Black holes with topologically nontrivial AdS asymptotics

TL;DR

This paper analyzes static black hole geometries with topologically nontrivial AdS asymptotics in higher-dimensional Lanczos-Lovelock gravities that share a unique AdS vacuum. It derives a general metric family with transverse constant-curvature sections, classifies horizon structures across gamma and parity of k, and computes thermodynamic quantities such as temperature, entropy, and the canonical partition function, showing that a negative cosmological constant is essential for stable equilibrium. The results reveal a rich thermodynamic landscape, including topology-dependent stability and, in the holographic view, a set of dual boundary CFTs corresponding to the different bulk actions I_k, with up to $\left\lfloor \frac{d-1}{2} \right\rfloor$ inequivalent theories. In particular, gamma=0 cases exhibit exact entropy matching with boundary CFTs on flat backgrounds, while gamma=±1 cases align in the high-temperature limit, supporting a broad AdS/CFT interpretation across bulk theories including Einstein-Hilbert and Chern-Simons gravities.

Abstract

Asymptotically locally AdS black hole geometries of dimension d > 2 are studied for nontrivial topologies of the transverse section. These geometries are static solutions of a set of theories labeled by an integer 0 < k < [(d-1)/2] which possess a unique globally AdS vacuum. The transverse sections of these solutions are d-2 surfaces of constant curvature, allowing for different topological configurations. The thermodynamic analysis of these solutions reveals that the presence of a negative cosmological constant is essential to ensure the existence of stable equilibrium states. In addition, it is shown that these theories are holographically related to [(d-1)/2] different conformal field theories at the boundary.

Figures (3)

• Figure 1: The horizons are located at the zeros of $f^{2}(r)$, which occur at the intersections of the parabolas $\left( \gamma +\frac{r^{2}}{l^{2}}\right)$ and the functions $\alpha \left( \frac{2G_{k}\mu}{r^{d-2k-1}}\right)^{1/k}$. These are displayed for $\gamma=0,\pm 1$, and different values of $\alpha$ and $\mu$. There exist a single horizon for $\alpha=1$ and $\mu \geq 0$. Two horizons arise either for $\alpha=1$ and $\mu_{c}<\mu<0$, or for $\alpha=-1$ and $0<\mu<\mu_{c}$. In the extreme case, both horizons coalesce for $\mu=\mu_{c}$.
• Figure 2: The temperature as a function of $r_{+}$ is depicted for generic theories with $\gamma=0,\pm 1$. For $\gamma=1$ the temperature has a minimum at $r_{+}=r_{c}$. When $\gamma=0$ the temperature is a linear function of $r_{+}$. For $\gamma=-1$ the temperature is an increasing function of $r_{+}$ that vanishes at $r_{+}=r_{c}$. For $r_{+}\gg l$ the temperature grows linearly with $r_{+}$ for all cases. For CS theories the three curves are replaced by the $\gamma=0$ straight line with a universal slope.
• Figure 3: The mass parameter ($\mu$) as function of the temperature is depicted for $\gamma=-1$ solutions. For even $k$, $\mu$ has a local maximum at $T=0$ and an absolute minimum at $T_{l}$. For odd $k$, the mass parameter has an absolute minimum at $T=0$ and an inflexion point at $T_{l}$ for $k\neq 1$. The specific heat vanishes at these critical points. For even $k$ the specific heat is negative for $T<T_{l}$. For odd $k\neq 1$, the inflexion point at $T=T_{l}$ signals the existence of a "volatile point". For the EH case ($k=1$) there is not such volatile state, since the specific heat has an absolute minimum at $T=0$.