R**2 correction to BMPV black hole entropy from Kerr/CFT correspondence

Following Kerr/CFT correspondence, we compute the entropy of five-dimensional supersymmetric rotating BMPV black holes.We successfully reproduce Iyer-Wald formula in the presence of Gauss-Bonnet terms from the viewpoint of microscopic CFT. This further supports the higher-derivative version of Kerr/CFT prescription proposed in arXiv:0903.4176 for four-dimensional extremal Kerr black holes.


Introduction
A microscopic entropy counting programme called Kerr/CFT correspondence was proposed by Guica et al. in [1]. They have examined 4D extremal Kerr black holes whose nearhorizon geometry has SL(2, R) × U (1) isometry. Further extension can be found in [2]- [21]. Remarkably, the tree-level black hole entropy S 0 predicted by Bekenstein-Hawking area law was reproduced by applying Cardy's formula to the dual 2D CFT: where T F T denotes Frolov-Thorne temperature [22].
Let us momentarily clarify the nature of Kerr/CFT correspondence. Compared with the BTZ/CFT case [23] pioneered earlier by Brown and Henneaux, 1 here one takes into account the asymptotic symmetry group (ASG) of the near-horizon black hole geometry instead of its asymptotically-far one. We will adopt the terminology asymptotic Killing vector (instead of Killing vector) especially for the former case under consideration. In addition, the central charge c in [23] did arise from the enhanced SL(2, R) L × SL(2, R) R , the isometry of the asymptotic 3D BTZ (or AdS 3 ), to two copies of (chiral and anti-chiral) Virasoro algebras. In contrast to this, c in (3.2) has a rather different nature. That is, given the above asymptotic Killing vector field like (3.1) infinitely many Fourier modes ζ n are identified with generators L n of Virasoro algebra whose central charge (3.2) is determined completely by the near-horizon metric and (3.1).
In this Letter, within the framework of Kerr/CFT correspondence we focus on the entropy of 5D BMPV black holes in the presence of R 2 -curvature corrections. 2 In [14,15], this direction has been explored in the context of 4D extremal Kerr ones. We still rely on Cardy's formula (1.1) but modify the central charge c due to the inclusion of Gauss-Bonnet terms. It is in [28,29,30] that the explicit expression of c in the presence of higher-derivative corrections has been spelt out. Combined with Frolov-Thorne temperature analyzed in [8,7] for BMPV black holes, we found that (1.1) coincides perfectly with Iyer-Wald formula [31,32,33].
We organize this Letter as follows. In Section 2, we describe some basic aspects of BMPV black holes. Iyer-Wald entropy formula in the presence of Gauss-Bonnet terms is also reviewed. In Section 3, we evaluate BMPV entropy microscopically via Kerr/CFT correspondence as stated above. We end up this Letter with a summary in Section 4.

