Localizing AlAdS$_5$ black holes and the SUSY index on $S^1 \times M_3$

TL;DR

This paper constructs supersymmetric Euclidean black holes in asymptotically AdS$_5$ with various boundary geometries, and matches field theory index calculations with gravity localization results, advancing understanding of AdS/CFT correspondence.

Contribution

It explicitly constructs Killing spinors for non-extremal black holes with diverse boundary geometries and connects field theory indices with gravity solutions via localization.

Findings

Computed supersymmetric index for various $M_3$ backgrounds.

Matched field theory results with gravity localization calculations.

Provided evidence for existence of corresponding gravity solutions.

Abstract

We consider complex, supersymmetric, non-extremal Euclidean black holes that are asymptotically locally AdS$_5$, with $S^1 \times M_3$ conformal boundary. We study field theory backgrounds consisting of various $M_3$, and explicitly construct Killing spinors that are anti-periodic around the Euclidean time circle. Focussing on elliptically/biaxially squashed three-spheres and Lens spaces, we compute the supersymmetric index of the $\mathcal{N}=4$ SYM in a Cardy-like limit. While such black holes have not been constructed for general $M_3$, we show that our field theory results can be recovered from a gravity computation using equivariant localization, just assuming the solutions exist.