Generalized Hawking-Page transitions

We construct holographic backgrounds that are dual by the AdS/CFT correspondence to Euclidean conformal field theories on products of spheres Sd1xSd2, for conformal field theories whose dual may be approximated by classical Einstein gravity (typically these are large N strongly coupled theories). For d(2) = 1 these backgrounds correspond to thermal field theories on Sd1, and Hawking and Page found that there are several possible bulk solutions, with two different topologies, that compete with each other, leading to a phase transition as the relative size of the spheres is modified. By numerically solving the Einstein equations we find similar results also for d(2)> 1, with bulk solutions in which either one or the other sphere shrinks to zero smoothly at a minimal value of the radial coordinate, and with a first order phase transition (for d(1) + d(2)<9) between solutions of two different topologies as the relative radius changes. For a critical ratio of the radii there is a (sub-dominant) singular solution where both spheres shrink, and we analytically analyze the behavior near this radius. For d(1) + d(2)<9 the number of solutions grows to infinity as the critical ratio is approached.