Nucl.Phys.B v.794

Nuclear Physics B 794 (2008) 112www.elsevier.com/locate/nuclphysb

Some implications of perturbative approachto AdS/CFT correspondence

Hikaru Kawai a,b, Takao Suyama a,

a Department of Physics, Kyoto University, Kitashirakawa, Kyoto 606-8502, Japanb Theoretical Physics Laboratory, The Institute of Physics and Chemical Research (RIKEN),

Wako, Saitama 351-0198, JapanReceived 21 September 2007; accepted 23 October 2007

Available online 28 October 2007

Abstract

We show some implications of the approach to AdS/CFT correspondence based on type IIB string inthe flat spacetime with D3-branes proposed in our previous paper. We discuss a correspondence for highenergy scattering amplitudes of N = 4 super-YangMills proposed recently. We also discuss AdS/CFTcorrespondence at finite temperature. Our approach provides clear understanding of these issues. 2007 Elsevier B.V. All rights reserved.

1. Introduction

In our previous paper [1], we proposed how to understand the reason why AdS/CFT corre-spondence holds. Our discussion there was based on the perturbative type IIB string in the flatspacetime with D3-branes introduced as boundaries of the worldsheets. We mainly discussedthe relation [2,3] between Wilson loops in N = 4 super-YangMills (SYM) in four dimensionsand minimal surfaces in the AdS5 S5 spacetime, and showed that the correspondence is aconsequence of an approximate symmetry which exists in the worldsheet theory, if the largeN limit is taken with the t Hooft coupling kept finite but large, and if the worldsheets rele-vant for evaluating the Wilson loops and the minimal surfaces are restricted within a region nearD3-branes.

* Corresponding author.E-mail addresses: [email protected] (H. Kawai), [email protected]

(T. Suyama).0550-3213/$ see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2007.10.016

2 H. Kawai, T. Suyama / Nuclear Physics B 794 (2008) 112

In this paper, we show that our approach is useful for understanding the other issues discussedin the literature. The first example, discussed in Section 2, is on the calculation of high energyscattering amplitudes of the SYM, which was proposed in [4] and recently confirmed on thestrong coupling side at the level of four-gluon scattering in [5]. It will be shown that this proposalis also justified by the scale invariance in a similar way to the case of the Wilson loop. Thesecond example is on AdS/CFT correspondence at finite temperature, discussed in Section 3.The discussions on Wilson loops can be carried out also in this case. From the point of view ofthe perturbative string with D3-branes, the issue on the entropy can be understood rather clearly.

2. High energy scattering amplitudes

Let us consider a scattering process of n massless open string states living on parallel ND3-branes. At a low energy scale, the dynamics of open strings is governed by N = 4 SYM infour dimensions. The scattering amplitude can be calculated from worldsheets with a number ofboundaries on which n vertex operators for the massless states are attached. We only considerthe case in which all the vertex operators are attached to a single boundary of the worldsheet.Let us take the large N limit with the t Hooft coupling = gsN fixed. Due to taking this limit,closed string handles of the worldsheets are suppressed, and therefore, the relevant topologyof the worldsheets is that of the disk with boundaries. These worldsheets correspond to planarribbon graphs of the SYM. Let (tak )ij be the color-dependent factor of the kth vertex operator.Then the amplitude considered here is proportional to Tr(ta1 ta2 tan) which is the only factordepending on the gauge group.

Suppose that one of the D3-branes is separated from the other N 1 D3-branes which are ontop of each other, and all of the external open string states are on the separated D3-brane. Therestriction of the external states only results in choosing a trivial prefactor, instead of a genericTr(ta1 ta2 tan), and therefore, it is easy to deduce the generic amplitude from this special one.We can also restrict the sum over worldsheets so that the boundaries, on which there is no vertexoperator, are on the coincident D3-branes. This restriction results in assigning N 1, not N , toeach color index loop, and therefore, this leads to a modification of the amplitude by an amountof order 1

N, which is negligible in the large N limit. Due to the separation of the D3-brane, the

strips of the string corresponding to the outermost loop are stretched between the D3-branes witha finite width, and therefore, it is a massive state that is propagating along this loop. The situationis depicted in Fig. 1. Fig. 2 shows the corresponding Feynman diagram.

The introduction of mass in this manner is expected to make the amplitude well defined inthe IR region. In terms of the N = 4 SYM, what we have done is the following. We first give anon-zero background to one of the scalar fields as

(2.1)

1 . . . n n + 1 . . .N1...

n

n + 1...

N

.. .

0. . .

0

,where the situation depicted in Fig. 1 corresponds to n = 1, and one can consider a situation forgeneric n. Here we assume n N , and consider only planar diagrams. We further assume that

H. Kawai, T. Suyama / Nuclear Physics B 794 (2008) 112 3

Fig. 1. Worldsheet configuration for the regularized amplitude.

Fig. 2. Feynman diagram corresponding to the worldsheet in Fig. 1.

the colors of external particles are taken from the n n matrices. Therefore, in the double linenotation, the index of the outermost index loop runs from 1 to n. Furthermore we can constrainthe other index loops to run from n + 1 to N . This constraint does not affect the amplitude ifn N . In this way, we give non-zero mass to all the propagators that belong to the outermostloop as in Fig. 2.

In order to illustrate the mechanism that mass of the outermost propagators regularizes theIR divergence, we consider the following integral which corresponds to the Feynman diagramdepicted in Fig. 3,

(2.2)I =

d4k11k21

1(p1 k1)2

1(p2 + k1)2

d4k2

1k22

1(p1 k1 k2)2

1(p2 + k1 + k2)2 .

Here we assume k2 = 0 for simplicity. If p21 = p22 = 0, the integral of k2 gives a term of orderlog(k21), and the k1 integral is IR divergent. The regularization we are discussing is to give massto all propagators in the outermost loop. However, it is sufficient to consider only one propagatorin this simple case. Actually, if we give mass to the outer gluon, I is regularized to

I =

d4k1 1 1reg 1k21 + 2 (p1 k1)2 (p2 + k1)2


(2.3)

d4k21k22

1(p1 k1 k2)2

1(p2 + k1 + k2)2 .

The k1 integral converges this time, although the k2 integral gives log(k21). The point is that themass of an outer propagator makes the whole diagram IR finite.1

Note that the mass introduced for the IR regularization must be much smaller than the stringscale, and therefore, the separation among D3-branes must be very small. Note also that themomenta of the external states are also much smaller than the string scale so as not to producemassive string states in the scattering process, although we call this process a high energyscattering.

To relate the scattering amplitude to a classical worldsheet configuration, let us make thefollowing change of variables in the worldsheet theory,

(2.4)X = XD,(2.5)XI = XID,

for = 1, . . . ,4 and I = 6, . . . ,10. This is nothing but the T-duality transformation of the world-sheet variables. The transformation of the fermionic variables is defined as

(2.6)Sa = SaD,(2.7)Sa = MabSbD.

We have employed the GreenSchwarz formalism and taken the light-cone gauge. Since we con-sider worldsheets in the flat spacetime, this transformation obviously preserves the worldsheetaction. The boundary condition for X turns into the Dirichlet boundary condition since (2.4)implies

(2.8)X = XD,at the boundaries. As a result, the D3-branes in the original setup turn into D-instantons. It isinteresting to notice that the constant modes xD of X

D are not fixed, since the boundary condition

is XD = 0. Therefore, xD should be integrated in the worldsheet path-integral which indicatesthat the D-instantons are distributed uniformly along x-directions.

Fig. 3. A Feynman diagram which could have an IR divergence.1 As far as we know, there is no proof for the justification of the regularization procedure described above. It is veryinteresting to prove that our procedure works for generic amplitudes.


It is very interesting to compare this situation with the large N reduction of the SYM. In the re-duced model, the momentum integrals for Feynman diagrams correspond to integrals of diagonalelements of matrices [6]. If the reduced model is regarded as an effective theory of D-instantons,then the diagonal elements of the matrices dictate the positions of the D-instantons. This chain ofcorrespondences also implies that the integration of the positions of the D-instantons is necessaryin the calculation of the high energy scattering amplitude.

In [5], classical solutions for the worldsheet are discussed in AdS5 S5 background, andthen the T-dual transformation is performed to obtain explicit expressions of the solutions. Thiscorresponds, in our perturbative approach, to performing an anisotropic scale transformation of[1] to go to the gravity region, and then perform the above transformation. In this paper, weproceed through a different way; we perform the above T-dual transformation first, and thenperform a scale transformation defined below.

