Statistical entropy of Calabi-Yau black holes

PHYSICAL REVIEW D, VOLUME 59, 064023

Statistical entropy of Calabi-Yau black holes

Mikhail Z. Iofa* and Leopoldo A. Pando Zayas†

Nuclear Physics Institute, Moscow State University, Moscow 119899, Russia~Received 30 April 1998; published 17 February 1999!

We compute the statistical entropy of nonextremal 4D and extremal 5D Calabi-Yau black holes and findexact agreement with the Bekenstein-Hawking entropy. The computation is based on the fact that the near-horizon geometry of equivalent representations contains as a factor the Ban˜ados-Teitelboim-Zanelli black holeand on subsequent use of Strominger’s proposal generalizing the statistical count of microstates of the BTZblack hole due to Carlip.@S0556-2821~98!05820-2#

PACS number~s!: 04.70.Dy, 11.25.Mj

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I. INTRODUCTION

The calculation of the statistical entropy of black holesreducing the problem to the counting of microstates ofBanados-Teitelboim-Zanelli~BTZ! black hole @1# has re-cently attracted the attention of many researchers. A funmental step was taken by Strominger@2# who realized thatthe microscopic calculation of the entropy of the BTZ blahole due to Carlip@3# can be performed equally well for anblack hole whose near-horizon geometry contains an asytotically three-dimensional anti–de Sitter (AdS3) region.This proposal is a generalization of a result obtained in@4#,where, using the fact that some solutions of type IIA canrelated via U duality to 4D~5D! black holes and to the product of BTZ3S2(S3) @5#, a microscopical count of the entropy of some black holes was performed. Strominger’s pposal has already been successfully applied to providmicroscopical interpretation for the entropy of several blaholes @6–11#. One of the most important features of threcent approach is that it does not rely on supersymmand can be applied to any consistent quantum theory of gity. This provides the possibility of microscopically calculaing the entropy of not only Bogomol’nyi-PrasadSommerfield~BPS! and near-BPS black holes, as in thD-brane approach, but of black holes arbitrarily away froextremality, as shown in@4#. In the light of this advantage iis natural to apply the near-horizon approach to N52 blackholes for which the brane explanation can break down duloop corrections as opposed to the N54 and N58 blackholes for which supersymmetry protects the counting.

The study of N52 black holes was pioneered in the wor@12# where some interesting observations were made athe structure of the moduli and where a way to constrexplicit solutions was opened. This approach providedbasis of many subsequent investigations where extremalsolutions were constructed~see@13# and references therein!.Recently@14# a rigorous derivation, accounting for tree-levand loop corrections, of the microscopic entropy of N52black holes was given. This latter description relies onM-brane content of the black holes. The M-theory interptation of these N52 black holes has also proved useful in tconstruction of the nonextremal solutions@15,16#. The non-

*Email address: [email protected]†Email address: [email protected]

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extremal ansatz proposed in@15# is based on the fact thaN52 extremal black holes are solutions of M-theory copactification on three-dimensional Calabi-Yau spaCY33S1 and are microscopically represented as three tyof M5-branes wrapping four-cycles in CY3 andS1. This sug-gested that for the nonextremal solution an analogy wnonextremal black holes of toroidally compactified M-theo@17# could also work. Still, the boost parameters characteing the nonextremal solutions must be subject to certain cstraints and explicit solutions were found only for a verestricted class of N52 string vacua. A refinement of thiconstruction was proposed in@16# where it was shown thathe ansatz goes through the equations of motion in the nhorizon region and is valid only to near-BPS saturated blholes. This is a sufficient condition to apply Stromingeproposal which concentrates on the near-horizon region.

In this paper we apply Strominger’s proposal to providemicroscopic interpretation of the Bekenstein-Hawking etropy of nonextremal 4D@15,16# and extremal 5D Calabi-Yau black holes@19# in terms of the entropy of the BTZblack hole that enters as a factor in the near-horizon geetry of some of the representations of the considered bholes. Exact agreement is found in both cases.

