2-线性化的上同调扭曲与推广的TQFT
Cohomological Twisting of 2-Linearization and Extended TQFT
摘要：
本文描述了一种称为“2-线性化”的绝对量子化结构与扩展拓扑量子场论（ETQFT）之间的关系。然后，我们描述了一个扩展的2-线性化过程，其中包括上同调扭曲。2-线性化过程将2-向量空间赋给（有限）广群，将它们之间的函子赋给广群的跨距，将它们之间的跨距赋给自然变换。将其应用于表示有限群G的拓扑规范理论的（离散）模空间的群胚，得到的ETQFT是与G相联系的无扭Dijkgraaf-Witten（DW）模型。这说明了因式分解的TQFT成“经典域论“在群胚值，和“量化函子”，这已经描述了自由，霍普金斯，卢里和Teleman。然后，我们描述了如何扩展到全DW模型，通过使用广义的对称monoidal bicategory的群胚和跨度，其中包括上循环。我们给出了2-线性化函子的一个推广，它作用于具有相关上同调数据的广群和跨距上。我们展示了DW模型作用中出现的分类空间BG上的3-上循环ω如何导出一个在此双范畴中取值的经典域论。
英文摘要：
In this paper, we describe a relation between a categorical quantization construction, called "2-linearization", and extended topological quantum field theory (ETQFT). We then describe an extension of the 2-linearization process which incorporates cohomological twisting. The 2-linearization process assigns 2-vector spaces to (finite) groupoids, functors between them to spans of groupoids, and natural transformations to spans between these. By applying this to groupoids which represent the (discrete) moduli spaces for topological gauge theory with finite group G, the ETQFT obtained is the untwisted Dijkgraaf-Witten (DW) model associated to G. This illustrates the factorization of TQFT into "classical field theory" valued in groupoids, and "quantization functors", which has been described by Freed, Hopkins, Lurie and Teleman. We then describe how to extend this to the full DW model, by using a generalization of the symmetric monoidal bicategory of groupoids and spans which incorporates cocycles. We give a generalization of the 2-linearization functor which acts on groupoids and spans which have associated cohomological data. We show how the 3-cocycle ω on the classifying space BG which appears in the action for the DW model induces a classical field theory valued in this bicategory.
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