Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS5 black holes

Abstract

We present a holographic derivation of the entropy of supersymmetric asymptotically AdS 5 black holes. We define a BPS limit of black hole thermodynamics by first focussing on a supersymmetric family of complexified solutions and then reaching extremality. We show that in this limit the black hole entropy is the Legendre transform of the on-shell gravitational action with respect to three chemical potentials subject to a constraint. This constraint follows from supersymmetry and regularity in the Euclidean bulk geometry. Further, we calculate, using localization, the exact partition function of the dual N = 1 SCFT on a twisted S 1 × S 3 with complexified chemical potentials obeying this constraint. This defines a generalization of the supersymmetric Casimir energy, whose Legendre transform at large N exactly reproduces the Bekenstein-Hawking entropy of the black hole.This makes it manifest that in the BPS limit, r + = r * , again the imaginary parts vanish, while the real parts coincide with the BPS values in (2.23). The energy E is also complex and it satisfies the relation

Probing the EVH limit of supersymmetric AdS black holes

“…Therefore, even though the original chemical potentials (2.42) become real in the limit r + = r 0 , the properly defined BPS potentials satisfy a complex constraint! This observation underlies all recent derivations of the AdS 5 BPS black hole entropy from the supersymmetric partition function of the N = 4 SYM theory [17][18][19].…”

“…In this section, we will define the various potentials for the supersymmetric black holes by taking the BPS limit of the potentials for the generic non-BPS black hole (2.12). The BPS limit can be taken through a zero temperature limit accompanied by specific limits for the various potentials as discussed in [46] (see also [18] for a more recent discussion in our context). The fact that one has to take limits of the chemical potentials as well can be illustrated by a simple example [46].…”

“…To find the BPS values of the chemical potentials, we follow [18]. We can start from the non-BPS black hole (2.8).…”

“…To resolve this issue, it is proposed in [19] (see also [18]) to restrict the theory to the hypersurface:…”

“…Recently, the entropy for certain supersymmetric asymptotically AdS 4 and AdS 5 black holes with finite horizon area has been accounted for from the dual CFT point of view [15][16][17][18][19][20][21][22][23][24][25][26]. In this paper, we employ the results on 1/16 th BPS AdS 5 black holes [27][28][29][30][31] to study the so-called extremal vanishing horizon (EVH) black holes [32][33][34][35][36][37][38].…”