Attractors and the quantum statistical relation for extreme (BPS or not) black holes

The attractor mechanism implies that the supersymmetric black hole (BH) near horizon (NH) solution is defined in terms of the conserved charges only and therefore is independent of asymptotic moduli. In these NH geometries, Sen recovered the BH entropy by means of a Legendre transformation where the electric fields are defined as conjugated variables to the electric charges. There is however another way to define the entropy and in general the “statistics mechanics of BPS BH” using a multi-scaling limit of non-extremal BH. In this formalism, direct contact with the dual field theory formulation (that motivates it) and also with the canonical BH thermodynamics is transparent and natural. In this article we make contact between these two different approaches giving a precise match between the corresponding statistical potentials conjugated to the charges, the Euclidean action and the entropy. In particular Sen’s results are recovered from the quantum statistical relation of BH thermodynamics where the role of the attractor mechanism is crucial. Our analysis is carried on for asymptotically flat and AdS BHs, as well as for supersymmetric and extreme non-BPS BHs.