Gluing gravitational blocks for AdS black holes

Abstract

We provide a unifying entropy functional and an extremization principle for black holes and black strings in AdS 4 × S 7 and AdS 5 × S 5 with arbitrary rotation and generic electric and magnetic charges. This is done by gluing gravitational blocks, basic building blocks that are directly inspired by the holomorphic blocks appearing in the factorization of supersymmetric partition functions in three and four dimensions. We also provide an explicit realization of the attractor mechanism by identifying the values of the scalar fields at the horizon with the critical points of the entropy functional. We give examples based on dyonic rotating black holes with a twist in AdS 4 × S 7 , rotating black strings in AdS 5 × S 5 , dyonic Kerr-Newman black holes in AdS 4 × S 7 and Kerr-Newman black holes in AdS 5 × S 5 . In particular, our entropy functional extends existing results by adding rotation to the twisted black holes in AdS 4 and by adding flavor magnetic charges for the Kerr-Newman black holes in AdS 4 . We also discuss generalizations to higher-dimensional black objects. arXiv:1909.10550v1 [hep-th] 23 Sep 20191 The BPS conditions impose a linear constraint on the magnetic charges and some non-linear con-5 This has been checked at large N in full generality for static mAdS 4 black holes [1,41], rotating black strings in AdS 5 [19], KN-AdS 5 black holes with equal angular momenta [4] and at large N but in the Cardy limit for general purely electric KN-AdS 4 and KN-AdS 5 [3,42]. It is still not known if the large N limit of the superconformal index of N = 4 SYM reproduces the entropy functional, and therefore the entropy, in the case of KN-AdS 5 black holes with unequal angular momenta.6 See also [43] for a different approach.10 In particular, in the regime considered in (4.15) the minus sign in the chemical potential is equivalent to a complex conjugate. This might explain the complex conjugate we detect in gravity, see (4.9). 11 We correct a misprint in the formula for the entropy in [21]. Note also that we have redefined the angular momentum as J there = − 1 2 J here (see footnote 21).An important symmetry of the equations of motion of supergravity is the electromagnetic duality. As the name suggests, the n V + 1 electric and magnetic gauge field strengths F Λ and G Λ (Λ = 0, . . . , n V ) can be transformed among each other under the symplectic group Sp(2(n V + 1), Z), resulting in a rotation of the electromagnetic charges,

Anomalies, black strings and the charged Cardy formula

“…A proposal for finding the saddle point of the topologically twisted index, which should capture the entropy in the static case, was discussed in [37]. It would be interesting to make it rigorous and generalize it to the rotating case, understanding in the process the role of holomorphic blocks and the relation to the entropy functions introduced in [25] and briefly discussed in appendix A.…”

“…We conclude in section 6 with comments and a discussion. In the appendix we rephrase our supergravity findings in the language of attractors and entropy functions, using the gravitational blocks introduced in [25].…”

“…For a more pedagogical introduction to symplectic covariance, the symplectic inner product •, • , the I 4 -formalism, along with various useful identities, see the appendices of [22,25,54,55]. The four real scalars that make up the bosonic content of the universal hypermultiplet parametrize the coset space SU(2, 1)/U(2).…”

“…We have checked explicitly that the entropy and scalar sections of the explicit solutions in section 4 precisely agree with the proposed entropy function here. For the rotating black strings in AdS 5 × S 5 this was already done in [25].…”

“…As briefly discussed in appendix A, the entropy function based on gravitational blocks introduced in[25] is the gravitational counterpart of (3.53).…”