It should be noted that the transformation we would like to perform is not exactly the T-dualitytransformation, since the x-directions are non-compact. The analysis below is thus valid onlywhen we restrict ourselves with the worldsheets which are topologically a disk with boundaries.Inclusion of handles, corresponding to considering a finite N case, would require a more compli-cated analysis. In the following, however, we call this the T-duality transformation, which wouldprobably make no confusion.

From now on, we consider disk worldsheets in the presence of N D-instantons. As mentionedabove, this system is equivalent to the original system including D3-branes only in the large Nlimit.

We consider the scale transformation discussed in [1] in this D-instanton system. The trans-formation properties of the T-dual variables can be easily derived from the original one. As willbe shown below, it is an isotropic scale transformation in this case. However, this transforma-tion will turn out to be an approximate symmetry in a similar sense to [1]. As a result, we willverify that the high energy scattering amplitude of N = 4 SYM can be calculated by a specificconfiguration of the worldsheet in AdS5, as is claimed in [5].

Recall that the scale transformation of the coordinate fields in the D3-brane setup is

(2.9)Xi( ) = MijXj ( ),(2.10)P i( ) = MijP j ( ),(2.11)Sa( ) = iMabSb( ),(2.12)Sa( ) = iMabSb( ),

where i, j, a, b run from 1 to 8, and the 8 8 matrix Mij and Mab are defined as

(2.13)Mij =[I44 0

0 I44

],

(2.14)Mab = ( 1 2 3 4)ab,where i are the SO(8) gamma matrices.

The T-duality transformation (2.4)(2.7) provides the scale transformation of the T-dual vari-ables as

(2.15)XiD( ) = XiD( ),(2.16)P iD( ) = P iD( ),

(2.17)SaD( ) = iSaD( ),


Fig. 4. LSZ-like reduction. The cross represents the insertion of S or |B0.

(2.18)SaD( ) = iSaD( ).Note that the transformation (2.4) exchanges the coordinates and the momenta which is clearin (2.8). In this T-dual situation, the scale transformation is indeed the ordinary isotropic scaletransformation. One can easily obtain the scale transformation of the oscillators as

(2.19)in = in,(2.20)in = in,(2.21)San = iSan,(2.22)San = iSan.

Using these transformation rules, it is easy to show that the boundary state of a D-instanton

(2.23)|B = exp[

n=1

(1ninin iSanSan

)]|B0,

(2.24)|B0 =(|i|i i|a| a)xI = 0p = 0

( n=1

|0n),

is invariant under the scale transformation. Note that

(2.25)p = 0= d4q x = q,describes the uniform distribution of the D-instanton.

To show the existence of the scale invariance in the D-instanton case, let us consider the freeenergy in the D-instanton background defined as

(2.26)F() =

n=0

Fn

n! n,

where Fn contains the contributions from worldsheets with n boundaries.It is easy to calculate the variation S of the worldsheet action under the scale transformation,

and it can be shown that the variation is a sum of vertex operators corresponding to the state |B0.Since the boundary state is invariant, the variation of Fn under the scale transformation is ob-tained by inserting the state |B0 on the worldsheet, that is,(2.27)Fn = Fn(|B0).


To evaluate Fn(|B0), consider a worldsheet path-integral Fn+1(z) with n+1 boundaries, oneof whose boundary is placed at xI = zI with I = 5, . . . ,10. Note that the N 1 D-instantons wehave discussed so far are placed at xI = 0. In other words, we place another set of D-instantonswhich are distributed parallel to the original D-instantons but their positions in xI -directions aredifferent. It is possible to obtain Fn(|B0) from Fn+1(z) through an LSZ-like procedure:

(2.28)Fn(|B0)=

d6zzFn+1(z),

where z is the Laplacian on R6. See Fig. 4 for an image of this procedure. The variation of thetotal free energy is therefore

(2.29)F () =

d6zzF (, z),

where

(2.30)F(, z) =

n=0

n

n! Fn+1(z).

Since a single D-instanton is separated from the hypersurface along which the other D-instantons are distributed, the worldsheet is stretched in the region 0 |xI | = xI xI r where

(2.31)r = max{ls , h}and h is the distance of the separated D-instanton from the hypersurface of the D-instantons. Ifwe take |zI | to be larger than r , then the corresponding worldsheet has a thin tube connectingthe boundary at xI = zI and the body of the worldsheet. The tube represents the propagationof closed string states, and the dominant contribution comes from the massless propagation. Asa result, F(, z) behaves as |zI |4 for large |zI |. For the range 0 |zI | r , we assume thatF(, z) varies slowly with |zI |.

Let us calculate the integral

(2.32)I =

d6zzf (z),

where

(2.33)f (z) ={

f (0) (0 |zI | r),r4

|zI |4 f (0) (|zI | > r)is a typical example of the function having the property assumed for F(, z). It is easy to checkthat I = 4vol(S5)r4f (0). Similarly, the RHS of (2.29) would be estimated as

(2.34)

d6zzF (, z) r4C(, r)F (, z = 0),where C(, r) is assumed to be of order one. Noticing that Fn+1(z = 0) = Fn+1, we obtain

(2.35)F () r4C(, r)F (),where

n (2.36)n=0 n!

Fn+1 = F ()


is used. In other words, the free energy transforms as

(2.37)F() F ( + r4C(, r)),which indicates the existence of the scale invariance if r 14 . Note that the sum of the infinitenumber of worldsheets with boundaries is crucial for the existence of this scale invariance.

It should be pointed out that our calculation has been done with an appropriate analytic con-tinuation of . At first, the free energy is defined as a power series of as (2.26). This definitionis valid when the effects from the D-instantons are small. However, to estimate the variation ofthe free energy, we assume the behavior of the sum F(, z), not each Fn(z), and therefore, thisestimate should be valid even beyond the convergence radius of the perturbative series (2.26).The result that there exists a scale invariance if r 14 agrees with the existence of an isometryof the corresponding metric when is large. To see the importance of the analytic continuation,let us consider the metric of D-instantons distributed along a four-dimensional hypersurface [7]

(2.38)ds2 = H() 12 ( dx dx + d2 + 2 d25 ),where

(2.39)H() = 1 + C4

.

In the region 14 , the metric (2.38) behaves as

(2.40)ds2 C[ dx

dx + d22

+ d25],

which is a metric on AdS5 S5. In this metric, the scale invariance is realized as the isometry(2.41)x = x, = .

From the perturbative string point of view, the metric (2.38) is given as a power series of .By summing them up, we obtain the non-trivial function H() 12 . The possibility to have sucha closed expression enables us to go beyond the convergence radius of the perturbative serieswhere the isometry exists.

We consider a worldsheet which is relevant to a high energy scattering of open string stateson the D3-branes. As is explained above, one of the D3-brane is separated from the other N 1D3-branes to implement the IR regularization, and all the external open string states are on theseparated D3-brane.

We make a scale transformation in the T-dual picture which is isotropic and brings the world-sheet to a region far away from the D-instantons. The situation is depicted in Fig. 5. If we take to be large, then the worldsheet can be brought to a place far enough from the hypersurface of theD-instantons so that the presence of the D-instantons is represented by the curved background(2.38) while the scale invariance is still valid. In the region 14 , the background (2.38) isapproximated by AdS5 S5. Then, the classical solution of the string is the minimal surfaceobtained in [5]. The size of the minimal surface is determined by the momenta of the externalopen string states, which become larger and larger by the scale transformation. It should be notedthat, before the scale transformation, the size of the worldsheet is smaller than the string scale, as

mentioned at the beginning of this section. After the scale transformation, this classical solutionwill dominate the summation over worldsheets. Note that, since the scale transformation for the


Fig. 5. Scale transformation in the T-dual picture.

D-instanton system is isotropic, the ratio of the size of the minimal surface to the distance fromthe hypersurface does not change. This implies that the worldsheet always exists near the bound-ary ( = 0) of AdS5. Notice that in the D-instanton case, the near horizon region correspondsto the region of the boundary of AdS5, as can be seen from the metric (2.38). This is in contrastwith the situation in [1] where D3-branes are placed at the center of the AdS5 spacetime fromthe gravity point of view.

Since the scale transformation is a symmetry as long as the worldsheet is restricted in theregion |xI | 14 , the classical action for the minimal surface provides the high energy scatteringamplitude with which we started. In this way, the scale invariance found in [1] provides theargument which verifies the relation claimed in [5].

One may think that the approximate symmetry is broken by the presence of the non-trivialdilaton background

(2.42)e = H().Recall that the dilaton coupling in the worldsheet action is a sub-leading order term of . Sincethe actual expansion parameter is

R2where R is a typical length scale of the target spacetime,

which is proportional to 14 in this case, the contribution from the dilaton background would be

negligible if we take a large .

