Localizing AlAdS₅ black holes and the SUSY index on S¹ × M₃

We consider complex, supersymmetric, non-extremal Euclidean black holes that are asymptotically locally AdS₅, with S¹ × M₃ conformal boundary. We study field theory backgrounds consisting of various M₃, and explicitly construct Killing spinors that are anti-periodic around the Euclidean time circle. Focussing on elliptically/biaxially squashed three-spheres and Lens spaces, we compute the supersymmetric index of the 𝒩 = 4 SYM in a Cardy-like limit. While such black holes have not been constructed for general M₃, we demonstrate that the supersymmetric indices can be recovered from gravity computations using equivariant localization, by extending the boundary Killing spinors to the bulk. We show that this involves gluing the black hole geometry with a supersymmetric, horizonless AlAdS₅ geometry, chosen such that the Casimir energy is removed from the supersymmetric partition function.