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 \$N=2\$ off-shell supergravity theory that is asymptotically \$AdS\_3
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.
Users
Please
log in to take part in the discussion (add own reviews or comments).