3. Finite temperature

Next, we consider D3-branes at finite temperature. This is realized by considering a Euclid-eanized D3-brane which wraps on a circle with the circumference . AdS/CFT correspondencehas also been considered in this case [8]. The argument on the scale transformation can be alsocarried out in the case of the finite temperature.

One of the crucial points of our perturbative approach is the transformation property of theboundary state of the wrapped D3-brane. Since we impose the anti-periodic boundary condi-tion for spacetime fermions in the Euclidean time direction on which the D3-brane wraps, thefermionic worldsheet variables flip their signs as the string winds around the direction. The scale

transformation for these winding sectors is the same as that for the zero-winding sector, exceptfor the fact that the modings of Sa and Sa are half-integral if the winding number is an odd


integer. Due to the presence of these winding sectors, the boundary state has the form

(3.1)|B =wZ

|B;w,

where w is the winding number, and

(3.2)|B;2k = exp[

n=1

(1nMijin

jn iMabSanSbn

)]|B0;2k,

(3.3)|B;2k + 1 = exp[

n=1

(1nMijin

jn iMabSan+ 12 S

b

n+ 12

)]|B0;2k + 1.

The term |B;2k has almost the same form with the supersymmetric boundary state [9], and thetransformation property is the same except for the obvious scaling of the circumference . Itmight look non-trivial to analyze the transformation property of the term |B;2k + 1, due to thehalf-integral moding of the fermionic oscillators. However, one can check that the calculationscan be done similarly with the case of |B;2k, and obtain

(3.4)|B;2k + 1 |B;2k + 1.Therefore, we can define the scale transformation so that the boundary state |B is scale invari-ant up to the scaling of the circumference . Then, AdS/CFT correspondence also follows atfinite temperature case, as in [1]. Note that the length scale of the background, if exists, is alsotransformed by the scale transformation. This fact implies that, if one considers a Wilson looprealized as in [1], then it is related to a minimal surface placed at the outside of the event horizonof a black hole background, since the boundary of the worldsheet is always at the outside of theevent horizon.

There are other researches in the case at finite temperature which discuss the entropy ofN = 4SYM and that of the AdSSchwarzschild black hole [8,10]. Due to the limitation of the explicitcalculations, the entropy is calculated in the classical gravity only when is large, and in theSYM only when is small. The corrections to these results have also been calculated in [1114]which suggest that the entropy in the gravity region is smoothly interpolated to the entropy inthe SYM region by varying . These are the arguments supporting that the SYM entropy and theblack hole entropy are the same for any . In the following, we will show the coincidence of theentropy at large .

Our point of view on AdS/CFT correspondence is based on D3-branes in the flat spacetime.The temperature is encoded in the radius of the Euclideanized time direction. The temperature-dependence thus comes from strings which wind around the time direction. Let us consider thefree energy of strings in this case. In the large N limit with kept fixed, the winding closedstrings with the genus h 1 provide contributions of order g2h2s 2h2N22h which is atmost of order N0. On the other hand, the winding open strings may provide contributions oforder N2. Therefore, in the large N limit, the temperature-dependent part of the free energy isdominated by open strings. When the temperature is small, then only massless open string statescontribute. As a result, the thermodynamical quantities which are obtained from the temperature-dependence of the free energy, for example the energy and the entropy, are equal to those of theSYM in the limiting case mentioned above.

From an observer at the asymptotically flat region, the thermal D3-branes are regarded as anon-extremal black hole(3.5)ds2 = H(r) 12 (f (r) dt2 + d x2)+ H(r) 12 (f (r)1 dr2 + r2 d25 ),


where

(3.6)H(r) = 1 + 4r4

, f (r) = 1 r40r4

.

The constant r0 is determined so that the Euclideanized version of (3.5) does not have a conicalsingularity. If is taken to be large, then r0 can be written as

(3.7)r0 = 12

.

Note that here is the circumference of the Euclidean time circle at the asymptotically flatregion. Therefore, this coincides with the one appeared in the D-brane setup. In this way, wecan equate the temperature of the gravity side with that of the gauge theory side.

The gravity description is valid when r0 is large. If we take to be large, we can also take to be large while keeping r0 still large. This means that there exists a parameter region in whichboth the gravity description and the SYM description are valid. (The description in terms of freeSYM cannot be valid in this region, of course.) The ADM mass of the black hole must be thesame with the total energy of the SYM plus the contribution from the tension of the D3-branes.Since the entropy is obtained from the thermodynamic relation

(3.8)SE

= 1T

,

we can conclude that the entropies of the SYM and the black hole must be the same, due tothe coincidence of the energy and the temperature, although it is difficult to perform an explicitevaluation of the entropy of the SYM with large .

The crucial points of our argument are that the temperature-dependence of the free energy isdominated by open strings in the large N limit, and that there is a region of parameters whereboth the gravity description and the SYM description are valid. It is also important that someof physical quantities, the energy and the temperature for example, in two descriptions can becompared directly with each other at the asymptotically flat region which is absent after takingthe near-horizon limit.

4. Discussion

We have shown that our point of view of AdS/CFT correspondence, based on the perturbativetype IIB string theory with D3-branes, can be applied to some situations discussed in the literatureof AdS/CFT correspondence. The amplitude of high energy scattering processes of the SYM canbe related to classical worldsheet configurations in the AdS5 space obtained in [5] by the scaletransformation, which clarifies the reason why such a correspondence holds. Thermal propertiesof the SYM are also discussed in our point of view. It can be shown that the reduction of openstring system on D3-branes to the SYM occurs even in the case of large , and some thermalproperties of the SYM can be related to some gravitational quantities, the connection being madeat the asymptotically flat region far from the D3-branes.

The successful applications of our point of view indicate that it would enable us to makefurther predictions for the correspondence which are not known so far. As long as the corre-spondence is based on our scale invariance, the coincidence of some quantities will have a firm

footing. We hope to provide some new and non-trivial correspondences between gauge theoryand gravity, possibly less supersymmetric, which could be checked by explicit calculations.


Acknowledgements

We would like to thank M. Fukuma, T. Matsuo, T. Takayanagi, T. Uematsu for valuable discus-sions. We would also like to thank L. Dixon for valuable comments on the IR regularization of theSYM amplitudes. This work is supported by the Grant-in-Aid for the 21st Century COE Cen-ter for Diversity and Universality in Physics from the Ministry of Education, Culture, Sports,Science and Technology (MEXT) of Japan. The research of T.S. is supported in part by JSPSResearch Fellowships for Young Scientists.

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Stationary black holes and attractor mechanism

Dumitru Astefanesei a, Hossein Yavartanoo b,

a Global Edge Institute, Tokyo Institute of Technology, Tokyo 152-8550, Japanb Center for Theoretical Physics and BK-21 Frontier Physics Division Seoul National University,

Seoul 151-747, South KoreaReceived 4 July 2007; accepted 22 October 2007


Abstract

We investigate the symmetries of the near horizon geometry of extremal stationary black hole in four-dimensional Einstein gravity coupled to Abelian gauge fields and neutral scalars. Careful consideration ofthe equations of motion and the boundary conditions at the horizon imply that the near horizon geometryhas SO(2,1) U(1) isometry. This compliments the rotating attractors proposal of hep-th/0606244 thathad assumed the presence of this isometry. The extremal solutions are classified into two families differen-tiated by the presence or absence of an ergo-region. We also comment on the attractor mechanism of bothbranches. 2007 Elsevier B.V. All rights reserved.

1. Introduction

The attractor mechanism plays a key role in understanding the entropy of non-supersymmetricextremal black holes in string theory [1,2]. In certain cases, the macroscopic entropy of ex-tremal non-supersymmetric attractor horizons can be matched to the weak coupling statisticalentropy despite the fact that these quantities do no seem to be protected by supersymmetry[37].

It was originally noticed in [8] that the extremal four-dimensional Kerr and KerrNewmanblack holes have an SO(2,1) U(1) isometry. The last year, the authors of [9], found evenmore four-dimensional extremal black holes had this isometry. Emboldened by this observation,

* Corresponding author.E-mail addresses: [email protected] (D. Astefanesei), [email protected] (H. Yavartanoo).0550-3213/$ see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2007.10.015

14 D. Astefanesei, H. Yavartanoo / Nuclear Physics B 794 (2008) 1327

they found that, for four-dimensional stationary extremal black holes, in a theory of gravitywith neutral scalar fields non-minimally coupled to Abelian gauge fields, one can generalisethe entropy function formalism of [10] simply by assuming an SO(2,1) U(1) near horizongeometry.

