On the SymTFTs of Finite Non-Abelian Symmetries
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914494069866496 |
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| author | Bergman, Oren Heckman, Jonathan J. Hübner, Max Migliorati, Daniele Yu, Xingyang Zhang, Hao Y. |
| author_facet | Bergman, Oren Heckman, Jonathan J. Hübner, Max Migliorati, Daniele Yu, Xingyang Zhang, Hao Y. |
| contents | The $(D+1)$-dimensional symmetry topological field theory (SymTFT$_{D+1}$) of a $D$-dimensional absolute quantum field theory (QFT$_D$) provides a topological characterization of symmetry data. In this framework, the SymTFT comes equipped with a physical boundary specifying a relative QFT, and a topological boundary which specifies the global form of symmetries. In general, there need not be a unique bulk theory which encodes this information but it is often helpful to have a more manifest presentation of symmetries in terms of bulk degrees of freedom. For the case of a finite non-Abelian symmetry group $G$, the bulk SymTFT may be described by a Dijkgraaf-Witten TFT with gauge group $G$. This makes manifest the ``electric'' presentation of the symmetry data but can obscure some of the magnetic data as well as non-Abelian structure present in the absolute QFT$_D$ such as symmetry operators which cannot fully detach from the topological boundary. We address these issues for 3D SymTFTs by constructing discrete BF-like theory Lagrangians for finite groups which admit a presentation as an extension by a finite Abelian group and a finite (possibly non-Abelian) group. This enables us to give a streamlined approach to reconstructing the fusion rules of the accompanying Drinfeld center, but also allows us to construct surface-attaching non-genuine line operators associated directly with non-Abelian group elements rather than just their conjugacy classes. We also sketch how our treatment generalizes to higher-dimensional SymTFTs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_12323 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the SymTFTs of Finite Non-Abelian Symmetries Bergman, Oren Heckman, Jonathan J. Hübner, Max Migliorati, Daniele Yu, Xingyang Zhang, Hao Y. High Energy Physics - Theory Strongly Correlated Electrons Category Theory The $(D+1)$-dimensional symmetry topological field theory (SymTFT$_{D+1}$) of a $D$-dimensional absolute quantum field theory (QFT$_D$) provides a topological characterization of symmetry data. In this framework, the SymTFT comes equipped with a physical boundary specifying a relative QFT, and a topological boundary which specifies the global form of symmetries. In general, there need not be a unique bulk theory which encodes this information but it is often helpful to have a more manifest presentation of symmetries in terms of bulk degrees of freedom. For the case of a finite non-Abelian symmetry group $G$, the bulk SymTFT may be described by a Dijkgraaf-Witten TFT with gauge group $G$. This makes manifest the ``electric'' presentation of the symmetry data but can obscure some of the magnetic data as well as non-Abelian structure present in the absolute QFT$_D$ such as symmetry operators which cannot fully detach from the topological boundary. We address these issues for 3D SymTFTs by constructing discrete BF-like theory Lagrangians for finite groups which admit a presentation as an extension by a finite Abelian group and a finite (possibly non-Abelian) group. This enables us to give a streamlined approach to reconstructing the fusion rules of the accompanying Drinfeld center, but also allows us to construct surface-attaching non-genuine line operators associated directly with non-Abelian group elements rather than just their conjugacy classes. We also sketch how our treatment generalizes to higher-dimensional SymTFTs. |
| title | On the SymTFTs of Finite Non-Abelian Symmetries |
| topic | High Energy Physics - Theory Strongly Correlated Electrons Category Theory |
| url | https://arxiv.org/abs/2603.12323 |