Author(s): Cui, Xingshan | Advisor(s): Wang, Zhenghan | Abstract: We give a construction of Turaev-Viro type (3+1)-TQFT out of a G-crossed braided spherical fusion category for G a finite group. The resulting invariant of 4-manifolds generalizes several known invariants in literature such as the Crane-Yetter invariant and Yetter's invariant from homotopy 2-types. Some concrete examples will be provided to show the calculations. If the category is concentrated only at the sector indexed by the trivial group element, a co-cycle in H^4(G,U(1)) can be introduced to produce another invariant, which reduces to the twisted Dijkgraaf-Witten theory in a special case. It can be shown that with a G-crossed braided spherical fusion category, one can construct a monoidal 2-category with certain extra structure, but these structures do not satisfy all the axioms of a spherical 2-category given by M. Mackaay. Although not proven, it is believed that our invariant is strictly different from other known invariants. It remains to see if the invariant has the power to detect any smooth structures.

中文翻译：

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来自$G$交叉编织类别的四维拓扑量子场理论

作者：崔兴山 | 顾问：王正涵 | 摘要：我们给出了一个 Turaev-Viro 型 (3+1)-TQFT 的构造，它来自 G 一个有限群的 G 交叉编织球面融合范畴。由此产生的 4 流形不变量概括了文献中的几个已知不变量，例如同伦 2 型的 Crane-Yetter 不变量和 Yetter 不变量。将提供一些具体的例子来显示计算。如果类别仅集中在由平凡群元素索引的扇区，则可以引入 H^4(G,U(1)) 中的共循环以产生另一个不变量，这简化为扭曲的 Dijkgraaf-Witten 理论一个特例。可以证明，使用 G 交叉编织球面融合范畴，可以构造具有一定额外结构的单面 2 范畴，但这些结构并不满足 M. Mackaay 给出的球形 2 类的所有公理。虽然没有得到证明，但我们相信我们的不变量与其他已知的不变量是严格不同的。不变量是否有能力检测任何光滑结构还有待观察。