The generalised entropy function is constructed, on an SO(2,1) U(1) symmetric back-ground, by taking the Legendre transform (with respect to the electric charges and angularmomentum) of the reduced Lagrangian evaluated at the horizon. Extremising the entropy func-tion is equivalent to the equations of motion and its extremal value corresponds to the entropy.Since the entropy function depends only on the near horizon geometry, its extremum and hencethe entropy is independent of the asymptotic data. This is precisely the attractor behaviour.However, if the entropy function has flat directions something interesting happens: while theextremum remains fixed, flat directions will not be fixed by near horizon data and can depend onthe asymptotic moduli.

There exist two distinct branches of stationary extremal black hole solutions which, in [9],are dubbed ergo- and ergo-free branches according to their properties.1 The first branch, alsoknown as the fast branch, can exist for angular momentum of magnitude larger than a certainlower bound and does have an ergo-region. On the other hand, the ergo-free branch can existonly for angular momentum of magnitude less than a certain upper bound. The ergo-free branchcan also be smoothly connected to a static extremal black hole.

The entropy function has no flat directions for the ergo-free branch: the scalar and all otherbackground fields at the horizon are independent of the asymptotic data. However, there is adrastic change for the ergo-branchthe entropy function has flat directions: despite the entropybeing independent of the moduli, the near horizon fields are dependent on the asymptotic data.

We find it significant that, the existence of an ergo-region allows energy to be extracted clas-sically either by the Penrose process for point particles or by superradiant scattering for fields.It is tempting to believe that the presence of the ergo-sphere is intimately related to the appear-ance of flat directions. One might say that the ergo-branch, not completely isolated from itsenvironment due to these processes, retains some dependence on the asymptotic moduli. Fromthis perspective, it is amazing that the black hole is isolated enough for the entropy to remainindependent.2

A consistent microscopic picture for KaluzaKlein (KK) black hole in agreement with themacroscopic analysis of rotating attractors [9] was provided in [3,4]. That is, the D-brane modelreproduces the entropy of KK black hole, while the mass gets renormalized from weak to strongcoupling just for the ergo-branch black hole solutions in agreement with the existence of theflat directions in the entropy function for this branch. Emparan and Maccarrone, [3], have alsoprovided a microscopic interpretation for the superradiant ergosphereeven if the temperature isvanishing, the extremal black holes with ergosphere correspond to states with both left- and right-moving excitations such that the open strings can combine and the emission of closed strings ispossible. The extraction of energy should reduce the angular momentum in such a way that theevent horizon area is increasing (it cannot decrease in the classical processes). Indeed, sincethe left-moving excitations have spin, the emitted closed string will necessarily carry angularmomentum away from the black hole.

1 The existence of two branches in the moduli space of extremal rotating black holes was discussed for the first timein [11].2 It is possible that the addition of higher derivative terms might lift these flat directions. It would be interesting to seewhether this would erase the ergo-sphere.

D. Astefanesei, H. Yavartanoo / Nuclear Physics B 794 (2008) 1327 15

In this note we fill up a gap in the proposal of [9] by proving that the near horizon geometryof extremal rotating black holes in EinsteinHillbert gravity coupled to Abelian gauge fields andneutral scalar fields has an enhanced SO(2,1) U(1) symmetry. Unlike the static case wherethe near horizon geometry is AdS2 S2, the AdS2 part does not decouple from the angularpart and the values of the moduli at the horizon have an angular dependence. Also, by addingangular momentum to static black holes, the SO(3) symmetry of the sphere is broken to U(1).However, the near horizon geometry is still universal in the sense that is still independent of thecoupling constants and is determined just by charges and angular momentum parameter. Theattractor mechanism is related to the extremality rather than to the supersymmetry property ofthe theory/solution. Indeed, the enhanced symmetry of the near horizon geometry and the longthroat of AdS2 is at the basis of the attractor mechanism for stationary black holes [9,10].

2. Generalities

We consider a theory of gravity coupled to a set of massless scalars and vector fields, whosegeneral bosonic action has the form

I[G,

i,AI]= 1

k2

M

d4xG

[R 2gij ()ij fAB()FAFB

(1) 12GfAB()F

AF

B

],

where FA with A = (0, . . . ,N) are the gauge fields, i with (i = 1, . . . , n) are the scalar fields,and k2 = 16G4. The moduli determine the gauge coupling constants and gij () is the metricin the moduli space. We use Gaussian units to avoid extraneous factors of 4 in the gauge fields,and the Newtonss constant is set to G4 = 1.

Varying the action we obtain the following equations of motion for the metric, moduli, andthe gauge fields:

(2)R 2gij ij = fAB(

2FAFB 12GFAF

B

)

(3)1G(Ggij j )= 14

fAB

iFAF

B + 18G

fAB

iFAF

B

,

(4)[G

(fABF

B + 12GfABF

B

)]= 0.

To get the equations of motion, we have varied the moduli and the gauge fields independently.The Bianchi identities for the gauge fields are FA[;] = 0.

We are interested in stationary black hole solutions to the equations of motion. In generalrelativity the boundary conditions are fixed. However, in string theory one can obtain interestingsituations by varying the asymptotic values of the moduli and so, in general, the asymptoticmoduli data should play an important role in characterizing these solutions. Indeed, the non-extremal black hole solutions are characterized by the usual conserved charges and also by thescalar chargesthe scalar charge is defined as the monopole in the multipoles expansion ofthe scalar field at the boundary. Thus all its properties are moduli dependent, e.g., the entropy

depends by the asymptotic values of the moduli. However, the entropy of extremal solutionsobtained by taking the smooth limit when the temperature is vanishing is independent of the


asymptotic moduli data. We will see in the next section that the enhanced symmetry of their nearhorizon geometry make them special in this regard.

Now let us write down the most general stationary black hole solution by using just its sym-metries.3 An asymptotically flat spacetime is stationary if and only if there exists a Killingvector field, , that is time-like at spatial infinityit can be normalized such that 2 = 1. Itwas also been shown that stationarity implies axisymmetry [12] and so the event horizon is aKilling horizon. Using the time-independence and axisymmetry we can write the most generalstationary metric with an axial Killing vector, , as

(5)ds2 = gtt dt2 + 2gt dt d + g d2 + grr dr2 + g d2.The event horizon of a stationary black hole is a Killing horizon of t + , where the

constant coefficient is the angular velocity of the horizon. It is convenient to rewrite the metric(5) in the ADM form

ds2 = N2 dt2 + ij(dxi +Ni dt)(dxj +Nj dt)

(6)= N2 dt2 + g(d +N dt)2 + grr dr2 + g d2,

and so we obtain:

N2 = (gt)2

g gtt , N = gt

g, ij = gij .

In this form, the event horizon corresponds to the surface N2 = 0. The shift vector, N , evaluatedat the horizon reproduces the angular velocity of the horizon:

= NH

= gtg

H

.

By eliminating the conical singularity in the Euclidean ( = it, r) sector, we obtain the temper-ature

(7)T = 1

= (N2)

4N2grr

H

.

3. Near horizon geometry of extremal black holes

We consider a generic covariant two derivative gravity Lagrangian that has three basic com-ponents: metric, scalars, and gauge fields. We show that, given a few simple assumptions, thenear horizon geometry of a stationary, extremal spinning black hole solutions of this Lagrangiannecessarily has the near horizon symmetry SO(2,1) U(1). To prove the previous statement,we make use of the following ingredients:

Symmetries: we assume time independence and axisymmetry; The black hole is extremalin other words the surface gravity (temperature) is zero; We expand the fields near the horizon and take a scaling limit; Gauge choices;3 The thermodynamics of the non-extremal black hole solutions using the method developed in [1315] will be pre-sented in [16].


Finiteness of certain physical quantities; Equations of motion; Spherical topology of the horizon.

3.1. Constraining the metric

As a warm-up exercise, we begin by examining 4-dimensional spherically symmetric blackholes by using the following ansatz:

(8)ds2 = a(r)2 dt2 + a(r)2 dr2 + b(r)2 d2.The near horizon geometry of the extremal black holes can be obtained in two steps: first, takethe extremal limit when the temperature is vanishing (this is a smooth limit on the Lorentziansection) and then obtain the near horizon geometry. We expand the metric components near theouter horizon and for the non-extremal solution (r+ = r) we obtain:

a2 = f (r) = (f0 + f1 + f22 + ),(9)b2 = ( + )

a2= +

f0 + f1 + f22 + and so the temperature is f0 = 4T . Here we used a coordinates system such that the horizon isat = r r+ = 0 and defined the non-extremality parameter = r+ r.

