Probing black holes with equivariant localization

TL;DR

This paper develops equivariant localization to compute probe brane actions in supergravity, applying it to D3-branes in black hole backgrounds, linking geometric data to field theory indices.

Contribution

It introduces a novel equivariant localization method for supergravity probe calculations, connecting geometric cycles to field theory observables.

Findings

Action reduces to equivariant integrals from toric data

Method applies to supersymmetric probe D3-branes in black hole backgrounds

Provides a way to compute non-perturbative corrections in superconformal indices

Abstract

We introduce equivariant localization as a method for computing the action of probe branes in supergravity backgrounds. We apply it to supersymmetric probe D3-branes in type IIB supersymmetric spacetimes obtained by uplifting the Kerr-Newman-AdS$_5$ black hole on a toric Sasaki-Einstein space. Depending on the cycles they wrap, such branes represent non-perturbative corrections to or defect operator insertions in the superconformal index of a large family of four-dimensional $\mathcal{N}=1$ quiver superconformal field theories. The resulting action reduces to equivariant integrals and can be evaluated entirely from toric data.