# Generalized Hawking-Page transitions

## TL;DR

This paper constructs and analyzes holographic backgrounds dual to conformal field theories on product spheres, revealing generalized Hawking-Page phase transitions with multiple solutions and topology changes, extending known results to higher dimensions.

## Contribution

It generalizes Hawking-Page transitions to higher-dimensional product spheres and provides numerical and analytical analysis of the phase structure and solutions.

## Key findings

- Multiple bulk solutions with different topologies exist for various sphere ratios.
- First-order phase transitions occur between solutions as the sphere sizes vary.
- Number of solutions diverges near a critical ratio for total dimension less than 9.

## Abstract

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