The extremal limit is obtained for 4T = f0 0 and to obtain the near horizon geometrywe also take = 0. By changing the coordinate = t/f1 one can easily obtain the AdS2 S2explicitly

(10)ds2 = 1f1

(2 d 2 + 1

2d2

)+ 1

f1d2.

We use a similar method to obtain the near horizon geometry of stationary extremal blackholes. However, the extremal limit in this case is more subtle since we should also consider thenon-diagonal component ( d dt) of the metric. Let us first rewrite the metric components in amore useful form:

N2 = (r r+)(r r)(r, ), N = + (r r+)(r, ),(11)grr = 1

(r r+)(r r)(r, ) ,where (r, ), (r, ), and (r, ) are regular functions. The temperature can be read off asbefore and using Eq. (7) we obtain

(12)T = r+ r4

()(),

where () and () are the values of (r, ) and (r, ) at the outer horizon. For a non-extremal black hole the temperature (surface gravity) is finite and constant on the horizon andso we obtain

()() = C, where C is a constant that depends on the charges P,Q and

the (mass and angular) parameters m,a. Expanding the ADM form of the metric near the outerhorizon we obtain the following metric:

(r r+)(r r)() dt2 + g[d + (r r+)() dt

]2

(13)+ 1

(r r+)(r r)()dr2 + g d2.


To obtain the near horizon geometry, we first construct the following family of metrics

(14)r r+ + r, t t,

where is an arbitrary parameter. There is a smooth limit 0 for which the near horizongeometry is obtained. Obviously, this is important for stationary field configurations where thereexist also terms of the form dr dt . This limit is especially useful when we consider the nearhorizon expansion of the gauge fields.

Taking the extremal limit (r+ r) and choosing a particular gauge we obtain

(15)ds2 = ()(r2 dt2 + dr

2

C2r2

)+ sin

2

()

(d + r() dt)2 + ()

C2d2,

where = t

. To obtain the above expression we use the appropriate coordinates system inwhich g = r2grr and the gauge freedom to write ()g() = sin2 . Here, we have consid-ered the metric in a rotating frame with respect to a distant observer with the angular velocityequal to that of the black hole. For a horizon with spherical topology, we require

(16)sin2

C22(){2, 0,( )2, ,

such that the deformed horizon, labelled by the coordinates (,), is a smooth deformation ofthe sphere. Unlike the static case, the fields at the horizon have an angular dependence and sosolving the attractor equations requires boundary conditions, i.e., the values of the fields at thepoles of the horizon.

Let us end up this subsection with an important comment about the extremal limit. For astationary black hole there are three intensive parameters associated to the horizon: the angularvelocity, the temperature, and the electric (magnetic) potential. Thus, there are two interestingextremal limits T = 0 when the angular velocity is or is not vanishing. In the discussion sectionwe comment further on the physics of the extremal black holes.

3.2. Constraining the scalars and gauge fields

Let us start by investigating the scalar and the gauge fields configuration in the near horizonlimit. For simplicity, we do not carry on the moduli and gauge fields indices in this subsectionwe specialize to one scalar and one gauge field configuration, but the generalization to a config-uration with more than one scalar and one gauge field is straightforward. Expanding the scalarsat the horizon, r = 0, we obtain

(17)(r, t) = r(() + r1() +O(r2)+ ).Requiring that the scalars are finite at the horizon, implies 0 and then by taking the scalinglimit, r r , 0, we find

(18) ={0(), = 0,0, > 0,

in the near horizon region. We assume that the near horizon effective gauge coupling f (()) iswell behaved and can be Taylor expanded around the poles, i.e.,(19)f (())= f0 + f1 +O(2).


Let us turn to the gauge fields and perform a similar analysis. We impose that the gauge fieldsare time-independent and start with the following ansatz

(20)A = At(r, ) dt +Ar(r, ) dr +A(r, ) d + A(r, ) d,that can be further simplified by choosing an appropriate gauge choice to fix A = 0 (or Ar = 0).We can expand the gauge fields about the horizon as

(21)A = r[at ()+O(r)]r dt + r[ar() +O(r)]drr

+ r [a() +O(r)]d.Requiring F 2 remains finite at the horizon implies ,, 0. We take ,, = 0 so thatafter taking the scaling limit r r , t t/, 0 we obtain a non-zero result. With thisassumption the scaling limit gives

(22)A = at ()r dt + ar()drr

+ a() d +O().The Einstein equations can be written as

(23)R 2 = f(

2FF 12gFF

).

The (r ) equation plays an important role in what follows: using the results from Appendix Aand the fact that = 0, we get

(24)sin2()

2()()() = 0,

which implies () is, in fact, a constant. This was the last step in our proofit is straightforwardto check that the metric (15) with () a constant function has the SO(2,1)U(1) isometry.

4. Attractor mechanism

In this section, we consider the attractor mechanism for static and stationary black holes. Forthe static black hole solutions, we show the equivalence of the entropy function formalism andthe effective potential method. Entropy function formalism was generalized to stationary blackholes in [9]. We comment on the role played by the enhanced symmetry of the near horizongeometry in decoupling the moduli equations of motion at the horizon from the bulk.

4.1. Static black holes

The Bianchi identity and equation of motion for the gauge fields can be solved by a fieldstrength of the form

(25)FA = f AB(QB fBCPC) 1b2

dt dr + PA sin d d,where PA,QA are constants that determine the magnetic and electric charges carried by thegauge field FA, and f AB is the inverse of fAB .

As discussed in [17], the equations of motion for the moduli are(26)r(a2b2gij r

j)= 1

2b2Veffi

,


where Veff(i) is a function of scalars fields i given by

(27)Veff(i) = f AB(QA fACPC

)(QB fBDPD

)+ fABPAPB.It is clear from Eq. (26) that Veff(i) is an effective potential for the scalar fieldsit plays animportant role in describing the attractor mechanism [17,18].

For the attractor mechanism it is sufficient for two conditions to be met. First, for fixedcharges, as a function of the moduli, Veff must have a critical point. Denoting the critical val-ues for the scalars as i = i0 we have,

(28)iVeff(i0) = 0.Second, the matrix of second derivatives of the potential at the critical point,

(29)Mij = 12ijVeff(i0),should have positive eigenvalues.

Once the two conditions mentioned above are met it was argued in [17] that the attractormechanism works and the entropy is given by the effective potential at the horizon.

The near horizon geometry is AdS2 S2 and so we can apply Sens entropy function [10]to investigate the attractor behaviour of static extremal solutions. All other background fieldsrespect the SO(2,1) SO(3) symmetry of AdS2 S2. We keep the analysis general in order tounderstand the role of Veff.

In [10], Sen found that the entropy of a spherically symmetric extremal black hole is theLegendre transform of the Lagrangian densitythe only requirements are gauge and generalcoordinate invariance of the action. In fact, this is similar with a generalization of the Waldsformalism for extremal black holes and it is based on the observation that there is a smoothextremal limit on the Lorentzian section of a charged black hole.

The entropy function is defined as

(30)F(u, v, e, p) = 2(eiqi f (u, v, e, p))= 2(eiqi

d d

GL),

where dGL is the Lagrangian density, qi = f/ei are the electric charges, us are the

moduli values at the horizon, pi and ei are the near horizon radial magnetic and electric fields,and v1, v2 are the sizes of AdS2 and S2, respectively. Thus, F/2 is the Legendre transform off with respect to the variables ei . Then, for an extremal black hole of electric charge Q andmagnetic charge P , Sen have shown that the equations determining u, v and e are given by:

(31)Fus

= 0, Fvi

= 0, Fei

= 0.

Thus, the black hole entropy is given by S = F(u, v, e, p) at the extremum (31). We observe thatthe entropy function, F(u, v, e, p), determines the sizes v1, v2 of AdS2 and S2, and also the nearhorizon values of moduli us and gauge field strengths ei .

Now, we are ready to apply this method to our action (1). The general metric of AdS2 S2can be written as

2(

2 2 1 2) ( 2 2 2) (32)ds = v1 d +

2d + v2 d + sin d .


The field strength ansatz is

(33)FA = FAr dr d + PA sin d d = eA dr d + PA sin d dand so

F(v1, v2, e, q,p) = 2[qAe

A f (v1, v2, e,p)],

(34)f (v1, v2, e,p) = 8k2

[v2 + v1 fAB

(v2v1

eAeB + v1v2

pApB)

2fABeAeB].

