charge sector of theN54 SYM theory at finite temperature, which in turn candescribed by an exactly soluble string model. The ideathen to compare the regime of the phase transition of blholes and the Hagedorn transition of strings, and t‘‘unify’’ the two pictures of the confinement or deconfinement phase transition inN54 SYM theory onR3S3: aHagedorn behavior atl!1, as discussed in@5#, and theHawking-Page black hole transition atl@1, as discussed in

FIG. 4. The chemical potentialmJ8 as a function of the Hagedorn temperatureTH for strings in the maximally supersymmetrplane wave with null RR five-form field strength.

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are

isks

@3#. In the rest of this section we explain that the regimethe criticality of large charge black holes appears to be vadifferent from the regime of the Hagedorn behaviorstrings in the PP wave background.

In Sec. II we studied the thermodynamics of large chablack holes in global AdS5 geometry. We found that thesnonextremal geometries have a phase transition which isa sense, a generalization of the Hawking-Page phase tration @4#. It was argued in@3# that the HP phase transitiorealizes the strong coupling gravitational dual to the confiment or deconfinement phase transition inN54 SU(N) su-persymmetric Yang-Mills theory on anR3S3 background.There is an important difference between the phase transfor the charged black holes and the HP one discussed in@3#.For the HP transition one has two geometries and compthe difference of their free energies~or Euclidean gravita-tional action!. In the limit of vanishing nonextremality, onof the geometries does not have a horizon, but is simplglobal Euclidean AdS5 with the appropriate periodicallyidentified time direction. In studying the phase transitiothis geometry is then used as a reference for the subtrac~‘‘regularization’’ in the language of@17#! of the Euclideanblack hole~BH! gravitational action. As we explained abovin the case of a charged BH this subtraction procedure leto an infinite expression for the ADM mass of the BHRather, the appropriate generalization would seem to besubtraction of the ‘‘charged’’ global AdS geometry, whichowever, does not have a horizon. Such a nonsingular geetry does not exist since it requires taking the limitm→0with q fixed, and this is a violation of the BPS bound. Thsituation motivated the use of the regularization proceduwhich does not require a reference background. Luckily, tcan be implemented with a straightforward application ofholographic renormalization ideas of@17#. The obviousdrawback in the absence of the reference background isfrom the gravity perspective, the phase transition is definin a ratherad hocmanner, i.e., as the vanishing of the geeralized free energy~36!. Nonetheless, this definition of thphase transition is well motivated from the perspective ofholographically dual gauge theory. We found@compare Eqs.~37!–~39!# that a large value of charge for the black holethe critical ~phase transition! temperature implies afinitenonzero value of the chemical potential. Also, a large chaat criticality implies a large critical temperature:

TcriticaluJ→`}J1/2,

mJ~Tcritical!uJ→`→ N2

2. ~52!

In Sec. III we recalled that, quite opposite to the regimethe criticality of large charge black holes~52!, the Hagedornregime of strings in the PP wave background for high Hadorn temperature requires large values of the chemicaltential ~51!.

V. CONCLUSION

We attempted to provide a more precise relation betwthe Hagedorn description and the Hawking-Page-like ph

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transition for the confinement-deconfinement transitionfour-dimensionalN54 SU(N) theory on S3, by introducinga chemical potential conjugate to the largeR charge. Thehope was that, although the exact quantization of stringAdS53S5 is not known, for largeR charge the essentiaphysics could be captured by a string dual to this largeRcharge sector, whichis exactly soluble. Unfortunately, wefound that this is not so. It is plausible that rephrasing tconfinement-deconfinement phase transition in the langu

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of the Hagedorn transition at strong coupling would requirmore detailed understanding of strings in AdS53S5.

ACKNOWLEDGMENTS

We are grateful to Finn Larsen, Jim Liu, EliezeRabinovici, and Cobi Sonnenschein for useful discussioL.A.P.Z. is especially thankful to D. Vaman for collaboratioon relevant topics. This work is supported in part by the UDepartment of Energy.

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gy

ld’’

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Documents

Hagedorn versus Hawking-Page transition in string theory