Magnetic charge and black hole supersymmetric quantum statistical relation

We study the thermodynamics in the Bogomol'nyi-Prasad-Sommerfield (BPS) limit of anti--de Sitter black holes realizing the topological twist. We use a limiting procedure that allows us to reach the extremal point along a trajectory in the space of supersymmetric Euclidean solutions. We show that on this space we can write a quantum statistical relation, which is well-defined in the BPS limit and relies on imposing a suitable constraint among the chemical potentials, due to supersymmetry and regularity. We stress the importance of this in relating the thermal partition function of the dual field theory to the topologically twisted index.