The attractor equations are:

(35)Fv1

= 0 1 v2v21

fABeAeB 1

v2fABp

ApB = 0,

(36)Fv2

= 0 1 + 1v1

fABeAeB v1

v22fABp

ApB = 0,

(37)Fi

= 0 fABi

(pApB eAeB)= 2 fAB

ieApB,

(38)FeA

= 0 qA = 16k2

(v2v1

fABeB fABpB

).

By combining the first two equations we obtain v = v1 = v2 = fAB(eAeB + pApB) that is ex-pecting also from our near horizon geometry analysis above. Its also easy to check that theentropy is given by F at the attractor critical point:

(39)S = F = 162

k2fAB

(eAeB + pApB)= v.

Using the electromagnetic field ansatz (25), it can be easily shown that S = Veff, qA = QA(the sign appears because of our convention for FAtr ), and (38) matches the critical pointcondition of Veff.

4.2. Stationary black holes

We have shown in the previous section that the near horizon geometry of extremal spinningblack holes has the symmetries of AdS2 S1 and can be written as

(40)ds2 g dx dx = v1()(r2 dt2 + dr

2

r2

)+ 2 d2 + 2v2()(d r dt)2

and the most general field configuration consistent with the SO(2,1)U(1) symmetry of AdS2 S1 is of the form:

i = ui(),(41)1

2FA dx

dx = (eA bA())dr dt + bA() d (d r dt),i Awhere , and ei are constants, and v1, v2, u , and b are functions of . Here is a periodic

coordinate with period 2 and takes value in the range 0 .


Based on this observation, a generalized entropy function was proposed in [9]

(42)F d d 2(J +QAeA

d

detGL),

and so there is one more attractor equation associated to the angular momentum J . Thus, theentropy and the near horizon background of a spinning extremal black hole are obtained byextremaizing this entropy function that depends only on the parameters labelling the near horizonbackground and the electric and magnetic charges and the angular momentum carried by theblack hole.

Interestingly, in all known cases, the appearance of flat directions in the entropy is associatedwith the presence of an ergo-sphere. Since not all moduli are fixed at the horizon the mass isnot guaranteed to be fixed. The microscopic analysis of [3] confirms that in fact the mass is notfixed and so there is a nice microscopic interpretation for the ergo-branch. On the other hand,the slowly spinning extremal black holes in the ergo-free branch lack the rotational superradi-ance but can produce superradiant amplification of KK electric charged waves. However, thisphenomenon cannot be easily seen in the CFT since it is related to a modification of the centralcharge.

One important question is if there is a similar effective potential for stationary black holes andif one can use a similar analysis as in the static case to study the attractor mechanism. Unfortu-nately, at this point, we have just shown that the equations of motion at the horizon decouple fromthe bulkwe hope to report a detailed analysis elsewhere. Here, let us just indicate the main stepin this analysis. We start by trying to solve the equations of motion and Bianchi identities forthe gauge fields. However, unlike the static case, we cannot obtain a general expression for thegauge fields, but rather their expressions in terms of two unknown functions (A and u):

FMrt = rAMt (r, ), FMt = AMt (r, ),(43)FM r = fMN (i)uN(r, )g , FM = fMN

(i)ruN(r, )g .

One can write down the expressions of the gauge fields in some concrete examples. However,an interesting exercise is to work with this general form of the gauge fields and try to extractas much information as possible from the equations of motion. For the moduli the equations ofmotion become

1G(Gi)= 1

2fMN

i

(rA

MrAN + r2AMAN

)

(44)+ 12fMN

i

(ruMruN + r2uMuN

).

However, in this case, the moduli have also an angular dependence and the equations do notdecouple. In principle one should be able to read off the effective potential from the right-handside of this equation, but that is not straightforward in this casean effective potential for con-stant scalar fields was proposed in [9]. The best thing we can do is to check what is happeningin the near horizon limit. After some tedious manipulations we found that in the near horizonlimit the moduli equations are decoupled from the bulk. The scalar fields at the horizon havealso an angular dependence and we obtain a system of distributions rather than functions. Thus,the boundary conditions, i.e., the values of the fields at the poles of the horizon, are important

and the equations are difficult to be solved in a general caseconcrete examples are presentedin [9].


5. Discussion

Recently, after the proposal of Sen [10], there was a lot of work on attractor mechanismand entropy function (see, e.g., [19]). Motivated by the generalization of the attractor mecha-nism to non-supersymmetric extremal stationary black holes, we investigated the near horizongeometry of spinning extremal black holes in a theory of gravity with uncharged scalar fieldsnon-minimally coupled to Abelian gauge fields. We found that the near horizon geometry ofthese black holes has the symmetry of AdS2 S1the AdS2 part does not decouple from theangular part. Consequently, the horizons are attractors for the moduli and their geometry is in-dependent of the boundary moduli data. One subtlety is that the extremal spinning black holesare further divided in two branches: ergo- and ergo-free branch, respectively. In both cases theSO(2,1) isometry of AdS2 is generated by the Killing vectors:

L1 = t , L0 = tt rr ,(45)L1 = (1/2)

(1/r2 + t2)t (tr)r + (/r),

but they have distinct properties. The former is characterized by an entropy function with flatdirections and for the latter there is no flat directions of the entropy function. If there are no flatdirections then, clearly, the entropy is independent of the moduli. On the other hand, if there areflat directions, then the extremization of the entropy function does not determine all the modulivalues at the horizon. Location of these parameters along the flat directions may depend on theasymptotic values of the moduli. But since the entropy function does not depend on the flatdirections, the entropy is still independent of the asymptotic values of the moduli, and so has anattractor behaviour.

Let us comment now on the physics of the two branches. In general, in the supergravityapproximation, the entropy is a function of the duality invariant combinations D(QA,PB),S =(|D| J 2) and the mass saturates an extremality bound that is independent of the angu-lar momentum parameter, J 2the plus sign corresponds to the ergo-free branch and the minussign to the ergo-branch. When D = J 2 the extremal horizon disappears and becomes a nakedsingularitythis situation resembles the static case with one charge. Except this situation, theextremal limit has finite area and zero surface gravity. The fastly spinning extremal black holeshave a non-zero horizon angular velocity and so their causal structure is similar with Kerr so-lution. Let us start with a non-extremal black hole that Hawking radiates. Clearly, Hawkingradiation carries away the angular momentum and so the black hole is slowing down. If theblack hole is radiating away all the angular momentum before reaching the extremal limit, thenthe corresponding solution will be in the ergo-free branch. On the other hand, if the black holereaches the extremal limit and the angular velociy is non-zero, then there is radiation due to theergo-region. If the evaporating process is fine tuned such that the extremal limit is reached whenJ = |D|, then the black hole behaves more as an elementary particle [20]there are potentialbarriers outside the horizon which increase without bound.

In this paper we also tried to extend the analysis of the effective potential to extremal spin-ning black holes. We have not been able to conclusively construct an explicit effective potential,mainly because of technical obstacles. In the static case one can explicitly check that the mod-

uli are fixed at the attractor horizon that is a critical point for the effective potential. A similaranalysis is difficult for the stationary case. However, by studying the equations of motion for the


moduli, we concluded that they decouple from the bulk at the horizon.4 A complete determina-tion of the scalar fields at the horizon needs also imposing the boundary conditions which are thevalues of the fields at the poles of the horizon.

The near horizon geometry of a stationary extremal black hole is universal and so the entropydoes not depend of couplings. The extremality condition is very powerfull to force an attractorbehaviour of the horizonit is independent of the supersymmetry of the theory/solution. Thisdoes not come as a surprise, though, since the near horizon geometry has an enhanced symmetryand the long throat of AdS2 is the main ingredient for the existence of the attractor mechanism.

Acknowledgements

We thank Kevin Goldstein for collaboration in the initial stages of this work and for furtherdiscussions. It is also a pleasure to thank Soo-Jong Rey, Ashoke Sen, and Sandip Trivedi foruseful conversations. D.A. would like to thank KIAS, Seoul for hospitality during part of thiswork. D.A. has presented this work at ISM06 Puri (December 2006), KIAS, Seoul (February2007), YITP, Kyoto (February, 2007), TITECH, Tokyo (May 2007) and he likes to thank theaudience at all these places for their positive feedback. The work of D.A. has been done withsupport from MEXTs program Promotion of Environmental Improvement for Independence ofYoung Researchers under the Special Coordination Funds for Promoting Science and Technol-ogy, Japan. D.A. also acknowledges support from NSERC of Canada. H.Y. would like to thankthe Korea Research Foundation Leading Scientist Grant (R02-2004-000-10150-0) and Star Fac-ulty Grant (KRF-2005-084-C00003). While this paper was being completed, Ref. [21] appearedwhich overlaps with the material presented in Section 3.

