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

Abstract:We use localization to evaluate the functional integral of string field theory on AdS 2 × S 2 background corresponding to the near horizon geometry of supersymmetric black holes in 4d compactifications with N = 2 supersymmetry. In particular, for a theory containing n v + 1 vector multiplets, we show that the functional integral localizes exactly onto an ordinary integral over a finite-dimensional submanifold in the field space labeling a continuous family of instanton solutions in which auxiliary fields in the vector multiplets are excited with nontrivial dependence on AdS 2 coordinates. These localizing solutions are universal in that they follow from the off-shell supersymmetry transformations and do not depend on the choice of the action. They are parametrized by n v + 1 real parameters {C I ; I = 0, . . . , n v } that correspond to the values of the auxiliary fields at the center of AdS 2 . In the Type-IIA frame, assuming D-terms evaluate to zero on the solutions for reasons of supersymmetry, the classical part of the integrand equals the absolute square of the partition function of the topological string as conjectured by Ooguri, Strominger, and Vafa; however evaluated at the off-shell values of scalar fields at the center of AdS 2 . In addition, there are contributions from one-loop determinants, brane-instantons, and nonperturbative orbifolds that are in principle computable. These results thus provide a concrete method to compute exact quantum entropy of these black holes including all perturbative and nonperturbative corrections and can be used to establish a precise relation between the quantum degeneracies of black holes and the topological string.