The BMPV black hole entropy
The BMPV black hole was first constructed in [34]. As shown by Kallosh et al. [35], it can get embedded in 5D N = 2 supergravity coupled to one vector multiplet, and preserves one-half supersymmetry. The metric of the BMPV black hole is Here, 1 4 dΩ 2 3 is the line element of a unit S 3 . The lowest scalar component of the vector multiplet can be set to some constant. Because of this fact, the gaugino equation implies a vanishing gauge field in the vector multiplet. On the other hand, the graviphoton gauge potential in the graviton multiplet is The conserved charges of the BMPV black hole (2.1) can be measured at the asymptotic spatial infinity according to [36]. Thus, the conserved angular momentum along ϕ and the electric charge measured at asymptotic infinity of the BMPV black hole (2.1) are where ∇ µ (ξ ϕ ) ν dx µ ∧ dx ν is abbreviated to ∇ξ ϕ with ξ ϕ being a Killing vector field ∂ ∂ϕ . Here, x µ represents the coordinates (t, r, θ, ψ, ϕ). We have also expressed the 5D Newton constant as G 5 . The ADM mass of the BMPV black hole, M ADM = 3πµ 4G 5 , is proportional to Q as required by supersymmetry. One can also see that there is no angular momentum along ψ.
Another realization of the BMPV black hole is through compactifying the ten-dimensional Type IIA string theory on a Calabi-Yau threefold X [37]. Wrapped on X is a D0-D2-D4-D6 brane system with (q 0 , q A , p A , p 0 ) with A = 1, · · · , h 1,1 (X) indicating their RR charges (or the numbers). In the case of the BMPV black holes, one has (q 0 , q A , p A , p 0 ) = (q 0 , q A , 0, 1). Due to one single D6-brane, the eleven-dimensional M-theory lift of this brane system amounts to placing the resultant 5D charged and rotational black hole at the center of a Taub-NUT space T N 4 (in order to preserve supersymmetry). Notify that q 0 ∝ J ϕ is the spin over the S 1 bundle of T N 4 . This stringy construction makes natural both the appearance of S 3 ⊂ T N 4 in (2.1) and the statement of no angular momentum along ψ. In addition, q A is related to Q by q A = 3QD ABC Y B Y C where D ABC is the triple intersection number of X and Y A stands for horizon values of scalar components in N = 2 vector multiplets. They are normalized by In order to consider the near-horizon limit, letr = r − √ µ. In the near-horizon limit r → 0, the BMPV black hole metric (2.1) becomes Obviously, when J = 0 (a = 0) (2.5) reduces to simply a direct product: AdS 2 × S 3 . For generic J = 0 cases, the bosonic isometry group of (2.5) gets broken down to SU (1, 1) × SU (2) × U (1) [38]. By completing the square term −( 2r √ µ dt + a 2 σ 3 ) 2 , the near-horizon topology of the BMPV black hole becomes AdS 2 fibered over S 3 [39]. This may be a sign of the existence of certain putative dual 2D chiral CFT. From (2.5) it is easily seen that the tree-level entropy is (2.6)

Entropy from the Iyer-Wald formula
Let us see how R 2 -curvature corrections modify (2.6). When it comes to the Gauss-Bonnet terms, the result has actually been calculated in [40]. Our goal here is to reproduce the known result and meantime set up some notational conventions to be used in the next section.
The 5D N = 2 supergravity can arise from compactifying 11D M-theory on Calabi-Yau threefolds X where the number of vector multiplets is h 1,1 (X). The resultant Einsteinframe low-energy effective action may necessarily contain higher-derivative corrections [41,42] such as the familiar Gauss-Bonnet term: where ξ is is proportional to c 2A Y A . As mentioned before, Y A denotes the scalar component in vector multiplets evaluated at the horizon. c 2A is given by where c 2 (X) is the second Chern class of X and ω A ∈ H 1,1 (X). What appeared in (2.1)-(2.2) correspond to the situation where one considers a special family of Calabi-Yau threefolds subject to h 1,1 (X) = 1. We will concentrate only on this case.
In order to carry out the computation in (2.9) with respect to the tree-level metric (2.5), it may be easier to take an orthonormal basis instead of dx µ . An orthonormal basis eâ = eâ µ dx µ , (â =t,r,θ,ψ,φ) of (2.5) is dt . (2.12) If one uses the orthonormal basis (2.12), the non-zero components for the binormal ǫâb simply become ǫtr = −ǫrt = 1 and the others are zero. Then, the Iyer-Wald formula can be simply written by When one explicitly inserts the near-horizon metric (2.5) to (2.13), one obtains the correction S 1 to the tree-level entropy S 0 (2.6), (2.14)

Entropy from the Kerr/CFT correspondence
Having obtained (2.14) from the geometric viewpoint, we go to provide a statistical (or microscopic) derivation by means of the Kerr/CFT prescription.

