Skein Construction of Balanced Tensor Products
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arXiv
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| Format: | Preprint |
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2025
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| author | Araújo, Manuel Guu, Jin-Cheng Hudson, Skyler |
| author_facet | Araújo, Manuel Guu, Jin-Cheng Hudson, Skyler |
| contents | The theory of tensor categories has found applications across various fields, including representation theory, quantum field theory (conformal in 2 dimensions, and topological in 3 and 4 dimensions), quantum invariants of low-dimensional objects, topological phases of matter, and topological quantum computation. In essence, it is a categorification of the classical theory of algebras and modules. In this analogy, the Deligne tensor product $\boxtimes$ is to the linear tensor $\otimes_{\mathbb{C}}$ as the balanced tensor product $\boxtimes_C$ is to the tensor over algebra $\otimes_A$, where $\mathbb{C}$ is a field, $A$ is a $\mathbb{C}$-algebra, and $C$ is a tensor category.
Before this work, several algebraic constructions for balanced tensor products were known, including categories of modules, internal Hom spaces, and generalized categorical centers. In this paper, we introduce a topological construction based on skein theory that offers a better mix of algebra and topology. This approach not only works for products of multiple module categories, but also provides the missing key to proving that the Turaev-Viro state sum model naturally arises from the 3-functor in the classification of fully extended field theories. Building on this result, we establish this long-anticipated proof in an upcoming work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05747 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Skein Construction of Balanced Tensor Products Araújo, Manuel Guu, Jin-Cheng Hudson, Skyler Mathematical Physics Category Theory Quantum Algebra Representation Theory Quantum Physics 81T45 (Primary) 16D90, 18D10, 17B37 (Secondary) The theory of tensor categories has found applications across various fields, including representation theory, quantum field theory (conformal in 2 dimensions, and topological in 3 and 4 dimensions), quantum invariants of low-dimensional objects, topological phases of matter, and topological quantum computation. In essence, it is a categorification of the classical theory of algebras and modules. In this analogy, the Deligne tensor product $\boxtimes$ is to the linear tensor $\otimes_{\mathbb{C}}$ as the balanced tensor product $\boxtimes_C$ is to the tensor over algebra $\otimes_A$, where $\mathbb{C}$ is a field, $A$ is a $\mathbb{C}$-algebra, and $C$ is a tensor category. Before this work, several algebraic constructions for balanced tensor products were known, including categories of modules, internal Hom spaces, and generalized categorical centers. In this paper, we introduce a topological construction based on skein theory that offers a better mix of algebra and topology. This approach not only works for products of multiple module categories, but also provides the missing key to proving that the Turaev-Viro state sum model naturally arises from the 3-functor in the classification of fully extended field theories. Building on this result, we establish this long-anticipated proof in an upcoming work. |
| title | Skein Construction of Balanced Tensor Products |
| topic | Mathematical Physics Category Theory Quantum Algebra Representation Theory Quantum Physics 81T45 (Primary) 16D90, 18D10, 17B37 (Secondary) |
| url | https://arxiv.org/abs/2501.05747 |