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AdS/CFT Duals of Topological Black Holes and the Entropy of Zero-Energy States

Roberto Emparan

TL;DR

This work extends the AdS/CFT correspondence to hyperbolic (topological) black holes, revealing new strong- versus weak-coupling features in the dual CFTs, notably entropy increases at fixed or zero energy and a highly degenerate zero-energy ground state. It shows that hyperbolic horizons map to thermal Rindler states, linking entanglement entropy to horizon thermodynamics and highlighting state-dependent discrepancies between weak-field (free) and strong-coupling (supergravity) results, especially for AdS_5/SYM. The findings point to exotic, possibly precursor-like degrees of freedom that contribute to entropy without increasing local energy, and suggest rich avenues for understanding the microstates of topological black holes and their CFT duals.

Abstract

The horizon of a static black hole in Anti-deSitter space can be spherical, planar, or hyperbolic. The microscopic dynamics of the first two classes of black holes have been extensively discussed recently within the context of the AdS/CFT correspondence. We argue that hyperbolic black holes introduce new and fruitful features in this respect, allowing for more detailed comparisons between the weak and strong coupling regimes. In particular, by focussing on the stress tensor and entropy of some particular states, we identify unexpected increases in the entropy of Super-Yang-Mills theory at strong coupling that are not accompanied by increases in the energy. We describe a highly degenerate state at zero temperature and zero energy density. We also find that the entanglement entropy across a Rindler horizon in exact AdS_5 is larger than might have been expected from the dual SYM theory. Besides, we show that hyperbolic black holes can be described as thermal Rindler states of the dual conformal field theory in flat space.

AdS/CFT Duals of Topological Black Holes and the Entropy of Zero-Energy States

TL;DR

This work extends the AdS/CFT correspondence to hyperbolic (topological) black holes, revealing new strong- versus weak-coupling features in the dual CFTs, notably entropy increases at fixed or zero energy and a highly degenerate zero-energy ground state. It shows that hyperbolic horizons map to thermal Rindler states, linking entanglement entropy to horizon thermodynamics and highlighting state-dependent discrepancies between weak-field (free) and strong-coupling (supergravity) results, especially for AdS_5/SYM. The findings point to exotic, possibly precursor-like degrees of freedom that contribute to entropy without increasing local energy, and suggest rich avenues for understanding the microstates of topological black holes and their CFT duals.

Abstract

The horizon of a static black hole in Anti-deSitter space can be spherical, planar, or hyperbolic. The microscopic dynamics of the first two classes of black holes have been extensively discussed recently within the context of the AdS/CFT correspondence. We argue that hyperbolic black holes introduce new and fruitful features in this respect, allowing for more detailed comparisons between the weak and strong coupling regimes. In particular, by focussing on the stress tensor and entropy of some particular states, we identify unexpected increases in the entropy of Super-Yang-Mills theory at strong coupling that are not accompanied by increases in the energy. We describe a highly degenerate state at zero temperature and zero energy density. We also find that the entanglement entropy across a Rindler horizon in exact AdS_5 is larger than might have been expected from the dual SYM theory. Besides, we show that hyperbolic black holes can be described as thermal Rindler states of the dual conformal field theory in flat space.

Paper Structure

This paper contains 9 sections, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Topological black holes
  3. CFT duals of spherical and planar black holes---a brief review
  4. AdS/CFT duality for hyperbolic black holes
  5. AdS$_5$/SYM on a hyperboloid
  6. Other dimensions
  7. Dual CFT description in Rindler space
  8. Finite 't Hooft coupling corrections
  9. Discussion

Figures (1)

  • Figure 1: Regions of Anti-deSitter space covered by the parametrizations $k=+1,0,-1$ ($\mu=0$ in eq. (\ref{['vk']})). With $k=+1$ all of AdS (the entire strip) is covered. (a) Portion covered in Poincaré (or horospheric) coordinates, $k=0$. (b) Portion covered by the hyperbolic slicing $k=-1$. Shown as a dashed line is a Cauchy surface.