II. CALABI-YAU BLACK HOLES

A. Nonextremal 4D solution

The N52 supergravity action in D54 includes in additionto the graviton multiplet,nv vector multiplets andnh hyper-multiplets. The hypermultiplet fields can be consistentaken to be constant leaving one with the following actio

S51

32pG4E A2g d4xFR22gAB]mzA]mzB

21

4Fmn

I ~!GI !mnG , ~1!

with the gauge fieldGI mn given by

GI mn5ReN IJFmnJ 2ImN IJ

! FmnJ , ~2!

and I ,J50,1, . . . ,nv . The complex scalar fieldszA (A51, . . . ,nv) parametrize a special Ka¨hler manifold withmetricgAB5]A] BK(z,z), whereK(z,z) is the Kahler poten-

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MIKHAIL Z. IOFA AND LEOPOLDO A. PANDO ZAYAS PHYSICAL REVIEW D 59 064023

tial. Both, the gauge field coupling and the Ka¨hler potentialare expressed in terms of the holomorphic prepotentialF(X):

e2K5 i ~XIFI2XIFI !

NIJ5F IJ12i~ Im FILXL!~ Im FJMXM !

Im FMNXMXN, ~3!

with FI5]F(X)/]XI andFMN5]2F(X)/]XM]XN. The sca-lar fieldszA are defined by

zA5XA

X0. ~4!

Throughout the paper we will consider the following preptential:

F~X!5dABCXAXBXC

X0, ~5!

wheredABC are the topological intersection numbers of tCalabi-Yau manifold. The ansatz discussed in@15,16# is thefollowing:

ds252e22Uf dt21e2U~ f 21dr21r 2dV2!

e2U5AH0dABCHAHBHC

f 512m

r

zA5 iH AH0e22U, HA5hAS 11m

rsinh2gAD

At05

rH 08

h0H0, Aw

C5r 2cosuHC8

HA5hAS 11m

rcoshgAsinhgAD

H05h0S 11m

rsinh2g0D

H05h0S 11m

rcoshg0sinhg0D , ~6!

where prime denotes derivation with respect tor. The non-zero components of the gauge field strengths are

Ftr0 5

H08

H02

, FwuA 5r 2sinuHA8. ~7!

As was already said, this ansatz is restricted to the stion of certain conditions onNAB that influence the values og ’s @15#. In the approach of@16# the technical restriction onNAB is traded to restricting the solution to the near-horiz

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region and the final form of the solution describes a neextremal solution. The Bekenstein-Hawking entropy of th4D solution is

S5pm2

G4Ah0cosh2g0dABChAcosh2gAhBcosh2gBhCcosh2gC,

~8!

and the asymptotic flatness condition ish0dABChAhBhC51.

B. Extremal 5D solution

The N52, D55 supersymmetric Lagrangian describinthe coupling of vector multiplets to supergravity is detemined by one function which is given by the intersectiform on a CY3:

V5dIJKXIXJXK. ~9!

The bosonic action is1

e21L521

2R2

1

4GIJFmn

IFmnJ21

2gi j ]mf i]mf j

1e21

8emnrsldIJKFmn

I FrsJ Al

K ~10!

whereR is the scalar curvature,FmnI 52] [mAn]

I is the Max-well field-strength tensor ande5A2g is the determinant ofthe Funfbeinem

n . The fieldsXI5XI(f) are the special coordinates satisfying

XIXI51, dIJKXIXJXK51 ~11!

where,XI , the dual coordinate is defined by

XI5dIJKXJXK. ~12!

The moduli-dependent gauge coupling metric is relatedthe prepotential via the relation

GIJ521

2

]

]XI

]

]XJ ~ lnV!uV51 . ~13!

The metricgi j is given by

gi j 5GIJ] iXI] jX

JuV51 ; S ] i[]

]f i D . ~14!

Here we will consider the static spherically symmetBPS black hole solution of N52 supergravity in D55 @19#;all other solutions are particular cases of this one. In@19# anelectrically charged BPS solution of supergravity coupledan arbitrary number of vector multiplets was constructThe metric is of the form

ds252e24Udt21e2U~dr21dV32!. ~15!

