A trilogy of mapping class group representations from three-dimensional quantum gravity
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911930110705664 |
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| author | Kim, Hyun Kyu |
| author_facet | Kim, Hyun Kyu |
| contents | For a punctured surface $\mathfrak{S}$, the author and Scarinci (arXiv:2112.13329) have recently constructed a quantization of a moduli space of Lorentzian metrics on the 3-manifold $\mathfrak{S} \times \mathbb{R}$ of constant sectional curvature $Λ\in \{-1,0,1\}$. The invariance of this quantization under the action of the mapping class group ${\rm MCG}(\mathfrak{S})$ of $\mathfrak{S}$ yields families of unitary representations of ${\rm MCG}(\mathfrak{S})$ on a Hilbert space, with key ingredients being three versions of the quantum dilogarithm functions depending on $Λ$. In this survey article, we review and elaborate on this result. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_02934 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A trilogy of mapping class group representations from three-dimensional quantum gravity Kim, Hyun Kyu Geometric Topology Mathematical Physics Representation Theory 57K20, 13F60, 81R60, 83C45, 20C35, 57K35, 20G42, 47B02 For a punctured surface $\mathfrak{S}$, the author and Scarinci (arXiv:2112.13329) have recently constructed a quantization of a moduli space of Lorentzian metrics on the 3-manifold $\mathfrak{S} \times \mathbb{R}$ of constant sectional curvature $Λ\in \{-1,0,1\}$. The invariance of this quantization under the action of the mapping class group ${\rm MCG}(\mathfrak{S})$ of $\mathfrak{S}$ yields families of unitary representations of ${\rm MCG}(\mathfrak{S})$ on a Hilbert space, with key ingredients being three versions of the quantum dilogarithm functions depending on $Λ$. In this survey article, we review and elaborate on this result. |
| title | A trilogy of mapping class group representations from three-dimensional quantum gravity |
| topic | Geometric Topology Mathematical Physics Representation Theory 57K20, 13F60, 81R60, 83C45, 20C35, 57K35, 20G42, 47B02 |
| url | https://arxiv.org/abs/2308.02934 |