Generalized Hawking-Page transitions

Abstract

We construct holographic backgrounds that are dual by the AdS/CFT correspondence to Euclidean conformal field theories on products of spheres Sd1×Sd2, for conformal field theories whose dual may be approximated by classical Einstein gravity (typically these are large N strongly coupled theories). For d2 = 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 d2> 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 d1 + d2< 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 d1 + d2< 9 the number of solutions grows to infinity as the critical ratio is approached.