Gravity Dual of Gauge Theory on S 2 × S 1 × R

We (numerically) construct new static, asymptotically AdS solutions where the conformal infinity is the product of time and S2 × S1. There always exist a family of solutions in which the S1 is not contractible and, for small S1, there are two additional families of solutions in which the S1 smoothly pinches off. This shows that (when fermions are antiperiodic around the S1) there is a quantum phase transition in the gauge theory as one decreases the radius of the S1 relative to the S2. We also compare the masses of our solutions and argue that the one with lowest mass should minimize the energy among all solutions with conformal boundary S2×S1×R. This provides a new positive energy conjecture for asymptotically locally AdS metrics. A simple analytic continuation produces AdS black holes with topology S2 × S1.