Which AdS3 configurations contribute to the SCFT2 elliptic genus?

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.