Phases of R-charged black holes, spinning branes and strongly coupled gauge theories

We study the thermodynamic stability of charged black holes in gauged supergravity theories in D = 5, D = 4 and D = 7. We find explicitly the location of the Hawking-Page phase transition between charged black holes and the pure anti-de Sitter space-time, both in the grand-canonical ensemble, where electric potentials are held fixed, and in the canonical ensemble, where total charges are held fixed. We also find the explicit local thermodynamic stability constraints for black holes with one non-zero charge. In the grand-canonical ensemble, there is in general a region of phase space where neither the anti-de Sitter space-time is dynamically preferred, nor are the charged black holes thermodynamically stable. But in the canonical ensemble, anti-de Sitter space-time is always dynamically preferred in the domain where black holes are unstable. We demonstrate the equivalence of large R-charged black holes in D = 5,D = 4 and D = 7 with spinning near-extreme D3-, M2- and M5-branes, respectively. The mass, the charges and the entropy of such black holes can be mapped into the energy above extremality, the angular momenta and the entropy of the corresponding branes. We also note a peculiar numerological sense in which the grand-canonical stability constraints for large charge black holes in D = 4 and D = 7 are dual, and in which the D = 5 constraints are self-dual.