用于(2+1)维拓扑序的边界和畴壁缺陷的广义余模管代数
Generalized comodule tube algebras for boundary and domain wall defects of (2+1)D topological order
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中文总结 AI 辅助
本研究将管代数框架推广到(2+1)维拓扑序的余维-2缺陷，提出缺陷管代数概念，表征边界与畴壁缺陷，拓展了拓扑缺陷的代数结构理论。
中文摘要 AI 辅助
管代数具有$C^*$弱Hopf代数结构，是表征拓扑相中拓扑激发的基础工具。本工作将管代数框架推广到由Turaev--Viro--Barrett--Westbury TQFT描述的(2+1)维有隙相中的余维-2缺陷，特别关注Levin--Wen弦网模型。我们证明这类余维-2缺陷由“缺陷管代数”描述，其自然具有与拓扑激发相关的弱Hopf管代数上的余模代数结构。边界和畴壁缺陷随后由相应的边界和畴壁缺陷管代数的表示来表征。特别地，畴壁缺陷管代数可视为与两个相邻边界缺陷相关的管代数的广义Drinfeld双重。更一般地，对于连接N个体相的畴壁，相关的畴壁缺陷管代数可视为N元组代数。这一视角自然推广到一般k-缺陷，即连接k个余维-1缺陷的余维-2缺陷，所得广义管代数具有多余模代数结构。
英文摘要
The tube algebra, which carries the structure of a $C^*$ weak Hopf algebra, is a fundamental tool for characterizing topological excitations in topological phases. In this work, we generalize the tube algebra framework to codimension-2 defects in $(2+1)$D gapped phases described by Turaev--Viro--Barrett--Westbury TQFTs, with particular emphasis on Levin--Wen string-net models. We show that such codimension-2 defects are described by a \emph{defect tube algebra}, which naturally carries the structure of a comodule algebra over the weak Hopf tube algebra associated with the topological excitations. Boundary and domain wall defects are then characterized by the representations of the corresponding boundary and domain wall defect tube algebras. In particular, a domain wall defect tube algebra can be regarded as a generalized Drinfeld double of the tube algebras associated with the two adjacent boundary defects. More generally, for a domain wall joining $N$ bulk phases, the associated domain wall defect tube algebra can be viewed as an $N$-tuple algebra. This perspective extends naturally to general $k$-defects, namely codimension-2 defects joining $k$ codimension-1 defects, for which the resulting generalized tube algebra carries the structure of a multicomodule algebra.
