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

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 $\partial\mathbb{H}^{n+1}_1$.