Which AdS3 configurations contribute to the SCFT2 elliptic genus?

TL;DR

This paper uses supersymmetric localization to identify specific AdS3 configurations that contribute to the elliptic genus in AdS3/CFT2, revealing a smooth interpolating geometry between BTZ black holes and boundary conditions.

Contribution

It demonstrates how the gravitational functional integral localizes onto specific supersymmetric AdS3 configurations, providing new insights into the AdS3/CFT2 correspondence and black hole microstates.

Findings

Localization onto supersymmetric AdS3 configurations

Identification of a smooth interpolating geometry for boundary conditions

Action matches the supersymmetric BTZ black hole, holomorphic in τ

Abstract

According to the AdS/CFT duality, the superconformal index of a superconformal field theory should have an AdS interpretation as a Euclidean functional integral with periodic boundary conditions on the fermions. Unlike the thermal case, the Euclidean continuation of the supersymmetric AdS black hole does not smoothly fill in these boundary conditions, leading us to ask the title question. In the context of AdS3/CFT2, we show using supersymmetric localization that the gravitational functional integral for the elliptic genus localizes onto asymptotically AdS3 configurations that are annihilated by a certain supercharge, in the relevant off-shell supergravity theory. For (0,4) superconformal field theories, we find such a localizing configuration in the 5d ${\cal N}=2$ off-shell supergravity theory that is asymptotically $AdS_3 \times S^2$. This configuration interpolates smoothly between the supersymmetric BTZ black hole in the interior and a constant gauge field configuration at the boundary, thus smoothly filling in the (++) boundary conditions for spinors on the boundary torus. It has an action equal to the supersymmetric BTZ black hole, holomorphic in the complex structure $\tau$ of the boundary torus. Our results have interesting implications for the black hole Farey tail in AdS3, as well as for higher dimensional AdS theories.