Localization on $AdS_3 \times S^2$ I: the 4d/5d connection in off-shell Euclidean supergravity

TL;DR

This paper develops a localization formalism for supergravity on $AdS_3 imes S^2$, connecting 4d and 5d off-shell solutions to analyze the dual elliptic genus of certain SCFTs.

Contribution

It introduces a new off-shell 4d/5d lift in Euclidean supergravity and applies localization to relate 4d and 5d solutions in the $AdS_3 imes S^2$ setting.

Findings

Localization reduces supergravity path integral to BPS solutions.

Constructed a Euclidean 4d/5d lift for off-shell solutions.

Identified a class of solutions lifting known 4d BPS solutions.

Abstract

We begin to develop the formalism of localization for the functional integral of supergravity on $AdS_3 \times S^2$. We show how the condition of supersymmetry in the Euclidean $\mathbb{H}^3/\mathbb{Z} \times S^2$ geometry naturally leads to a twist of the S$^2$ around the time direction of $AdS_3$. The twist gives us a five-dimensional Euclidean supergravity background dual to the elliptic genus of $(0,4)$ SCFT$_2$ at the semiclassical level. On this background we set up the off-shell BPS equations for one of the Killing spinors, such that the functional integral of five-dimensional Euclidean supergravity on $\mathbb{H}^3/\mathbb{Z} \times S^2$ localizes to its space of solutions. We obtain a class of solutions to these equations by lifting known off-shell BPS solutions of 4-dimensional supergravity on $AdS_2 \times S^2$. In order to do this consistently, we construct and use a Euclidean version of the off-shell 4d/5d lift of arxiv:1112.5371, which could be of independent interest.