Appendix A. The Maxwell equations in the near horizon limit

In this appendix we explicitly obtain the equations of motion for the gauge fields in the nearhorizon limit. These expressions are useful in Section 3.2for simplicity, we specialize again toa configuration with one scalar and one gauge field, but the generalization is straightforward.

The non-zero components of the Maxwell tensor are given by

(A.1)Ft i = (at (), rat (),0)

Fi = (ar ()/r,0, a())}

where i {r, ,}.

Raising the indices we obtain

(A.2)F t i = C2

2()

(at (),

()

r, 0

),

(A.3)F i = C2

2()

(C2rar (), 0,

2()

sin2 a()+ ()()

),

(A.4)Fr = C2

2()()rat (),

where () = at () ()a().4 This is not a sufficient condition for the attractor mechanism to exist. However, a rigurous proof was given in [9] byusing the entropy function formalism.


Maxwells equations are

(A.5)(GfF)= 0.

From the r-component of Maxwells equation we obtain

(A.6)(1() sin()f ()ar ()

)= 0,which can be integrated to give

(A.7)ar () = 1()

f sin 1

,

where we have assumed that the effective gauge coupling at the north pole, f ( = 0), is wellbehaved. Now, for F 2 to be finite at = 0, we require 1 = 0, i.e., ar = 0. This in turn meansthat ar() does not contribute to the Maxwell tensor and can be gauged away.

Similarly from the t-component of Maxwells equation we obtain

(A.8)(1 sin()f ()()

)= 0,which, by an argument similar to the one for ar above, implies is zero. Some important relationsused in this derivation are

(A.9)G = C2() sin and

(A.10)

(s)2 = g= 1

()

([r1t ]2 + C2r22r +C22 )+ ()sin2

2,

(A.11)F 2 = 2C2

2(a2t 2 + C2a2r )+ 2C

2

sin2 (a)2.

References

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Two-dimensional black holes in a higher derivativegravity and matrix model

Kwangho Hur, Seungjoon Hyun , Hongbin Kim, Sang-Heon YiDepartment of Physics, College of Science, Yonsei University, Seoul 120-749, Republic of Korea

Received 24 August 2007; accepted 23 October 2007


Abstract

We construct perturbatively a class of charged black hole solutions in type 0A string theory with higherderivative terms. They have extremal limit, where the solution interpolates smoothly between near horizonAdS2 geometry and the asymptotic linear dilaton geometry. We compute the free energy and the entropy ofthose solutions using various methods. In particular, we show that there is no correction in the leading termof the free energy in the large charge limit. This supports the duality of the type 0A strings on the extremalblack hole and the 0A matrix model in which the tree level free energy is exact without any correctionsin the leading order. 2007 Elsevier B.V. All rights reserved.

PACS: 04.70.Bw; 04.70.-s; 04.70.Dy; 11.25.Tq; 11.25.Pm

Keywords: Black holes; Noncritical 0A string theory; 0A matrix model

1. Introduction

Black holes are interesting objects in gravity and string theory, which may be awaiting thecomplete understanding on the quantum nature of gravity. Since black holes have a large curva-ture value region near the singularity, the non-perturbative formulation of gravity theory or itsnon-perturbative stringy generalization may be needed to understand the full quantum natureof black holes. Though full non-perturbative descriptions of M/string theories are not avail-

* Corresponding author.E-mail addresses: [email protected] (K. Hur), [email protected] (S. Hyun), [email protected]

(H. Kim), [email protected] (S.-H. Yi).0550-3213/$ see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2007.10.018

K. Hur et al. / Nuclear Physics B 794 (2008) 2845 29

able yet, there are several interesting toy string models whose non-perturbative descriptionsare known. One such class of toy models is the one of matrix models which correspond to thenon-perturbative formulation of noncritical string theories. Specifically, various two-dimensionalstring theories on the flat background can be reformulated in terms of matrix models. In somecases, the dual matrix models are believed to be a complete non-perturbative formulation of thecorresponding noncritical string theories. Then, one may naturally ask whether there are blackhole solutions in a noncritical string theory, and if there are, how they can be incorporated andunderstood in the context of dual matrix model.

It was proposed in [1] that the type 0A matrix model with = 0 and non-zero RR fluxesq+ = q = q is dual to 0A string theory on the extremal black hole [2]. Later on, the type 0Amatrix model was generalized [3] to incorporate two kinds of RR fluxes and found that it dependsonly on the combined flux1 Q = q+ + q. Since the generalized matrix model is the same as thematrix model with just one kind of RR flux and there is no evidence for the existence of blackholes in the type 0A matrix model side, it was argued that the matrix model is not related to the0A strings on black holes [3]. Furthermore, the curvature radius of the black hole solution is theorder of string length scale, independent of charge, and therefore the low energy gravity cannotbe trusted and it is not clear whether the black hole exists at all. Nevertheless, there are somepursuits of matching between the matrix model and type 0A strings on black holes or AdS space[46] with partial success.

In this paper we will try to extend these efforts by including lowest order corrections in thelow energy effective gravity for type 0A string theory. As we mentioned, the curvature radius ofthe black hole is the order of string length scale, and therefore higher derivative terms should betaken into account. One of the motivation of this work is to determine how the behavior of theblack hole geometry is modified under the higher derivative correction.

We find the perturbative evidence of the existence of charged black holes even with the higherderivative terms in the type 0A string theory, which interpolate smoothly between the near hori-zon geometry and the asymptotic geometry. In the case of the extremal black hole, the nearhorizon geometry is AdS2, as usual. This is in contrast to the four-dimensional cases where it isnot easy to find the interpolating solutions. We compute the exact entropy of this extremal blackhole using Sens formalism and find the condition on the coefficient a of the higher derivativeterm in order to have the extremal black hole solution. We also compute the free energy of thenon-extremal black holes using Euclidean action approach and Walds Noether charge method.In the extremal limit, the leading term, in the large charge limit, of the free energy turns out tobe unaffected under the -correction. This agrees with the result from the matrix model, whichsupports the duality of those two models.

The organization of the paper is as follows. In Section 2, we briefly review some relevantaspects on the type 0A string theory and its dual 0A matrix model. We also review the blackhole solutions in type 0A string theory. In Section 3, we construct charged black hole solutionsin the presence of higher derivative terms in the metric. We construct solutions, perturbativelyin the coefficient a, and find the black hole geometries which interpolate near horizon regionand asymptotic region, which is linear dilaton geometry. In Section 4, we compute the free en-ergy and the entropy of the charged black hole solutions, for both extremal and non-extremalcases using various approaches. The computation supports the duality between the 0A matrix1 There is an additional term which depends on the difference between RR fluxes, but it was irrelevant to argumentsfor black holes [3].

30 K. Hur et al. / Nuclear Physics B 794 (2008) 2845

model and the 0A strings on the extremal black hole geometry. In Section 5, we draw someconclusions. In Appendix A, we review some relevant results in the Noether charge method andin Appendix B, we derive a generic relation between the temperature dependence of the radiusof the time-like Killing horizon for a non-extremal black hole and the curvature radius of thenear-horizon geometry of the corresponding extremal black hole.

2. Black holes in the 0A string theory and the 0A matrix model

The nonchiral projection of two-dimensional fermionic string theories gives rise to two type 0string theories, so-called type 0A and type 0B. Only NSNS and RR sectors survive under theprojection, and thus the type 0 theories contain bosonic fields only. In type 0A string theory, theNSNS sector contains a graviton, a dilaton and a tachyon, while the RR sector includes twoone-form gauge fields [7,8]. It was known that the low energy effective theory of the 0A stringsadmits charged black hole solutions [2]. Based on the computation of the free energy, it wasargued that the 0A string theory on those black holes is dual to the 0A matrix model [1,9,10]. InSection 2.1, we review the low energy effective theory of 0A strings and its black hole solutions.In Section 2.2, we give some relevant features in the 0A matrix model for the comparison withblack hole side results.

2.1. Black holes at the lowest order in : Review

The low energy effective action of type 0A string theory at the lowest order in is givenby [7]

I0 =

d2xg

[1

22e2

(R + 4 + 8

f1(T )(T )2 + f2(T )

)

(1) 2

4f3(T )

(F+

)2 24

f3(T )(F

)2 q+F+ qF],

where F denote field strengths of two RR gauge fields and q denote the corresponding charges,respectively. The theory admits the following linear dilaton geometry as a vacuum solution:

= k,(2)ds2 = dt2 + d2,

where all other fields vanish. It is also known that there exist charged black hole solutions in themodel [1,2]. When the tachyon field T is turned off, RR gauge fields can be easily solved as

(3)F+01 = F01 =q

2, T = 0,

which corresponds to the configuration of the background D0-branes with charges given byq = q .