1For the complete action, including the fermionic part, seeexample@18#.

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STATISTICAL ENTROPY OF CALABI-YAU BLACK HOLES PHYSICAL REVIEW D59 064023

Again our main interest is in the form of the solution in thnear-horizon region. Here the general principle establishe@12# that allows one to compute the area of the horizonsN52 extremal black holes as an extremum of the cencharge is enough to obtain the near-horizon geometry ofsolution. Namely

e2Uuhor51

3XAHAU

hor

51

3XAU

hor

qA

r 25

Z0

3r 2, ~16!

hereZ5qAXA is the central charge of the superalgebra aZ0 its value at the horizon. This way, the near-horizon gometry is

ds252S 11Z0

3r 2D 22

dt21S 11Z0

3r 2D ~dr21r 2dV32!.

~17!

If the value of the moduli at the horizon can be consistenkept fixed throughout the spacetime, as is the case in s4D solutions, then the obtained metric is the double-extreblack hole solution@18#. The Bekenstein-Hawking entropy i

S5p2

2G5S Z0

3 D 3/2

. ~18!

III. STATISTICAL ENTROPY

To relate the Bekenstein-Hawking entropy of the blaholes reviewed in the previous section to the counting bain the BTZ black hole entropy it is necessary to show thaan equivalent representation the near-horizon geomecontain a BTZ black hole as a factor. For the nonextrem4D and extremal 5D black hole we consider here the cosponding representations are as in@4#; for the 4D black holewe find a 5D representation whose near horizon region tathe form BTZ3S2 and for the 5D black hole BTZ3S3.

A. Statistical entropy of 4D black holes

To relate the 4D Calabi-Yau black hole of the previosection to the BTZ it is convenient to start with the followin5D metric:

ds525~h0dABCHAHBHC!21

3S 2dt21dy21m

r~coshg0dt1sinhg0dy!2D

1~h0dABCHAHBHC!2/3S dr2

12m

r

1r 2dV22D . ~19!

Upon compactification over the compact directiony oneobtains the metric of the nonextremal ansatz~6!. This metricis the direct generalization to the Calabi-Yau case oftoroidal compactification of three M5-branes that intersorthogonally over a common string that carries moment@17#. As a solution of M-theory this metric must have a

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11D lifting, a nice fact noted in@7# is that the 6D part ofthe 11D M-theory solution decouples and then one cconsider the 5D part as a solution of compactification either aT6 or a CY3. To see that this is indeed truit suffices to check that upon compactification over t6D part ~following the notation of @17#! it amounts to(F1F2F3)2(2/3)6(F2F3)2(F1F3)2(F1F2)251 as a factor ofthe 5D metric. The near-horizon region isr→0 or moreexactly

mhasinh2ga

r@1 ~20!

for any a50,A. In this region, defining

l 52~h0dABCpApBpC!1/3 ~21!

with pA5mhAsinh2gA , we have

ds525

2r

l S 2dt21m

r~coshg0dt1sinhg0dy!2D

1l 2

4r 2S 12m

r D dr21l 2

4dV2

2 , ~22!

where it is explicitly shown that the near-horizon region icludes a factorS2. To check that the remaining threedimensional part is a BTZ black hole, it is useful to make tfollowing change of variables which follows@4# and @7#:

t5l

Rt, f5

1

Ry, r25

2R2

l~r 1m sinh2g0!, ~23!

whereR is the compactification radius ofy. In these variablesthe 5D metric is

ds525dsBTZ

2 1l 2

4dV2

2

dsBTZ52N2dt21r2~df1Nfdt!21N22dr2

N25r2

l 21

m2R4sinh22g0

r2l 42

2mR2cosh2g0

l 3

Nf5mR2sinh 2g0

r2l 2. ~24!

The geometry of the BTZ black hole is described in@1# fromwhere we find that the mass and the angular momentum

MBTZ52mR2cosh 2g0

l 3

JBTZ5mR2sinh 2g0

4G3l 2. ~25!

