Non-compact 3D TQFT and non-semisimplicity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914224190521344 |
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| author | Lagiotis, Theodoros |
| author_facet | Lagiotis, Theodoros |
| contents | We define a once extended non-compact 3-dimensional TQFT $\mathcal{Z}$ from the data of a (potentially) non-semisimple modular tensor category. This is in the framework of generators and relations of [Bartlett et al., arxiv:1509.06811 (2015)], having disallowed generating 2-morphisms whose source is the empty. Moreover, we show that the projective mapping class group representations this TQFT gives rise to, are dual to those of [Lyubashenko, arXiv:hep-th/9405167 (1994)] and [De Renzi et al., arXiv:2010.14852 (2020)]. We develop a method to decompose a closed 3-manifold in terms of 2-morphism generators. We use this to compute the value of $\mathcal{Z}$ on 3-manifolds, explaining why it should recover Lyubashenko's 3-manifold invariants [Lyubashenko, arXiv:hep-th/9405167 (1994)]. Finally, we explain that the value of the non-compact TQFT on the solid torus recovers the data of a modified trace [Geer et al., arXiv:0711.4229 (2007)]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_23698 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-compact 3D TQFT and non-semisimplicity Lagiotis, Theodoros Quantum Algebra Category Theory Geometric Topology We define a once extended non-compact 3-dimensional TQFT $\mathcal{Z}$ from the data of a (potentially) non-semisimple modular tensor category. This is in the framework of generators and relations of [Bartlett et al., arxiv:1509.06811 (2015)], having disallowed generating 2-morphisms whose source is the empty. Moreover, we show that the projective mapping class group representations this TQFT gives rise to, are dual to those of [Lyubashenko, arXiv:hep-th/9405167 (1994)] and [De Renzi et al., arXiv:2010.14852 (2020)]. We develop a method to decompose a closed 3-manifold in terms of 2-morphism generators. We use this to compute the value of $\mathcal{Z}$ on 3-manifolds, explaining why it should recover Lyubashenko's 3-manifold invariants [Lyubashenko, arXiv:hep-th/9405167 (1994)]. Finally, we explain that the value of the non-compact TQFT on the solid torus recovers the data of a modified trace [Geer et al., arXiv:0711.4229 (2007)]. |
| title | Non-compact 3D TQFT and non-semisimplicity |
| topic | Quantum Algebra Category Theory Geometric Topology |
| url | https://arxiv.org/abs/2512.23698 |