Is Crane--Yetter fully extended?
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908394654269440 |
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| author | Stehouwer, Luuk |
| author_facet | Stehouwer, Luuk |
| contents | We revisit the question of whether the Crane-Yetter topological quantum field theory (TQFT) associated to a modular tensor category admits a fully extended refinement. More specifically, we use tools from stable homotopy theory to classify extensions of invertible four-dimensional TQFTs to theories valued in symmetric monoidal 4-categories whose Picard spectrum has nontrivial homotopy only in degrees 0 and 4. We show that such extensions are classified by two pieces of data: an equivalence class of an invertible object in the target and a sixth root of unity. Applying this result to the 4-category $\mathbf{BrFus}$ of braided fusion categories, we find that there are infinitely many equivalence classes of fully extended invertible TQFTs reproducing the Crane-Yetter partition function on top-dimensional manifolds, parametrized by a $\mathbb{Z}/6$-extension of the Witt group of nondegenerate braided fusion categories.
This analysis clarifies common claims in the literature and raises the question of how to naturally pick out the $SO(4)$-fixed point data on the framed TQFT which assigns the input braided fusion category to the point so that it selects the Crane-Yetter state-sum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04864 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Is Crane--Yetter fully extended? Stehouwer, Luuk Mathematical Physics Algebraic Topology Category Theory Quantum Algebra 81T45 We revisit the question of whether the Crane-Yetter topological quantum field theory (TQFT) associated to a modular tensor category admits a fully extended refinement. More specifically, we use tools from stable homotopy theory to classify extensions of invertible four-dimensional TQFTs to theories valued in symmetric monoidal 4-categories whose Picard spectrum has nontrivial homotopy only in degrees 0 and 4. We show that such extensions are classified by two pieces of data: an equivalence class of an invertible object in the target and a sixth root of unity. Applying this result to the 4-category $\mathbf{BrFus}$ of braided fusion categories, we find that there are infinitely many equivalence classes of fully extended invertible TQFTs reproducing the Crane-Yetter partition function on top-dimensional manifolds, parametrized by a $\mathbb{Z}/6$-extension of the Witt group of nondegenerate braided fusion categories. This analysis clarifies common claims in the literature and raises the question of how to naturally pick out the $SO(4)$-fixed point data on the framed TQFT which assigns the input braided fusion category to the point so that it selects the Crane-Yetter state-sum. |
| title | Is Crane--Yetter fully extended? |
| topic | Mathematical Physics Algebraic Topology Category Theory Quantum Algebra 81T45 |
| url | https://arxiv.org/abs/2506.04864 |