5d Black Hole as Emergent Geometry of Weakly Interacting 4d Hot Yang-Mills Gas

We demonstrate five-dimensional anti-de Sitter black hole emerges as dual geometry holographic to weakly interacting N=4 superconformal Yang-Mills theory. We first note that an ideal probe of the dual geometry is the Yang-Mills instanton, probing point by point in spacetime. We then study instanton moduli space at finite temperature by adopting Hitchin's proposal that geometry of the moduli space is definable by Fisher-Rao "information geometry". In Yang-Mills theory, the information metric is measured by a novel class of gauge-invariant, nonlocal operators in the instanton sector. We show that the moduli space metric exhibits (1) asymptotically anti-de Sitter, (2) horizon at radial distance set by the Yang-Mills temperature, and (3) after Wick rotation of the moduli space to the Lorentzian signature, a singularity at the origin. We argue that the dual geometry emerges even for rank of gauge groups of order unity and for weak `t Hooft coupling.