Abelian TQFTS and Schrödinger local systems
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866912034350694400 |
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| author | Andreev, Aleksei Beliakova, Anna Blanchet, Christian |
| author_facet | Andreev, Aleksei Beliakova, Anna Blanchet, Christian |
| contents | We construct an action of 3-cobordisms on the finite dimensional Schrödinger representations of the Heisenberg group by Lagrangian correspondences. In addition, we review the construction of the abelian Topological Quantum Field Theory (TQFT) associated with a $q$-deformation of $U(1)$ for any root of unity $q$. We prove that for3-cobor\-disms compatible with Lagrangian correspondences, there is a normalization of the associated Schrödinger bimodule action that reproduces the abelian TQFT. The full abelian TQFT provides a projective representation of the mapping class group $\mathrm{Mod}(Σ)$ on the Schrödinger representation,which is linearizable at odd root of 1. Motivated by homology of surface configurations with Schrödinger representation as local coefficients, we define another projective action of $\mathrm{Mod}(Σ)$ on Schrödinger representations. We show that the latter is not linearizable by identifying the associated 2-cocycle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_10725 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Abelian TQFTS and Schrödinger local systems Andreev, Aleksei Beliakova, Anna Blanchet, Christian Geometric Topology Quantum Algebra 57K16, 57R56 We construct an action of 3-cobordisms on the finite dimensional Schrödinger representations of the Heisenberg group by Lagrangian correspondences. In addition, we review the construction of the abelian Topological Quantum Field Theory (TQFT) associated with a $q$-deformation of $U(1)$ for any root of unity $q$. We prove that for3-cobor\-disms compatible with Lagrangian correspondences, there is a normalization of the associated Schrödinger bimodule action that reproduces the abelian TQFT. The full abelian TQFT provides a projective representation of the mapping class group $\mathrm{Mod}(Σ)$ on the Schrödinger representation,which is linearizable at odd root of 1. Motivated by homology of surface configurations with Schrödinger representation as local coefficients, we define another projective action of $\mathrm{Mod}(Σ)$ on Schrödinger representations. We show that the latter is not linearizable by identifying the associated 2-cocycle. |
| title | Abelian TQFTS and Schrödinger local systems |
| topic | Geometric Topology Quantum Algebra 57K16, 57R56 |
| url | https://arxiv.org/abs/2306.10725 |