| Summary: | We use equivariant localization to construct off-shell entropy functions for supersymmetric black holes in N = 2, D = 4 gauged supergravity coupled to
matter. This allows one to compute the black hole entropy without solving
the supergravity equations of motion and provides a novel generalization of the
attractor mechanism. We consider magnetically charged black holes in AdS4
which have an AdS2 × M2 near horizon geometry, where M2 is a sphere or
a spindle, and we also obtain entropy functions for ungauged supergravity as
a simple corollary. We derive analogous results for black strings and rings in
D = 5 supergravity which have an AdS3 × M2 near horizon geometry, and in
this setting we derive an off-shell expression for the central charge of the dual
N = (0, 2), d = 2 SCFT.
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