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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.10113 |
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| _version_ | 1866913654693167104 |
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| author | Großkopf, Paul |
| author_facet | Großkopf, Paul |
| contents | Homotopy Quantum Field Theories as variants of Topological Quantum Field Theories are described by functors from some cobordism category, enriched with homotopical data, to a symmetric monoidal category $\mathcal{V}$. A new notion of HQFTs is introduced using target pairs of spaces $(X,Y)$ acounting for basepoints being sent to points in $Y$. Such $(X,Y)$-HQFTs are classified in dimension 1 by dualizable representations of $\mathcal{G}:=Π_1(X,Y)$, the relative fundamental groupoid. For dimension 2, the notion of crossed loop Frobenius $(\mathcal{G},\mathcal{V})$-categories is introduced, generalizing crossed Frobenius $G$-algebras, where $G$ is only a group. After stating generalities of these multi-object generalizations, a classification theorem of 2-dimensional $(X,Y)$-HQFTs via crossed loop Frobenius $(\mathcal{G},\mathcal{V})$-categories is proven. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10113 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | 2D HQFTs and Frobenius $(\mathcal{G},\mathcal{V})$-categories Großkopf, Paul Quantum Algebra Homotopy Quantum Field Theories as variants of Topological Quantum Field Theories are described by functors from some cobordism category, enriched with homotopical data, to a symmetric monoidal category $\mathcal{V}$. A new notion of HQFTs is introduced using target pairs of spaces $(X,Y)$ acounting for basepoints being sent to points in $Y$. Such $(X,Y)$-HQFTs are classified in dimension 1 by dualizable representations of $\mathcal{G}:=Π_1(X,Y)$, the relative fundamental groupoid. For dimension 2, the notion of crossed loop Frobenius $(\mathcal{G},\mathcal{V})$-categories is introduced, generalizing crossed Frobenius $G$-algebras, where $G$ is only a group. After stating generalities of these multi-object generalizations, a classification theorem of 2-dimensional $(X,Y)$-HQFTs via crossed loop Frobenius $(\mathcal{G},\mathcal{V})$-categories is proven. |
| title | 2D HQFTs and Frobenius $(\mathcal{G},\mathcal{V})$-categories |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2501.10113 |