Equivariant localization in supergravity

TL;DR

This paper introduces a method using equivariant localization to evaluate physical observables in supergravity solutions with R-symmetry, simplifying calculations by relying on topological data and fixed point theorems.

Contribution

It develops a framework connecting supersymmetric supergravity solutions with equivariantly closed forms, enabling the computation of physical quantities via fixed point theorems without solving equations.

Findings

Expressions depend only on topological data and R-symmetry vector.

Applicable to holographic examples like black hole entropy and central charges.

Calculations are simplified by avoiding direct supergravity solutions.

Abstract

We show that supersymmetric supergravity solutions with an R-symmetry Killing vector are equipped with a set of equivariantly closed forms. Various physical observables may be expressed as integrals of these forms, and then evaluated using the Berline-Vergne-Atiyah-Bott fixed point theorem. We illustrate with a variety of holographic examples, including on-shell actions, black hole entropies, central charges, and scaling dimensions of operators. The resulting expressions depend only on topological data and the R-symmetry vector, and hence may be evaluated without solving the supergravity equations.