Equivariant localization for $D=5$ gauged supergravity

TL;DR

This paper develops a method using equivariant localization to compute on-shell actions, supersymmetric Casimir energy, and indices in 5D gauged supergravity by reducing to 4D, avoiding explicit solutions.

Contribution

It introduces a novel equivariant localization approach for $D=5$ gauged supergravity solutions, enabling calculations without explicit supergravity solutions.

Findings

Computed supersymmetric Casimir energy and index of dual SCFTs.

Demonstrated the formalism with explicit examples.

Enabled calculations via dimensional reduction and localization.

Abstract

We consider supersymmetric solutions of $D=5$ Euclidean gauged supergravity coupled to an arbitrary number of vector multiplets. We consider solutions that admit both the R-symmetry Killing vector, $\mathcal{K}$, constructed as a bilinear in the Killing spinor, as well as an additional Killing vector $\ell$. Using $\ell$ to perform a dimensional reduction to $D=4$, $\mathcal{N}=2$ gauged supergravity, we show how the $D=5$ on-shell action can be computed using equivariant localization. We illustrate the formalism with some examples, computing the supersymmetric Casimir energy and the supersymmetric index of the dual SCFT without using the explicit supergravity solutions.