有限群的酉2-表示与拓扑量子场论
On unitary 2-representations of finite groups and topological quantum field theory
摘要：
本论文包含了关于有限群的酉2-表示及其2-特征标，以及融合范畴的枢轴结构的各种结果。其动机是扩展拓扑量子场论（TQFT），其中有限群的酉2-表示的2-范畴被认为是无扭有限群模型中的“分配给点的2-范畴”。第一个结果是有限群的酉2-表示的2-范畴上的单位变换的辫状幺半范畴计算为具有融合张量积的群上的共轭等变向量丛范畴。这一结果与Baez和Dolan的扩展TQFT假设是一致的，因为它确定了分配给圆的范畴可以作为分配给点的2-范畴的“同一性的更高迹”来获得。第二个结果是关于2-表示的2-特征标，这是Ganter和Kapranov独立引入的一个概念。证明了酉2-表示的2-特征标可以关于2-表示的态射成为函子，并且2-特征标实际上是酉2-表示的复化Grothendieck范畴到群上酉等变向量丛范畴的酉完全忠实函子.最后的结果是关于融合范畴上的枢纽结构，并对Etingof，Nikshych和Ostrik提出的一个猜想进行了展望。证明了融合范畴上的枢转结构不存在，除非hom-set上的某些对合是正或负单位映射，在这种情况下，枢转结构与范畴上的单位函子的扭么半群自然变换是同一回事.此外，当且仅当这些标记可以被移除时，枢转结构可以被制成球形。
英文摘要：
This thesis contains various results on unitary 2-representations of finite groups and their 2-characters, as well as on pivotal structures for fusion categories. The motivation is extended topological quantum field theory (TQFT), where the 2-category of unitary 2-representations of a finite group is thought of as the `2-category assigned to the point' in the untwisted finite group model. The first result is that the braided monoidal category of transformations of the identity on the 2-category of unitary 2-representations of a finite group computes as the category of conjugation equivariant vector bundles over the group equipped with the fusion tensor product. This result is consistent with the extended TQFT hypotheses of Baez and Dolan, since it establishes that the category assigned to the circle can be obtained as the `higher trace of the identity' of the 2-category assigned to the point. The second result is about 2-characters of 2-representations, a concept which has been introduced independently by Ganter and Kapranov. It is shown that the 2-character of a unitary 2-representation can be made functorial with respect to morphisms of 2-representations, and that in fact the 2-character is a unitarily fully faithful functor from the complexified Grothendieck category of unitary 2-representations to the category of unitary equivariant vector bundles over the group. The final result is about pivotal structures on fusion categories, with a view towards a conjecture made by Etingof, Nikshych and Ostrik. It is shown that a pivotal structure on a fusion category cannot exist unless certain involutions on the hom-sets are plus or minus the identity map, in which case a pivotal structure is the same thing as a twisted monoidal natural transformation of the identity functor on the category. Moreover the pivotal structure can be made spherical if and only if these signs can be removed.
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