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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.24834 |
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| _version_ | 1866908845346914304 |
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| author | Debray, Arun Ye, Weicheng Yu, Matthew |
| author_facet | Debray, Arun Ye, Weicheng Yu, Matthew |
| contents | We develop a systematic framework for constructing (3+1)-dimensional topological orders or topological quantum field theories (TQFTs) that realize specified anomalies of finite symmetries, as encountered in gauge theories with fermions or in fermionic lattice systems. Our approach generalizes the symmetry-extension construction to the fermionic setting, and is grounded in recent advances in the categorical classification of anomalous TQFTs in (3+1)d. In this framework, symmetry-extension data of a supercohomology theory are translated into a fusion 2-category, on which the anomalous TQFT is built. Building on this machinery, we demonstrate explicit calculations for various symmetry groups and their associated anomalies, with the help of a hastened Adams spectral sequence for computing supercohomology groups which we will detail in a planned sequel. Finally, we prove that all supercohomology anomalies can be realized by fermionic topological orders, whereas beyond-supercohomology anomalies cannot, resolving a question of Córdova--Ohmori for fermionic (3+1)d systems with finite symmetries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24834 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | How to Build Anomalous (3+1)d Topological Quantum Field Theories Debray, Arun Ye, Weicheng Yu, Matthew Mathematical Physics Strongly Correlated Electrons High Energy Physics - Theory Algebraic Topology We develop a systematic framework for constructing (3+1)-dimensional topological orders or topological quantum field theories (TQFTs) that realize specified anomalies of finite symmetries, as encountered in gauge theories with fermions or in fermionic lattice systems. Our approach generalizes the symmetry-extension construction to the fermionic setting, and is grounded in recent advances in the categorical classification of anomalous TQFTs in (3+1)d. In this framework, symmetry-extension data of a supercohomology theory are translated into a fusion 2-category, on which the anomalous TQFT is built. Building on this machinery, we demonstrate explicit calculations for various symmetry groups and their associated anomalies, with the help of a hastened Adams spectral sequence for computing supercohomology groups which we will detail in a planned sequel. Finally, we prove that all supercohomology anomalies can be realized by fermionic topological orders, whereas beyond-supercohomology anomalies cannot, resolving a question of Córdova--Ohmori for fermionic (3+1)d systems with finite symmetries. |
| title | How to Build Anomalous (3+1)d Topological Quantum Field Theories |
| topic | Mathematical Physics Strongly Correlated Electrons High Energy Physics - Theory Algebraic Topology |
| url | https://arxiv.org/abs/2510.24834 |