Localization of the free energy in supergravity

TL;DR

This paper derives a universal formula for calculating the gravitational free energy in certain supergravity theories, enabling computations without solving complex equations and confirming results with dual field theories.

Contribution

It introduces a general localization formula for free energy in $D=4$, $ ext{N}=2$ gauged supergravity, simplifying calculations and providing new results consistent with holographic duals.

Findings

Derived a general free energy formula for supersymmetric solutions

Validated the formula with known and new supergravity results

Enabled computations without solving supergravity equations

Abstract

We derive a general formula for the gravitational free energy of Euclidean supersymmetric solutions to $D=4$, $\mathcal{N}=2$ gauged supergravity coupled to vector multiplet matter. This allows one to compute the free energy without solving any supergravity equations, just assuming the solutions exist. As well as recovering some known results in the literature with ease, we also present new supergravity results that match with holographically dual field theory computations.