Localization of the 5D supergravity action and Euclidean saddles for the black hole index

TL;DR

This work develops a systematic equivariant-localization framework for the five-dimensional ungauged supergravity action, including manifolds with boundary, and applies it to compute the on-shell action of Euclidean supersymmetric saddles that contribute to the gravitational index. By introducing an extra U(1) symmetry and leveraging Gibbons-Hawking base constructions, the authors express the action as a sum over fixed-point nuts (and bolts) with boundary terms shown to cancel in a SUSY-compatible scheme, and they relate 5D results to 4D localization via dimensional reduction. They construct and analyze two-center, multi-charge supersymmetric saddles, deriving their on-shell actions and thermodynamics, and show how these saddles interpolate between extremal BPS black holes and horizonless microstate geometries in the β→0 limit. The results reveal a transparent gravitational-block structure for the action and illuminate the interplay between boundary data, regularity, and analytic continuation in determining physically meaningful saddles and their charges. The framework paves the way for extensions to AdS, higher dimensions, and higher-derivative corrections, with potential applications to exact black hole entropy counts from a gravitational path integral perspective.

Abstract

We investigate equivariant localization of the gravitational on-shell action in odd dimensions, focusing on five-dimensional ungauged supergravity. We analyze the conditions for cancellation of boundary terms, so that the full action integral is given in terms of the odd-dimensional analog of the nuts and bolts of Gibbons-Hawking. We specialize to supersymmetric configurations with an additional ${\rm U}(1)$ symmetry preserving the supercharge and provide a formula for the localized on-shell action. We construct asymptotically flat Euclidean supersymmetric non-extremal solutions with two independent rotations and an arbitrary number of electric charges, providing black hole saddles of the gravitational path integral that computes a supersymmetric index, and evaluate their action equivariantly. We find that these Euclidean saddles interpolate between supersymmetric extremal black holes and two-center horizonless microstate geometries. The interpolation involves dialing the temperature and implementing different analytic continuations. The corresponding on-shell action does not depend on temperature but is affected by the analytic continuations.