One may use this to integrate out RR gauge fields, and obtain the action of the form

(4)I 0 =

d2xg

[1

22e2

(R + 4 + 4k2

)+ ].2 Here we denoted the original cosmological constant as k = 2/ and new effective cosmological

constant, coming from the gauge field contributions, as = q2/(2). Therefore the low


energy effective theory of type 0A string theory reduces to the two-dimensional dilaton gravitywith two kinds of cosmological constants, one of which is related to the charges of RR gaugefields. The theory admits the charged black hole solutions in which the dilaton is taken to beproportional to the spatial coordinate ,

(5)0 = kwhile the metric is of the form

(6)ds2 = l0() dt2 + d2

l0(),

with the factor l0() given by2

(7)l0() = 1 e2k 12k (M0 ).As will be clear, M0 may be regarded as a mass of the black hole. One can also obtain theextremal black hole solution where the position of the horizon and the mass are given in termsof the charge as

(8)e2kex = 4k2

, Mex = ex + 2ke2kex = 2k[

1 ln(

4k2

)].

The near horizon geometry of the extremal black hole becomes AdS2.

2.2. The free energy in the 0A matrix model

In this section we give the expression of free energy in the 0A matrix model [3,7]. The freeenergy, F lnZ, of the 0A matrix model with the Fermi energy level and RamondRamond(RR) flux q compactified at the radius R (or at the temperature 1/(2T ) = /(2)) is given by

(9)3F0A3

=(

2

)2R Im

[ 0

dtt/2

sinh(t/2)

2 t/2R

sinh(

2 t/2R)

ei

2 t 12 qt

].

In this integral form non-perturbative effects for R and are included. Since it is enough for usto consider the perturbative effects, we perform the series expansion on x/ sinhx and integrateterm by term. This gives us the perturbatively expanded form as

3F0A3

=(

2

)2R Re

[ m=0

(1)m+1(2m)!(

2

i q2

)2m1

(10)m

n=0

|1 212(mn)||1 212n||B2(mn)||B2n|[2(m n)]!(2n)!

(/2R

)2n].

Note that index m corresponds to the genus expansion in the 0A string theory as (

2 i q2 )1

in the matrix model plays the role of the string coupling gs in the type 0A theory. On the otherhand, the summation over n corresponds to the -expansion.2 We set 22 = 1 from now on.


Since the thermodynamic free energy, F0A is defined by(11)F0A F0A,

we obtain

F0A =

2

1

Re

[1

2

(

2

i q2

)2ln(

2

i q2

)

+ 124

{1 +

(/2R

)2}ln(

2

i q2

)

(12)+

m=2(1)m2(2m 3)!

(

2

i q2

)22mFm(R)

],

where

(13)Fm(R) m

n=0

|1 212(mn)||1 212n||B2(mn)||B2n|[2(m n)]!(2n)!

(/2R

)2n.

To match the matrix model results with those of the type 0A string theory on the two-dimensional extremal black hole, we should take some appropriate limits in the above. Firstof all, we should take the infinite radius limit (i.e., zero temperature limit) with = 0. Further-more, we should double the RR flux q to get the effects from two kinds of RR flux q+ = q = q .After all these limits are taken, we obtain the tree level part of the thermodynamic free energy as

(14)F tree0A =

2

1

Re[1

2

(

2

iq)2

ln(

2

iq)]

=0=

2

q2

4lnq2.

Note that we have suppressed the ambiguous quadratic terms on q , which may have correc-tions. They are related to the divergent parts which exist in the free energy expressions and maybe regularized by the explicit cutoff. One may also note that there is no further correction inthe tree level free energy of the type 0A matrix model, which is not the case for higher genusones. In the next section, we will consider higher derivative corrections to the low energy effec-tive action of type 0A string theory and check that it gives the same free energy with the type 0Amatrix model.

3. Black holes in the higher derivative type 0A gravity

Now we would like to include the higher derivative correction, i.e., higher order correctionto the action given in the previous section. We restrict ourselves to the higher derivative terms inthe NSNS sector fields only.3 The unique higher derivative term for the metric in two dimen-sions, which appear in the next order in correction, is an R2 term. Furthermore, the resultsfrom function computation [11] tell us that we do not need to consider higher derivative termsin . Henceforth, it is enough to add the following correction term to the action (4):

(15)I1 = 122

d2xge2(aR2),3 One may include the generic higher derivative terms in the RR sector as well.


where a is a certain dimensionless number which may be fixed by the computation of correc-tions in string theory.

3.1. Black hole solutions

The equations of motion of the metric and the dilaton are found to be

0 = R + 2 + g[

22 4()2 + 4k2 + 12e2

] 4a

k2e2

(e2R

),

(16)0 = R + 42 4()2 + 4k2 + 2ak2

R2.

The metric for the black hole solutions of the above equations of motion is chosen to be theSchwarzschild type as

(17)ds2 = l() dt2 + d2

l().

In general, we have three equations of motion in which only two of them are independent. Aftersimple manipulation, the equations of motion for the metric component gtt and dilaton in theabove Schwarzschild type metric are given by

0 = l 6l 4l + 8l( )2 8k2 e2 + 4ak2

(2ll ll),

0 = l 4l 4l + 4l( )2 4k2 2ak2

(l)2,

where = dd

and l = dld

. It is not easy to obtain a black hole solution analytically. Instead, weuse perturbative approach to find out charged black hole solutions. We require that the solutionsreduce to the well-known black hole solutions as a 0. We also require that the geometry ap-proaches, asymptotically, to the linear dilaton geometry (2). The word perturbative here meansthe perturbative expansion in terms of variable a in Eq. (15). Let us recall a is just a numberfixed by correction and not a parameter we may vary arbitrarily. Furthermore there is no guar-antee that it is small. Nevertheless we may still regard it as an adjustable variable and try to doa perturbative analysis. The partial justification of this approach is given by the comparison ofresults from this approach with those from the exact one in the near horizon limit of extremalblack hole.

First of all, the analysis of the asymptotic behavior of the dilaton and the metric leads to

() = k +O(e4k),(18)l() = 1 e2k 1

2k(M ) +O(e4k),

where we denote M as a mass of the black hole. As will be shown in later section, there is anambiguity in defining the ADM mass in two dimensions, due to the divergencies. However thedifference M Mex, where Mex is the corresponding quantity in the extremal black hole withthe same charge, is well defined as the energy above the extremal black hole, and thus it maybejustified to call M as the mass of the black hole. The mass of the black hole depends on a andmay be written generically in the formM = M0 + aM1 + a2M2 + .


We introduce, for clarity, a dimensionless mass m and a dimensionless cosmological constant as

m = Mk

, = k2

.

Through the perturbative expansion in a, we obtain the solutions of the equations of motion,up to second order in a, as

(19) = k 16a2e4k(m0 k + )2 +O(a3),

(20)l() = l0() + al1() + a2l2() +O(a3),

where

l0() = 1 + e2k[1

2(m0 k)

],

l1() = e2k[m1

2

]+ e4k[2 + 2(m0 k)2],

l2() = e2k[m2

2

]+ e4k[962 + 4(m0 k)(m1 48) 128(m0 k)2]

(21)+ e6k[83 + 82(m0 k) + 48(m0 k)2 + 16(m0 k)3].At first glance, one may think that mi s are independent parameters describing the above

perturbative solutions. However, this is not the case and the mass of a black hole is expressedin terms of just a single parameter m as can be seen from the fact that all the expressions of thegiven solution can be rewritten as the function of a single variable m up to the given order. Interms of mass parameter m, the solution can be rewritten as

(22) = k 16a2e4k(m k + )2 +O(a3),l() = 1 e

2k

2(m k) + ae4k[2 + 2(m k)2]

32a2e4k[32 + 6(m k) + 4(m k)2]+ 8a2e6k[3 + 2(m k) + 6(m k)2 + 2(m k)3]

(23)+O(a3).This solution has several distinct features. First of all, the only combination which appear in

the metric and the dilaton is of the form

(24)p(m k)qe2(p+q)k.Secondly, the coefficients of an in the dilaton include, generically, terms proportional toe2lk , with l = 2,3, . . . , n, while the coefficients of an in the metric function l() contain termsproportional to e2lk , with l = 2,3, . . . , n + 1. These generic features can be shown to be truein all orders of a from the recursive structure in the equations of motion. From these properties,we find an additional characteristic feature of the solution: the solution depends only on on

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