Twisting and localization in supergravity: equivariant cohomology of BPS black holes

TL;DR

This paper develops a formalism for supersymmetric localization in supergravity, enabling systematic computation of quantum black hole entropy through equivariant cohomology and one-loop determinants.

Contribution

It introduces a twisted supergravity framework with off-shell supersymmetry, providing a method to compute quantum entropy of black holes from first principles.

Findings

Constructed a twisted supergravity formalism with off-shell supersymmetry.

Computed the one-loop determinant for off-shell fluctuations of the Weyl multiplet.

Validated the approach with results consistent with large-charge on-shell analysis.

Abstract

We develop the formalism of supersymmetric localization in supergravity using the deformed BRST algebra defined in the presence of a supersymmetric background as recently formulated in arxiv:1806.03690. The gravitational functional integral localizes onto the cohomology of a global supercharge $Q_\text{eq}$, obeying $Q_\text{eq}^2=H$, where $H$ is a global symmetry of the background. Our construction naturally produces a twisted version of supergravity whenever supersymmetry can be realized off-shell. We present the details of the twisted graviton multiplet and ghost fields for the superconformal formulation of four-dimensional N=2 supergravity. As an application of our formalism, we systematize the computation of the exact quantum entropy of supersymmetric black holes. In particular, we compute the one-loop determinant of the $Q_\text{eq} \mathcal{V}$ deformation operator for the off-shell fluctuations of the Weyl multiplet around the $AdS_2 \times S^2$ saddle. This result, which is consistent with the corresponding large-charge on-shell analysis, is needed to complete the first-principles computation of the quantum entropy.