Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories

We study the thermodynamics of four-dimensional Kerr-Newman-AdS black holes both in the canonical and the grand-canonical ensemble. The stability cond itions are investigated, and the complete phase diagrams are obtained, whic h include the Hawking-Page phase transition in the grand-canonical ensemble . In the canonical case, one has a first-order transition between small and large black holes, which disappears for a sufficiently large electric char ge or angular momentum. This disappearance corresponds to a critical point in the phase diagram. Via the AdS/CFT conjecture, the obtained phase struct ure is also relevant for the corresponding conformal held theory living in a rotating Einstein universe, in the presence of a global background U(1) c urrent. An interesting limit arises when the black holes presence some supe rsymmetry. These BPS black holes correspond to highly degenerate zero-tempe rature stares in the dual CFT, which lives in an Einstein universe rotating with the speed of light.