Non compact (2+1)-TQFTs from non-semisimple spherical categories
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916208848142336 |
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| author | Costantino, Francesco Geer, Nathan Patureau-Mirand, Bertrand Virelizier, Alexis |
| author_facet | Costantino, Francesco Geer, Nathan Patureau-Mirand, Bertrand Virelizier, Alexis |
| contents | This paper contains three related groupings of results. First, we consider a new notion of an admissible skein module of a surface associated to an ideal in a (non-semisimple) pivotal category. Second, we introduce the notion of a chromatic category and associate to such a category a finite dimensional non-compact (2+1)-TQFT by assigning admissible skein modules to closed oriented surfaces and using Juhász's presentation of cobordisms. The resulting TQFT extends to a genuine one if and only if the chromatic category is semisimple with nonzero dimension (recovering then the Turaev-Viro TQFT). The third grouping of results concerns sided chromatic maps in finite tensor categories. In particular, we prove that every spherical tensor category (in the sense of Etingof, Douglas et al.) is a chromatic category (and so can be used to define a non-compact (2+1)-TQFT). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_04509 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non compact (2+1)-TQFTs from non-semisimple spherical categories Costantino, Francesco Geer, Nathan Patureau-Mirand, Bertrand Virelizier, Alexis Quantum Algebra Geometric Topology 57M27, 57R56 This paper contains three related groupings of results. First, we consider a new notion of an admissible skein module of a surface associated to an ideal in a (non-semisimple) pivotal category. Second, we introduce the notion of a chromatic category and associate to such a category a finite dimensional non-compact (2+1)-TQFT by assigning admissible skein modules to closed oriented surfaces and using Juhász's presentation of cobordisms. The resulting TQFT extends to a genuine one if and only if the chromatic category is semisimple with nonzero dimension (recovering then the Turaev-Viro TQFT). The third grouping of results concerns sided chromatic maps in finite tensor categories. In particular, we prove that every spherical tensor category (in the sense of Etingof, Douglas et al.) is a chromatic category (and so can be used to define a non-compact (2+1)-TQFT). |
| title | Non compact (2+1)-TQFTs from non-semisimple spherical categories |
| topic | Quantum Algebra Geometric Topology 57M27, 57R56 |
| url | https://arxiv.org/abs/2302.04509 |