Equivariant localization in supergravity

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.