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A volume correspondence between anti-de Sitter space and its boundary

Lizhao Zhang

Abstract

Let $\mathbb{H}^{n+1}_1$ be the $(n+1)$-dimensional anti-de Sitter space (AdS), in this paper we propose to extend $\mathbb{H}^{n+1}_1$ conformally to another copy of $\mathbb{H}^{n+1}_1$ by gluing them along the boundary at infinity, and denote the resulting space by \emph{double anti-de Sitter space} $\mathbb{DH}^{n+1}_1$. We propose to introduce a volume $V_{n+1}(P)$ (possibly complex valued) on polytopes $P$ in $\mathbb{DH}^{n+1}_1$ whose facets all have non-degenerate metrics (called \emph{good} polytopes), and show that it is well defined and invariant under isometry, including the case that $P$ contains a non-trivial portion of $\partial\mathbb{H}^{n+1}_1$. For $n$ even, $V_{n+1}(P)$ is shown to be completely determined by the intersection of $P$ and $\partial\mathbb{H}^{n+1}_1$, which leads to the following important applications: it induces a new intrinsic (conformal) \emph{volume} on good polytopes in $\partial\mathbb{H}^{n+1}_1$ that is invariant under conformal transformations of $\partial\mathbb{H}^{n+1}_1$, and establishes an AdS-CFT type correspondence between the volumes on $\mathbb{DH}^{n+1}_1$ and $\partial\mathbb{H}^{n+1}_1$.

A volume correspondence between anti-de Sitter space and its boundary

Abstract

Let be the -dimensional anti-de Sitter space (AdS), in this paper we propose to extend conformally to another copy of by gluing them along the boundary at infinity, and denote the resulting space by \emph{double anti-de Sitter space} . We propose to introduce a volume (possibly complex valued) on polytopes in whose facets all have non-degenerate metrics (called \emph{good} polytopes), and show that it is well defined and invariant under isometry, including the case that contains a non-trivial portion of . For even, is shown to be completely determined by the intersection of and , which leads to the following important applications: it induces a new intrinsic (conformal) \emph{volume} on good polytopes in that is invariant under conformal transformations of , and establishes an AdS-CFT type correspondence between the volumes on and .
Paper Structure (24 sections, 24 theorems, 80 equations, 8 figures)

This paper contains 24 sections, 24 theorems, 80 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Double anti-de Sitter space
  4. Main results
  5. Strategy overview
  6. Preliminaries
  7. A new model for $\mathbb{DH}^{n+1}_1$
  8. An isometric embedding
  9. A gluing procedure
  10. A sign function $h(x)$
  11. Isometries of $\mathcal{R}_D$
  12. Half-spaces in $\mathcal{R}_D$
  13. A definition of $V_{n+1}(P)$ on $\mathbb{DH}^{n+1}_1$
  14. Definition of $\mu(P)$ on $\mathcal{R}$
  15. Proof of Theorem \ref{['theorem_polytope_volume_finite_invariant']} (assuming Theorem \ref{['theorem_mu_volume_invariant_AdS']})
  16. ...and 9 more sections

Key Result

Theorem 1.4

Let $P\in\mathcal{H}_0$ in $\mathbb{DH}^{n+1}_1$, then $V_{n+1}(P)$ is well defined and invariant under isometry.

Figures (8)

  • Figure 1: The double anti-de Sitter space $\mathbb{DH}^{n+1}_1$ is obtained by gluing $\mathbb{H}^{n+1}_1$ to $\mathbb{H}^{n+1}_{1,-}$ by identifying their opposite ends projectively, e.g., $A$ with $A'$, and $B$ with $B'$, etc.
  • Figure 2: An isometric embedding of $\mathcal{R}$ and $\mathcal{R}_{-}$ into $\mathbb{DH}^{n+1}_1$
  • Figure 3: A closed null geodesic is formed by connecting a null line $l$ in $\mathcal{R}$ to $l_{-}$ in $\mathcal{R}_{-}$ through their opposite ends, by identifying $A$ with $A'$, and $B$ with $B'$ respectively
  • Figure 4: The face $ax^2+b\cdot x+c=0$ (when $a\ne 0$) with discriminant $D=b^2-4ac$
  • Figure 5: The top, bottom, and side faces in $\mathcal{R}$
  • ...and 3 more figures

Theorems & Definitions (72)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 3.1
  • Definition 3.2
  • Definition 3.2
  • ...and 62 more