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

We study the thermodynamic stability of charged black holes in gaugedsupergravity theories in D=5, D=4 and D=7. We find explicitly the location ofthe Hawking-Page phase transition between charged black holes and the pureanti-de Sitter space-time, both in the grand-canonical ensemble, where electricpotentials are held fixed, and in the canonical ensemble, where total chargesare held fixed. We also find the explicit local thermodynamic stabilityconstraints for black holes with one non-zero charge. In the grand-canonicalensemble, there is in general a region of phase space where neither the anti-deSitter space-time is dynamically preferred, nor are the charged black holesthermodynamically stable. But in the canonical ensemble, anti-de Sitterspace-time is always dynamically preferred in the domain where black holes areunstable. We demonstrate the equivalence of large R-charged black holes in D=5, D=4 andD=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 energyabove extremality, the angular momenta and the entropy of the correspondingbranes. We also note a peculiar numerological sense in which thegrand-canonical stability constraints for large charge black holes in D=4 andD=7 are dual, and in which the D=5 constraints are self-dual.Comment: 30 pages, 6 figure