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To relate the 3D Newton’s constant to the 4D one we cfollow for example@7,9#, where the only ingredients needeare the fact that the Ricci scalar in BTZ3S2 decompose asthe sum of the Ricci scalar of each factor and that theaction can be written in terms of the 4D Newton’s constaThe final result is

G35G4

2R

l 2. ~26!

To provide a statistical explanation for the entropy of blaholes one always needs a central charge. The relevant cecharge for the BTZ black hole was initially found in@20#where it was obtained as the central charge of the Virasalgebra of the conformal field theory generated by the abra of diffeomorphisms of asymptotically AdS3 spaces. Aswas already pointed out in@9#, to move from a BTZ geom-etry to an asymptotically AdS3 one needs to go to the regioof large r and therefore moves away from the initial neahorizon geometry. A more appropriate approach is the on@21# where the same central charge is obtained from thegebra of global charges and it can be found for any valuethe radius. In either approach the central charge is

c53l

2G3. ~27!

The zero modes of the Virasoro generators are related tomass and angular momentum of the BTZ black hole as~seethe second reference in@1#!

MBTZ58G3

l~L01L0!

JBTZ5L02L0 . ~28!

Now using Cardy’s formula for the degeneration of statesa conformal field theory of given central charge in oscillalevels one finds

S52pSAcnR

61AcnL

6 D5

p

4G3„Al ~ lM BTZ18G3JBTZ!

1Al ~ lM BTZ28G3JBTZ!…. ~29!

Inserting Eqs.~25! and~26! we obtain the following entropy

S5pm2

G4Ah0cosh2g0dABChAsinh2gAhBsinh2gBhCsinh2gC

~30!

which coincides with the geometrical entropy of the nonetremal 4D Calabi-Yau black hole~8! for any value ofg0 andfor large values ofg I which means, in physical terms, foany electric charge and for large magnetic charges. Inbrane picture this is the so called dilute gas regimeg I@1@22#. This way one finds that in this limit the Bekenstei

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Hawking entropy of this black hole can be given a statistiinterpretation in terms of the degrees of freedom associawith the conformal theory of the BTZ black hole.

B. Statistical entropy of 5D black holes

The ideology to solve this problem for the extremal N52D55 black holes is exactly the same as for the 4D blaholes and therefore most of the details are omitted. Befwriting a 6D metric which upon compactification on oncompact dimension gives the 5D black hole metric~17!, letus consider the candidates obtained from analogy with todal compactifications of M-theory. The 5D black hole in troidal compactifications of M-theory can be obtainedcompactification of two M-theory configurations: three othogonally intersecting M2-branes and a M2-brane orthonally intersecting a M5-brane~see@17# and @23# for a de-tailed analysis!. The technical reason for which the threM2-brane picture cannot be used to obtain the statisticaltropy of the three-charge 5D black hole is that it is not neessary to introduce a boost between time and a compacordinate. This boost is what determines the AdS3 geometrythat we need to be able to apply Strominger’s proposal.justify the use of the 2'5 picture we still need to prove thathe 5D part that lifts the 6D metric to a solution of M-theodecouples as was the case in the 4D case. The metric oconfiguration of a M2 intersecting a M5 with a boost alothe common string is@17,23#

ds112 5T2/3F1/3~2K21f dt21Kdy1

2!

1T21/3F22/3~ f 21dr21r 2dV32!1T2/3F22/3dy2

2

1T21/3F1/3~dy321dy4

21dy521dy6

2!, ~31!

with f 512m2/r 2 and T21,F21 harmonic functions. It canbe seen that in the limit we are interested in, all chargequal Eq.~17!, the 5D part described by (y2 ,y3 ,y4 ,y5 ,y6)decouples; in fact this part decouples for any 5D black hhaving two equal charges (F5T). Thus we conclude that the6D metric that upon compactification gives the 5D blahole ~17! can be lifted as a solution of M-theory. In thconclusions we will comment on what may be the relationthis 6D solution to the actual compactification of M-theoon CY3 we are considering. The 6D metric which upon compactification over the compact directiony gives the 5D blackhole metric~17! is

ds625TS 2dt21dy21

m2

r 2~coshsdt1sinhsdy!2D

1T21S dr2

f1r 2dV3

2D . ~32!

