Projective symmetries of three-dimensional TQFTs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918079597903872 |
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| author | Van Dyke, Jackson |
| author_facet | Van Dyke, Jackson |
| contents | Quantum field theory has various projective characteristics which are captured by what are called anomalies. This paper explores this idea in the context of fully-extended three-dimensional topological quantum field theories (TQFTs).
Given a three-dimensional TQFT (valued in the Morita 3-category of fusion categories), the anomaly identified herein is an obstruction to gauging a naturally occurring orthogonal group of symmetries. In other words, the classical symmetry group almost acts: There is a lack of coherence at the top level. This lack of coherence is captured by a "higher (central) extension" of the orthogonal group, obtained via a modification of the obstruction theory of Etingof-Nikshych-Ostrik-Meir [ENO10]. This extension tautologically acts on the given TQFT/fusion category, and this precisely classifies a projective (equivalently anomalous) TQFT. We explain the sense in which this is an analogue of the classical spin representation. This is an instance of a phenomenon emphasized by Freed [Fre23]: Quantum theory is projective.
We also establish a general relationship between the language of projectivity/anomalies and the language of topological symmetries. We also identify a universal anomaly associated with any theory which is appropriately "simple". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01637 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Projective symmetries of three-dimensional TQFTs Van Dyke, Jackson Quantum Algebra Other Condensed Matter Mathematical Physics Algebraic Topology Quantum field theory has various projective characteristics which are captured by what are called anomalies. This paper explores this idea in the context of fully-extended three-dimensional topological quantum field theories (TQFTs). Given a three-dimensional TQFT (valued in the Morita 3-category of fusion categories), the anomaly identified herein is an obstruction to gauging a naturally occurring orthogonal group of symmetries. In other words, the classical symmetry group almost acts: There is a lack of coherence at the top level. This lack of coherence is captured by a "higher (central) extension" of the orthogonal group, obtained via a modification of the obstruction theory of Etingof-Nikshych-Ostrik-Meir [ENO10]. This extension tautologically acts on the given TQFT/fusion category, and this precisely classifies a projective (equivalently anomalous) TQFT. We explain the sense in which this is an analogue of the classical spin representation. This is an instance of a phenomenon emphasized by Freed [Fre23]: Quantum theory is projective. We also establish a general relationship between the language of projectivity/anomalies and the language of topological symmetries. We also identify a universal anomaly associated with any theory which is appropriately "simple". |
| title | Projective symmetries of three-dimensional TQFTs |
| topic | Quantum Algebra Other Condensed Matter Mathematical Physics Algebraic Topology |
| url | https://arxiv.org/abs/2311.01637 |