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

Abstract

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

simplified the paper, removed some non-essential contents. 30 pages, 8 figures