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arXiv · 2503.20905

Non-semisimple WRT at the boundary of Crane-Yetter

Benjamin Haïoun
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Abstract

We prove the slogan, promoted by Walker and Freed-Teleman twenty years ago, that "The Witten-Reshetikhin-Turaev 3-TQFT is a boundary condition for the Crane-Yetter 4-TQFT" and generalize it to the non-semisimple case following ideas of Jordan, Reutter and Walker. To achieve this, we prove that the Crane-Yetter 4-TQFT and its non-semisimple version arXiv:2306.03225 are once-extended TQFTs, using the main result of arXiv:2412.14649. We define a boundary condition, partially defined in the non-semisimple case, for this 4D theory. When the ribbon category used is modular, possibly non-semisimple, we check that the composition of this boundary condition with the values of the 4-TQFT on bounding manifolds reconstructs the Witten-Reshetikhin-Turaev 3-TQFTs and their non-semisimple versions arXiv:1912.02063, in a sense that we make precise.

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Benjamin Haïoun. 2025-03-26. Non-semisimple WRT at the boundary of Crane-Yetter. https://arxiv.org/abs/2503.20905

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