Black holes from the gravitational path integral: supersymmetric indices and precision holography

TL;DR

This paper explores the gravitational path integral approach to counting microstates of supersymmetric black holes, revealing new saddle solutions and confirming the holographic correspondence through exact entropy calculations.

Contribution

It formulates protected observables directly within gravity via the Euclidean path integral, introduces novel saddle solutions, and matches gravitational results with dual field theory indices.

Findings

Exact agreement between corrected black hole entropy and superconformal index.

Discovery of a broad family of supersymmetric saddle solutions with diverse topologies.

Application of equivariant localization to compute on-shell actions in flat boundary conditions.

Abstract

The counting of microstates of supersymmetric black holes with anti-de Sitter or flat asymptotics is obtained by computing a supersymmetric index in a weakly coupled string theory or a dual superconformal field theory. These indices are protected observables, whose value can be reliably extrapolated from weak to strong coupling, where the gravitational description applies. In this Thesis, after a broad introductory review, we discuss recent progress in formulating such protected observables directly within the gravitational theory, via the Euclidean path integral. In the semiclassical limit the index reduces to a sum over complex Euclidean saddles. These saddles are supersymmetric but ''non-extremal'', and arise in both anti-de Sitter and flat spaces. In the holographic setting, we investigate four-derivative corrections to the thermodynamics of AdS$_5$ black holes. Using off-shell methods, we construct the corrected action of five-dimensional gauged supergravity. We then evaluate the corrected on-shell action of supersymmetric AdS$_5$ black holes and find exact agreement with a Cardy-like limit of the superconformal index of the dual conformal field theory. By a Legendre transform of the action, we obtain the corrected black hole entropy, and we confirm this result by applying Wald's formula to the corrected near-horizon geometry. We then turn to the gravitational index with asymptotically flat boundary conditions. We uncover a broad family of novel saddles and present a general classification based on their rod structure, which characterizes their topology. These solutions may feature multiple horizons or three-dimensional bubbles with lens space topology, and allowing for conical singularities yields further geometries, involving spindles and branched spheres. For the simpler geometries, the on-shell action is computed using an odd-dimensional version of equivariant localization.