Topological Black Holes in Anti-de Sitter Space

TL;DR

The paper extends the AdS/CFT framework to a broad class of topological black holes in $d$-dimensional anti-de Sitter space, where the event horizon is an Einstein manifold with curvature $k\in\{1,0,-1\}$. It constructs these solutions with the metric $ds^{2} = -f(r)dt^{2} + f^{-1}(r)dr^{2} + r^{2} h_{ij}dx^{i}dx^{j}$ and $f(r)= k - \frac{\omega_{d}M}{r^{d-3}} + \frac{r^{2}}{l^{2}}$, proving they are Einstein with negative cosmological constant when $R_{ij}(h)=(d-3)k h_{ij}$; the horizon topology can be spherical, toroidal, or hyperbolic, and the spacetimes are asymptotically locally AdS. The thermodynamics are computed, yielding the entropy $S = \frac{\mathrm{Vol}(M^{d-2})}{4G} r_{+}^{d-2}$ and energy $E = M - M_{\mathrm{crit}}$, with a temperature dependent phase structure: a Hawking-Page transition for $k=1$, while $k=0$ and $k=-1$ black holes are thermodynamically stable at all temperatures and reproduce the expected holographic entropy scaling at high temperature. Through AdS/CFT, these results provide a microscopic explanation for the entropy behavior of the dual CFT on $S^{1}\times M^{d-2}$ and extend holographic phase structure to topologically nontrivial horizons, highlighting potential links to horizon topology invariants.

Abstract

We consider a class of black hole solutions to Einstein's equations in d dimensions with a negative cosmological constant. These solutions have the property that the horizon is a (d-2)-dimensional Einstein manifold of positive, zero, or negative curvature. The mass, temperature, and entropy are calculated. Using the correspondence with conformal field theory, the phase structure of the solutions is examined, and used to determine the correct mass dependence of the Bekenstein-Hawking entropy.