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

We consider complex, supersymmetric, non-extremal Euclidean black holes that are asymptotically locally AdS$_5$, with $S^1 \times M_3$ conformal boundary. We study field theory backgrounds consisting of various $M_3$, 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 $\mathcal{N}=4$ SYM in a Cardy-like limit. While such black holes have not been constructed for general $M_3$, 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 AlAdS$_5$ geometry, chosen such that the Casimir energy is removed from the supersymmetric partition function.