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According to the AdS/CFT duality, the superconformal index of asuperconformal field theory should have an AdS interpretation as a Euclideanfunctional integral with periodic boundary conditions on the fermions. Unlikethe thermal case, the Euclidean continuation of the supersymmetric AdS blackhole does not smoothly fill in these boundary conditions, leading us to ask thetitle question. In the context of AdS3/CFT2, we show using supersymmetriclocalization that the gravitational functional integral for the elliptic genuslocalizes onto asymptotically AdS3 configurations that are annihilated by acertain supercharge, in the relevant off-shell supergravity theory. For (0,4)superconformal field theories, we find such a localizing configuration in the5d ${\cal N}=2$ off-shell supergravity theory that is asymptotically $AdS_3\times S^2$. This configuration interpolates smoothly between thesupersymmetric BTZ black hole in the interior and a constant gauge fieldconfiguration at the boundary, thus smoothly filling in the (++) boundaryconditions for spinors on the boundary torus. It has an action equal to thesupersymmetric BTZ black hole, holomorphic in the complex structure $\tau$ ofthe boundary torus. Our results have interesting implications for the blackhole Farey tail in AdS3, as well as for higher dimensional AdS theories.

Which AdS3 configurations contribute to the SCFT2 elliptic genus?