A volume correspondence between anti-de Sitter space and its boundary
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914154518937600 |
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| author | Zhang, Lizhao |
| author_facet | Zhang, Lizhao |
| contents | 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 $\partial\mathbb{H}^{n+1}_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09034 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A volume correspondence between anti-de Sitter space and its boundary Zhang, Lizhao Metric Geometry 51M10, 51F99, 51M25, 52B11 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 $\partial\mathbb{H}^{n+1}_1$. |
| title | A volume correspondence between anti-de Sitter space and its boundary |
| topic | Metric Geometry 51M10, 51F99, 51M25, 52B11 |
| url | https://arxiv.org/abs/2301.09034 |