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Observations on the Holographic Duals of 4-D Extremal Black Holes

We argue that extremal black holes of N=4 Poincare supergravity coupled to conformal matter have a quantum-exact dual 5-d description in the maximally supersymmetric extension of the Randall-Sundrum theory. This dual is a the classical, static supergravity solution describing a string with both Neveu-Schwarz and Ramond charge with respect to different antisymmetric tensor doublets. We also discuss the issue of the singularity present in a class of such supergravity solutions found by Cvetic, Lu and Pope.


Introduction
The 5-d Randall-Sundrum (RS) compactifications [1,2] have a holographic interpretation in terms of a dual 4-d theory consisting of dynamical gravity coupled to a strongly-interacting field theory (see [3] and references therein; see also [4,5]). Its supersymmetric extension has been studied by many authors. In particular, the findings in [6,7,8] will have a direct bearing on this paper. The N=4 extension of the model in [2], usually called RS2, allows for compelling quantitative checks due to non-renormalization theorems [9,5]. An interesting problem in this context is to find the 5-d configuration dual to a 4-d Schwarzschild black hole on the UV brane.
The linearized analysis of [10] has not been extended to an exact all-order solution, in spite of various attempts in the literature.
Recently, an intriguing reason for this failure has been proposed by Emparan, Fabbri and Kaloper [11], who have pointed out that the 5-d configuration dual to the Schwarzschild black hole must also reproduce its Hawking radiation at the classical level, so that it cannot be stationary, since the 4-d black hole necessarily radiates away in an asymptotically Minkowsky background.
In any extension of the RS2 scenario with N≥ 2 supersymmetry, the 4-d dual possesses absolutely stable extremal black holes, that preserve some of the supersymmetry and do not Hawking-radiate. This suggests the possibility that such 4-d black holes can be represented in 5-d AdS space by classical, static configurations. Indeed, solutions were found in [7] in the N=2 case, and in [8] in the N=4 case. They are strings ending on the UV brane, and extending in a straight line from the brane to the AdS 5 horizon 2 . Their drawback is that they always have null singularities on the AdS 5 horizon. In this paper, we use general properties of maximally extended supersymmetry to argue that, whether or not the solutions in [7,8] are unique, any other solution corresponding to an extremal 4-d black hole of N=4 Poincaré supergravity must describe a string ending on the UV brane. and it must also be quantum-mechanically stable.
In the next Section, we briefly recall some relevant aspects of 4-d, N=4 Poincaré supergravity, and its extremal black holes. In Section 3 we identify the duals of these black holes in N=8, AdS 5 supergravity [12,13]. In particular, we point out that the duals are strings carrying charges under some of the 12 antisymmetric forms of the 5-d gauged supergravity, which are doublets of SL(2, R) and in the 6 of SU(4) ∼ SO (6). The fact that the antisymmetric forms are doublets of SL(2, R) means that they define charges of both Ramond and Neveu-Schwarz type. If the string carries a single charge, it is 1/2 supersymmetric; otherwise, it is 1/4 supersymmetric. This property has a nice holographic counterpart. The antisymmetric forms are dual to the 6 graviphotons of the 4-d Poincaré theory, and 4-d black holes can carry both electric and magnetic 2 By AdS 5 horizon we mean, of course, the horizon of the Poincaré coordinate patch. those with both electric and magnetic charge (under different U(1)s) are 1/4 BPS [14,15]. This fact is yet another proof that the SL(2) electric-magnetic duality in 4-d, N=4 supergravity [16] maps into the SL(2) symmetry of type IIB superstrings. We conclude in Section 4 with some remarks on the puzzle arising from the explicit form of the known 5-d supegravity metrics associated with BPS black holes.

