Black holes from the Gravitational Path Integral: Supersymmetric indices and precision holography

The counting of microstates of certain 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 exact value can be reliably extrapolated from weak to strong coupling, where the gravitational description applies. In this Thesis, we review recent progress in formulating such protected observables directly within the gravitational theory, via the Euclidean path integral. It has been recently understood that, under suitable boundary conditions, the latter computes a gravitational index, providing a reliable counterpart to the microscopic count. We study the gravitational index in the semiclassical limit, where it reduces to a sum over complex Euclidean saddles weighted by their on-shell action. 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 AdS5 black holes. Using off-shell superconformal methods, we construct the corrected action of five-dimensional gauged supergravity, both with and without vector multiplet couplings, thereby providing an effective model that reproduces the ’t Hooft anomalies of generic holographic superconformal field theories, including the first subleading terms in the large-N expansion. We then evaluate the corrected on-shell action of supersymmetric AdS5 black holes and find exact agreement with a Cardy-like limit of the superconformal index of the dual conformal field theory. Crucially, the two-derivative solution is enough to perform this computation. From a Legendre transform of the action we obtain the corrected microcanonical entropy of supersymmetric and extremal black holes, and, in the minimal theory, we independently 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 saddles with U(1)^3 symmetry 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 S^3, S^2 x S^1, or lens space topology, and allowing for conical singularities yields further geometries, involving spindles and branched spheres. We focus on configurations with either a single horizon, or a horizon accompanied by an exterior bubble. We determine their on-shell actions and study their thermodynamics. For the simpler geometries, the on-shell action is computed using an odd-dimensional version of equivariant localization. These saddles interpolate, by tuning the inverse temperature ß and performing suitable analytic continuations, between supersymmetric extremal black holes (spherical black holes, black rings and black lenses) and horizonless bubbling geometries. In both limits their on-shell action, which is independent of ß, remains finite and well-defined.