A quantization of moduli spaces of 3-dimensional gravity
| Autor: | Kim, Hyun Kyu, Scarinci, Carlos |
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| Rok vydání: | 2021 |
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| Zdroj: | Commun. Math. Phys. 405, 144 (2024) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00220-024-05012-8 |
| Popis: | We construct a quantization of the moduli space $\mathcal{GH}_\Lambda(S\times\mathbb{R})$ of maximal globally hyperbolic Lorentzian metrics on $S\times \mathbb{R}$ with constant sectional curvature $\Lambda$, for a punctured surface $S$. Although this moduli space is known to be symplectomorphic to the cotangent bundle of the Teichm\"uller space of $S$ independently of the value of $\Lambda$, we define geometrically natural classes of observables leading to $\Lambda$-dependent quantizations. Using special coordinate systems, we first view $\mathcal{GH}_\Lambda(S\times\mathbb{R})$ as the set of points of a cluster $\mathscr{X}$-variety valued in the ring of generalized complex numbers $\mathbb{R}_\Lambda = \mathbb{R}[\ell]/(\ell^2+\Lambda)$. We then develop an $\mathbb{R}_\Lambda$-version of the quantum theory for cluster $\mathscr{X}$-varieties by establishing $\mathbb{R}_\Lambda$-versions of the quantum dilogarithm function. As a consequence, we obtain three families of projective unitary representations of the mapping class group of $S$. For $\Lambda <0$ these representations recover those of Fock and Goncharov, while for $\Lambda\geq 0$ the representations are new. Comment: 64 pages, 2 figures. revised version accepted for publication in Commun. Math. Phys |
| Databáze: | arXiv |
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