AI 中文总结
本文研究反射正拓扑场论的拓扑缺陷，定义反射缺陷TQFT并证明其关联的二维缺陷双范畴具O(2)-dagger双范畴结构，反射正情形下可配备与3-希尔伯特空间相关的额外结构。
AI 中文摘要
量子场论中的拓扑缺陷被认为会组装成带有额外结构的高阶范畴。Davydov、Kong和Runkel已针对二维和三维定向拓扑量子场论中的缺陷，将这一结论精确化；Carqueville、Meusburger和Schaumann则分别完成了相关工作。本文研究当拓扑场论额外具备反射正性时，这些范畴上存在的额外结构。我们将反射缺陷TQFT定义为源自配边范畴的对称幺半函子，该函子将定向反转与复共轭相互关联；若其满足额外的正性条件，则为反射正的。我们的主要结果是，在二维情形下，与反射缺陷TQFT相关的缺陷双范畴$\boldsymbol{\textit{T}_\textit{Z}}$具有$O(2)$- dagger双范畴的自然结构，我们明确定义了这一结构。若该理论是反射正的，$\boldsymbol{\textit{T}_\textit{Z}}$可配备与3-希尔伯特空间定义密切相关的额外结构（二者在有限性与完备性条件下一致）。
英文摘要
Topological defects in quantum field theories are believed to assemble into higher categories with extra structure. This has been made precise for defects in $2$- and $3$-dimensional oriented topological quantum field theories by Davydov, Kong, and Runkel and by Carqueville, Meusburger, and Schaumann, respectively. In this paper we study the extra structure present on these categories when the topological field theory is additionally reflection positive. We define reflection defect TQFTs as symmetric monoidal functors out of a defect bordism category that intertwine orientation reversal with complex conjugation; they are reflection positive if they satisfy an additional positivity condition. Our main result is that in two dimensions the bicategory of defects $\mathcal{T}_\mathcal{Z}$ associated to a reflection defect TQFT carries the natural structure of an $O(2)$-dagger bicategory, a structure we define explicitly. If the theory is reflection positive, $\mathcal{T}_\mathcal{Z}$ can be equipped with additional structure closely related to the definition of a 3-Hilbert space (the two agree up to some finiteness and completeness conditions).
Comments22 pages, 10 figures, comments welcome
