Generalized Hawking-Page transitions

Abstract

Abstract We construct holographic backgrounds that are dual by the AdS/CFT correspondence to Euclidean conformal field theories on products of spheres $$ {S}^{d_1}\times {S}^{d_2} $$ S d 1 × S d 2 , 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 $$ {S}^{d_1} $$ S d 1 , 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.