N=4, 4-d Poincaré Supergravity and Its Extremal Black Holes
N=4 Poincaré Supergravity in four dimensions [17] has two types of multiplets: gravitational, and matter. The gravitational multiplet contains 16 bosons plus 16 fermions: Here is the string dilaton, and a is the RR 10-d axion.
Matter, which in what follows play no significant role, is in N=4, Lie-algebra valued Yang-Mills multiplets. In the solutions we will discuss, these fields are set consistently to zero.
The N=4 pure supergravity black holes are the axion-dilaton black holes studied in [14,15,18,19]. In particular, 1/4 BPS black holes have nonzero entropy, given by the Bekenstein-Hawking formula, and their axion-dilaton scalar evolves towards a fixed value at the horizon [20,22,23]. In a generic BPS black hole background of pure N=4 supergravity, the central charge where q Λ , p Λ are the asymptotic charges coupled to the 6 graviphotons. K = − log(iS − iS) is the Kähler potential of the SU(1, 1)/U(1) axion-dilaton sigma model. Z Λ changes by an overall phase under the SL(2, Z) symmetry of type IIB 10-d supergravity.
The eigenvalues Z 1 , Z 2 of the central-charge matrix are given by [23, 24] The modulus |Z 1 | is the ADM mass (in Planck units) of the extremal black hole. If p 2 q 2 = (p·q) 2 , |Z 1 | = |Z 2 |, and the resulting configuration is 1/2 BPS and has zero entropy. If p 2 q 2 > (p · q) 2 , the resulting configuration is 1/4 BPS, and it has a nonzero entropy given by the attractor equation [20,21,22, 23] where Z * 1 is the value of Z 1 at the stationary point These equations fix the value of the dilaton and the axion to Substituting these values into Eq. (4), we arrive at We note that by an SO(6) rotation, followed by an SL(2, R) transformation, we can set q Λ = (q, 0, 0, ...0), p Λ = (0, p, 0..., 0), which gives exp(−2φ * ) = |q/p|, a * = 0. Then, we can re-write the equation for |Z 1 | as

The 5-d Duals of 4-d Extremal Black Holes
The 1/2 and 1/4 BPS black holes we have briefly described here are extremal, so that they do not decay by Hawking radiation. Since they also possess some residual supersymmetry, they belong to short multiplets of N=4, and are thus completely stable, both classically and quantum mechanically. Therefore, their 5-d duals must be classical, static configurations of AdS 5 supergravity.
To identify these configuration, we must first understand which 5-d fields give rise to the graviphotons. The crucial observation is that they cannot be the SU(4) 5-d gauge fields, since they have no zero mode, and, moreover, they do not transform under the SU(4) symmetry as the 4-d graviphotons. As we noticed previously, the graviphotons of 4-d Poincaré supergravity belong to the 6 of SO(6). Moreover, their corresponding electric and magnetic fields are doublets under SL(2, R). These are precisely the quantum numbers of the 12 antisymmetric tensors of the gauged N=8 supergravity in AdS 5 3 . It is therefore natural to identify the 4-d graviphotons with the zero modes of these antisymmetric tensors. This identification is indeed the correct one, as shown in [7,6] for the N=4, AdS 5 reduction, and in [8] for the N=8 reduction. Since antisymmetric fields couple to strings, instead of point particles, it is obvious that the 5-d configuration dual to extremal black holes must be strings. The ones corresponding to regular, nonzero-entropy black holes must carry charge under at least two antisymmetric forms, one of NS and the other of R type. The form of the N=8 Poincaré superalgebra, and its possible AdS 5 extension, also implies that the 5-d strings are in the 6 of SO(6) [25]. To see this, we write the relevant part of the 4-d super-Poincaré algebra, namely the anticommutator of two same-chirality supercharges This expression corresponds to the 5-d anticommutator This equation shows as promised that 5-d strings are in the antisymmetric representation of SU(4). Supergravity solutions corresponding to 5-d string have been found in [8]. In that paper, it was shown that it is possible to lift the SL(2, Z) black holes of N=4 Poincaré supergravity to full solution of the equations of motion of N=8, AdS 5 supergravity (indeed, to solutions of 10-d type IIB supergravity). The significance of those solutions has been questioned, since they exhibit null singularities at the AdS 5 horizon.
What is clear from our analysis of the R-symmetry of the 4-d Poincaré theory, is that any 5-d configuration corresponding to a BPS black hole must be a string, perpendicular to the UV brane. The fact that these configurations have a source extending into the 5-d bulk may seem strange; nevertheless, it does not contradict any property of the holographic duality, and the R-symmetry properties of the graviphotons require it to be so. Before discussing further the explicit solutions of refs. [7,8], we must notice that Eq. (5) can be exactly reinterpreted as the extremization of the 5-d string tension [26], since, schematically L × String Tension in 5d ∼ BH Mass in 4-d.
Here L is the radius of curvature of AdS 5 .
In order to have 1/4 BPS states it is essential that the 5-d two-forms are nonsinglets under the R-symmetry. Therefore, unlike the case of 10 dimensions [27], (p, q)-strings can also be 1/4 BPS, instead of just 1/2 BPS. We conclude this Section by pointing out that unlike 5-d Poincaré supergravity, AdS 5 supergravity does not allow for duality transformations exchanging antisymmetric forms with vectors. 4