Fusion 3-Categories for Duality Defects
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914011872755712 |
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| author | Bhardwaj, Lakshya Décoppet, Thibault Schafer-Nameki, Sakura Yu, Matthew |
| author_facet | Bhardwaj, Lakshya Décoppet, Thibault Schafer-Nameki, Sakura Yu, Matthew |
| contents | We study the fusion 3-categorical symmetries for quantum theories in (3+1)d with self-duality defects. Such defects have been realized physically by half-space gauging in theories with 1-form symmetries $A[1]$ for an abelian group $A$, and have found applications in the continuum and the lattice. These fusion 3-categories will be called (generalized) Tambara-Yamagami fusion 3-categories $(\mathbf{3TY})$. We consider the Brauer-Picard and Picard 4-groupoids to construct these categories using a 3-categorical version of the extension theory introduced by Etingof, Nikshych and Ostrik. These two 4-groupoids correspond to the construction of duality defects either directly in 4d, or from the 5d Symmetry Topological Field Theory (SymTFT). The Witt group of non-degenerate braided fusion 1-categories naturally appears in the aforementioned 4-groupoids and represents enrichments of standard duality defects by (2+1)d TFTs. Our main objective is to study graded extensions of the fusion 3-category $\mathbf{3Vect}(A[1])$. Firstly, we use invertible bimodule 3-categories and the Brauer-Picard 4-groupoid. Secondly, we use that the Brauer-Picard 4-groupoid of $\mathbf{3Vect}(A[1])$ can be identified with the Picard 4-groupoid of its Drinfeld center. Moreover, the Drinfeld center of $\mathbf{3Vect}(A[1])$, which represents topological defects of the SymTFT, is completely described by a sylleptic strongly fusion 2-category formed by topological surface defects of the SymTFT. These are classified by a finite abelian group equipped with an alternating 2-form. We relate the Picard 4-groupoid of the corresponding braided fusion 3-categories with a generalized Witt group constructed from certain graded braided fusion 1-categories using a twisted Deligne tensor product. We perform explicit computations for $\mathbb{Z}/2$ and $\mathbb{Z}/4$ graded $\mathbf{3TY}$ categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13302 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fusion 3-Categories for Duality Defects Bhardwaj, Lakshya Décoppet, Thibault Schafer-Nameki, Sakura Yu, Matthew Category Theory High Energy Physics - Theory We study the fusion 3-categorical symmetries for quantum theories in (3+1)d with self-duality defects. Such defects have been realized physically by half-space gauging in theories with 1-form symmetries $A[1]$ for an abelian group $A$, and have found applications in the continuum and the lattice. These fusion 3-categories will be called (generalized) Tambara-Yamagami fusion 3-categories $(\mathbf{3TY})$. We consider the Brauer-Picard and Picard 4-groupoids to construct these categories using a 3-categorical version of the extension theory introduced by Etingof, Nikshych and Ostrik. These two 4-groupoids correspond to the construction of duality defects either directly in 4d, or from the 5d Symmetry Topological Field Theory (SymTFT). The Witt group of non-degenerate braided fusion 1-categories naturally appears in the aforementioned 4-groupoids and represents enrichments of standard duality defects by (2+1)d TFTs. Our main objective is to study graded extensions of the fusion 3-category $\mathbf{3Vect}(A[1])$. Firstly, we use invertible bimodule 3-categories and the Brauer-Picard 4-groupoid. Secondly, we use that the Brauer-Picard 4-groupoid of $\mathbf{3Vect}(A[1])$ can be identified with the Picard 4-groupoid of its Drinfeld center. Moreover, the Drinfeld center of $\mathbf{3Vect}(A[1])$, which represents topological defects of the SymTFT, is completely described by a sylleptic strongly fusion 2-category formed by topological surface defects of the SymTFT. These are classified by a finite abelian group equipped with an alternating 2-form. We relate the Picard 4-groupoid of the corresponding braided fusion 3-categories with a generalized Witt group constructed from certain graded braided fusion 1-categories using a twisted Deligne tensor product. We perform explicit computations for $\mathbb{Z}/2$ and $\mathbb{Z}/4$ graded $\mathbf{3TY}$ categories. |
| title | Fusion 3-Categories for Duality Defects |
| topic | Category Theory High Energy Physics - Theory |
| url | https://arxiv.org/abs/2408.13302 |