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Anti-de Sitter space and black holes

Maximo Banados, Andres Gomberoff, Cristian Martinez

TL;DR

This work extends the concept of AdS black holes from 2+1 dimensions to higher dimensions by constructing topological black holes via identifications in AdS spaces. It derives both Lorentzian and Euclidean realizations in 4D and 5D, analyzes their causal structure, and connects them to thermal AdS physics; it further embeds these solutions in five-dimensional Chern-Simons gravity and a CS supergravity framework, studying global charges, holonomies, and Killing spinors. The authors show vanishing global charges in pure $SO(4,2)$ CS gravity but nonzero charges in $SO(4,2)\times U(1)$ when a boundary 2-form is fixed, and they compute holonomies through explicit group elements, including extreme and Euclidean cases. They also explore generalized topologies $M^k\times W_{D-k}$ and discuss solutions with and without a cosmological constant, highlighting rich structures beyond standard Schwarzschild-AdS geometries and their potential implications for AdS/CFT and higher-dimensional gravity theories.

Abstract

Anti-de Sitter space with identified points give rise to black-hole structures. This was first pointed out in three dimensions, and generalized to higher dimensions by Aminneborg et al. In this paper, we analyse several aspects of the five dimensional anti-de Sitter black hole including, its relation to thermal anti-de Sitter space, its embedding in a Chern-Simons supergravity theory, its global charges and holonomies, and the existence of Killing spinors.

Anti-de Sitter space and black holes

TL;DR

This work extends the concept of AdS black holes from 2+1 dimensions to higher dimensions by constructing topological black holes via identifications in AdS spaces. It derives both Lorentzian and Euclidean realizations in 4D and 5D, analyzes their causal structure, and connects them to thermal AdS physics; it further embeds these solutions in five-dimensional Chern-Simons gravity and a CS supergravity framework, studying global charges, holonomies, and Killing spinors. The authors show vanishing global charges in pure CS gravity but nonzero charges in when a boundary 2-form is fixed, and they compute holonomies through explicit group elements, including extreme and Euclidean cases. They also explore generalized topologies and discuss solutions with and without a cosmological constant, highlighting rich structures beyond standard Schwarzschild-AdS geometries and their potential implications for AdS/CFT and higher-dimensional gravity theories.

Abstract

Anti-de Sitter space with identified points give rise to black-hole structures. This was first pointed out in three dimensions, and generalized to higher dimensions by Aminneborg et al. In this paper, we analyse several aspects of the five dimensional anti-de Sitter black hole including, its relation to thermal anti-de Sitter space, its embedding in a Chern-Simons supergravity theory, its global charges and holonomies, and the existence of Killing spinors.

Paper Structure

This paper contains 27 sections, 97 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The four dimensional topological black hole
  3. Identifications and causal structure
  4. Kruskal coordinates
  5. Schwarzschild metric
  6. The vacuum solution
  7. Euclidean black hole and thermal anti-de Sitter space
  8. The five dimensional topological black hole
  9. The spinless 5D black hole
  10. Euclidean anti-de Sitter space
  11. A black hole with two times
  12. Minkowskian rotating black hole
  13. The extreme and zero mass black hole
  14. Euclidean Global charges in five dimensions
  15. Global charges in Chern-Simons gravity
  16. ...and 12 more sections

Figures (2)

  • Figure 1: Anti-de Sitter space. Each point in the figure represent the pair ${x_3,x_4}$ with $x_3^2-x_4^2$ (the norm of the Killing vector) fixed.
  • Figure 2: Penrose Diagram