It is still necessary to set

T21511Z0

3r 2, ~33!

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STATISTICAL ENTROPY OF CALABI-YAU BLACK HOLES PHYSICAL REVIEW D59 064023

and take the limitm2→0, s→` with m2sinh2s→Z0/3. Per-forming the change of variable

r25R2

l 2~r 21m2sinh2s!, f5

y

R, t5

l

Rt, ~34!

with R the compactification radius and takingl 5(Z0/3)1/2

one finds that the 6D metric can be written as

ds525dsBTZ

2 1l 2

4dV3

2

dsBTZ52N2dt21r2~df1Nfdt!21N22dr2

N25r2

l 21

m4R4sinh22s

4r2l 62

m2R2cosh 2s

l 4

Nf5m2R2sinh 2s

2r2l 3. ~35!

The mass and angular momentum of this solution are

MBTZ5m2R2cosh 2s

l 4

JBTZ5m2R2sinh 2s

8G3l 3. ~36!

The relation between the Newton’s constants is

1

G35

1

G5

p l 3

R. ~37!

Using Eqs.~36!, ~37!, and~29! the statistical entropy is

S5p2

2G5l 2m coshs, ~38!

which coincides with the entropy~18! in the limit of larges,which is indeed needed to obtain the extremal limit.

IV. CONCLUSIONS

Here it has been shown that the Bekenstein-Hawkingtropy of Calabi-Yau black holes can be given a statistiinterpretation using Strominger’s proposal. For the 4D blahole the nonextremal case was considered for an ansatzis more general than the actual black hole metric. As shoin @15,16# some other conditions must be included thatstrict the ansatz. Exact agreement was found in the ‘‘dil

J.

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gas’’ regime or more precisely for large values of the manetic charges and arbitrary values of the electric chargethe 5D case the extremal solution was treated using theeral form of the near-horizon metric presented in@19# andbased on the beautiful results of@12#. Hopefully for the non-extremal 5D black holes a counting similar to the one carrout here for 4D could be performed; still some workneeded to consistently find the explicit form of the 5D noextremal Calabi-Yau black holes. In the present work ostep has been made in this direction noting that the M-brpicture that is consistent with the counting of the microstselects the microscopic content of the black hole as a bostate of M2 and M5-branes and not that of only M2-branIt could also be conjectured that a twelve-dimensional themay be of importance in the construction of nonextremalblack holes since in the counting use was made of a thethat has aS1 factor and upon compactification yields the 5black hole. The 5D black hole in the picture used in thpaper seems to be a solution of a twelve-dimensional thecompactified on CY33S1. Another sensible picture is that oM-theory compactified on CY3 that can effectively be described as having aS1 factor.

SinceN52 black holes receive quantum corrections itworth explaining to what extent this microscopic counticould include them. Unfortunately a rigorous statistical eplanation of the quantum corrections as presented in@14#does not seem to be available in this picture. Still, it is wonoting that certain quantum corrections can be includedthe approach used in this paper. Namely those that presthe polynomial degree of three of the prepotential. Oneample of this type of quantum corrections was presented@24# and has the form

dABCHAHBHC5H1H2H31a~H3!3. ~39!

The metric for this quantum corrected solution is

g002 54S h01

q0

r D S S h11p1

r D S h21p2

r D S h31p3

r D1aS h31

p3

r D 3D ; ~40!

redefining l in Eq. ~21! one can consistently include thiquantum correction to the geometric entropy in the statistdescription presented here.

ACKNOWLEDGMENTS

This work was partially supported by the RFFR Grant N98-02-16769. L.A.P.Z. is grateful to N. Pando Girard freading this paper.

al

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Statistical entropy of Calabi-Yau black holes