阿贝尔陈-西蒙斯理论中缺陷的函子理论
A Functorial Theory of Defects in Abelian Chern-Simons Theory
- Yau Mathematical Sciences Center and Department of Mathematics, Tsinghua University（清华大学丘成桐数学科学中心及数学系）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究通过范畴TQFT和Alterfold构造统一描述阿贝尔陈-西蒙斯理论中的缺陷，识别有限二次模为关键不变量，并澄清对偶性，为拓扑序缺陷提供模型。
中文摘要 AI 辅助
近期工作已将阿贝尔陈-西蒙斯理论构造为范畴拓扑量子场论（TQFT），从而能够自然地纳入范畴缺陷，并构建阿贝尔陈-西蒙斯TQFT的缺陷扩展。我们首先在中心情形和双重尖点模情形下识别了阿贝尔陈-西蒙斯理论的Turaev-Viro实现，阐明了单一体实现与典范双重实现之间的区别。另一方面，Alterfold构造提供了相关的拓扑边界、畴壁和凝聚扇区，建立了明确的Alterfold/陈-西蒙斯词典。我们证明了有限二次模是控制体理论、其拓扑对称性、定向反转不变性及缺陷的不变量。我们进一步证明，多分量阿贝尔BF理论作为非对角阿贝尔陈-西蒙斯理论的扩展TQFT出现，自然地置于同一扩展框架内。最后，我们证明了近期提出的阿贝尔陈-西蒙斯对偶性并不定义真正的TQFT对偶。这些结果为阿贝尔拓扑序中的缺陷提供了具体模型，并为非阿贝尔情形指明了路径。
英文摘要
Recent work has constructed Abelian Chern-Simons theories as categorical TQFTs, allowing us to naturally incorporate categorical defects and construct defect extensions of Abelian Chern-Simons TQFTs. We first identify the Turaev-Viro realizations of Abelian Chern-Simons theory in the center and doubled pointed modular cases, clarifying the distinction between single bulk realizations and canonical doubled ones. Alternatively, the Alterfold construction supplies the associated topological boundaries, domain walls, and condensation sectors, establishing an explicit Alterfold/Chern-Simons dictionary. We show that the finite quadratic module is the invariant controlling the bulk theory, its topological symmetries, orientation-reversal invariance, and defects. We further show that multicomponent Abelian BF theory arises as the extended TQFT of an off-diagonal Abelian Chern-Simons theory, placing it naturally within the same extended framework. Finally, we demonstrate that recently proposed Abelian Chern-Simons dualities do not define a genuine TQFT duality. These results provide a concrete model for defects in Abelian topological orders and suggest a route toward the non-Abelian case.