Central charge of dual 2D CFT
As shown in [1], because a suitable asymptotic boundary condition is imposed on the nearhorizon metric there appears a series of vector fields which preserves them. We may call them asymptotic Killing vector fields. One has to tell the difference between them and usual Killing vector fields as advocated in Section 1. The asymptotic boundary condition for the near-horizon metric (2.5) has been worked out in [7]. It is preserved by the following vector field: To yield the Virasoro algebra one has to add a central term. The central extension can be accomplished by using an asymptotic charge associated to the asymptotic Killing vector field (3.1). The general formula for the asymptotic charge with respect to a given Lagrangian density including higher-derivative terms has been obtained in [28,29,30]: where by | n 3 we mean that only terms proportional to n 3 will be extracted, and L ζn denotes the Lie derivative with respect to ζ n (3.1). X αβ and W α are defined by Furthermore, the explicit form of the E-term in (3.2) was obtained in [15]; namely, where φ is any field including g µν . Furthermore, E[L ζn φ, L ζ −n φ; φ] is defined as (3.5) where denotes the interior product. From the direct evaluation of (3.5) by using (3.4), one explicitly has Indeed, as shown in [15], by using the aforementioned definition of c, the Iyer-Wald formula gets completely reproduced at least in the 4D extremal Kerr cases. Here, nevertheless we clarify its validness even for the 5D BMPV black hole. As a matter of fact, it is highly non-trivial to check whether (3.2) still holds for the BMPV black hole because its near-horizon topology, S 1 -fibration over AdS 2 × S 2 , definitely differs from that of the 4D extremal Kerr black holes.

Calculation
Again, let us take the orthonormal basis (2.12) during evaluating (3.2) for the sake of computational convenience. One can rewrite (3.1) by a basis eâ = eâ µ ∂ µ where eâ µ = (eâ µ ) −1 . Then, the explicit form of ζ n in terms of the basis eâ is Equipped with the above asymptotic Killing vector (3.7), one manages to have whose columns (rows) are labeled in order byt,r,θ,ψ andφ.
By combining both (3.8) and Ztrrφ = 0, the first term in (3.2) becomes Equipped with Cardy's formula (1.1) and the Frolov-Thorne temperature [8,7] T F T = − µ − a 2 2πa , (3.10) we have It precisely coincides with the Iyer-Wald result (2.13). Surely, we have assumed the validness of the Frolov-Thorne temperature even in the presence of the R 2 -curvature corrections. 4 In view of (3.10), it seems that when a > 0 the above T F T of some putative 2D CFT become negative. This is a quite common issue encountered during applying Kerr/CFT correspondence. The sign dependence on a is merely an illusion because it has no real effect on a physical quantity like entropy S 1 which is independent of the sign of a. Certainly, another choice of ζ associated with ψ is possible but leads to a vanishing central charge. This situation which resembles the tree-level case encountered in [7] may be attributable to zero angular momentum along ψ.
Let us examine other contributions to the central charge c in (3.2). We will see that they cancel one another out eventually. Since we have already reproduced the Iyer-Wald result only from the first term of (3.2), the contributions from the other terms have to cancel one another out. In (3.2), the explicit evaluation can show that the second and third terms as a whole give Consequently, a perfect cancellation happens as expected. Namely, we have confirmed that the entropy obtained from the central charge (3.2) precisely reproduces the Iyer-Wald formula (2.9).

Summary
Our results are summarized as follows. Plugging into Cardy's formula of 2D CFT (1.1) the central charge spelt out in [28,29,30] and Frolov-Thorne temperature analyzed in [8,7], we obtained the entropy of BMPV black holes when Gauss-Bonnet terms are present. This computation, though semiclassical, can be regarded as a microscopic derivation in contrast to Iyer-Wald formula.
Because there are two independent (ϕ, ψ), it is ϕ-direction associated with non-zero angular momentum that gives us the finite central charge. This resembles much the tree-level (Einstein-Hilbert action) situation encountered in [7]. It will be interesting to understand this phenomenon further within a more general framework.