标题： 科伦-耶特完全扩展了吗？ 显示英文标题

标题： Is Crane--Yetter fully extended?

摘要： 我们重新探讨了与模张量范畴相关的Crane-Yetter拓扑量子场论（TQFT）是否可以被完全扩展细化的问题。更具体地说，我们利用稳定同伦理论的工具，对可逆四维TQFT向以对称单态4-范畴为值的理论的扩展进行分类，其中Picard谱在度数0和4上有非平凡同伦。 我们证明了这种扩展由两部分数据分类：目标中的一个可逆对象的等价类以及一个六次单位根。 将这一结果应用于辫子融合范畴的4-范畴 $\mathbf{BrFus}$，我们发现存在无穷多个等价类的完全可逆TQFT，它们在顶级流形上重现Crane-Yetter划分函数，这些等价类由非退化辫子融合范畴的Witt群的一个 $\mathbb{Z}/6$-扩张参数化。 这一分析澄清了文献中的常见主张，并提出了如何自然选择框架TQFT上的 $SO(4)$-固定点数据的问题，该TQFT将输入的辫子融合范畴赋值到点上，以便选择Crane-Yetter状态求和。 显示英文摘要

摘要： 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.