Equivariant localization in supergravity in odd dimensions

TL;DR

This paper develops a localization formula for supersymmetric solutions in five-dimensional supergravity with boundaries, connecting the on-shell action to toric data and Sasakian volume, and applies it to black hole entropy.

Contribution

It introduces a new localization approach for odd-dimensional supergravity solutions, relating on-shell actions to toric and Sasakian geometric data, and derives black hole entropy from topological information.

Findings

Derived a formula for on-shell action in terms of toric data and Sasakian volume.

Connected regularized on-shell action to the effective compact manifold's toric data.

Reproduced the entropy function of supersymmetric black holes using topological data.

Abstract

We discuss a localization formula for certain integrals on odd-dimensional manifolds with boundaries, equipped with a Killing vector, and employ this to localize the regularised on-shell action of a large class of supersymmetric solutions of five dimensional minimal gauged supergravity. Specifically, we consider asymptotically AdS_5 solutions in the time-like class, in which the transverse K\"ahler foliation is assumed to be toric. We find that the background subtraction regularization method leads to an intriguing formula for the on-shell action, in terms of an analytic continuation of the Martelli-Sparks-Yau Sasakian volume. In particular, we show that the regularised on-shell action is a function of the toric data of an effective compact five-dimensional manifold, as well as of the supersymmetric Killing vector, outside the corresponding dual cone. As our main example we provide a derivation of the well-known entropy function of supersymmetric and rotating black holes in AdS_5, using only topological data.