Categorical 4-manifold invariants from trisection diagrams
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909921520386048 |
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| author | Meusburger, Catherine Mulevicius, Vincentas Torzewska, Fiona |
| author_facet | Meusburger, Catherine Mulevicius, Vincentas Torzewska, Fiona |
| contents | We use Gay and Kirby's description of 4-manifolds in terms of trisections and trisection diagrams to define a new 4-manifold invariant. The algebraic data are an indecomposable finite semisimple bimodule category over a pair of spherical fusion categories, equipped with a bimodule trace, and a pivotal functor from another spherical fusion category into the spherical fusion category of its bimodule endofunctors and natural transformations between them. The 4-manifold invariant has a simple description in terms of a diagrammatic calculus for this data, in which the three spherical fusion categories correspond to the three colours of the trisection diagram.
It includes the Hopf algebraic 4-manifold invariants by Chaidez, Cotler and Cui, which arise when the bimodule category is the category of finite-dimensional complex vector spaces. We also recover the 4-manifold invariants of Bärenz and Barrett defined by a pivotal functor from a spherical fusion category into a modular fusion category. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_19384 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Categorical 4-manifold invariants from trisection diagrams Meusburger, Catherine Mulevicius, Vincentas Torzewska, Fiona Quantum Algebra Mathematical Physics Geometric Topology We use Gay and Kirby's description of 4-manifolds in terms of trisections and trisection diagrams to define a new 4-manifold invariant. The algebraic data are an indecomposable finite semisimple bimodule category over a pair of spherical fusion categories, equipped with a bimodule trace, and a pivotal functor from another spherical fusion category into the spherical fusion category of its bimodule endofunctors and natural transformations between them. The 4-manifold invariant has a simple description in terms of a diagrammatic calculus for this data, in which the three spherical fusion categories correspond to the three colours of the trisection diagram. It includes the Hopf algebraic 4-manifold invariants by Chaidez, Cotler and Cui, which arise when the bimodule category is the category of finite-dimensional complex vector spaces. We also recover the 4-manifold invariants of Bärenz and Barrett defined by a pivotal functor from a spherical fusion category into a modular fusion category. |
| title | Categorical 4-manifold invariants from trisection diagrams |
| topic | Quantum Algebra Mathematical Physics Geometric Topology |
| url | https://arxiv.org/abs/2511.19384 |