标题： 二维张量范畴之间域墙的弱霍普夫管代数 显示英文标题

标题： Weak Hopf tube algebra for domain walls between 2d gapped phases of Turaev-Viro TQFTs

摘要： 我们通过构建域墙管代数来研究Turaev-Viro型拓扑量子场论（TQFT）的二维间隙相之间的域墙。我们首先分析与双模范畴相关的域墙管代数，然后将该构造扩展到$N$基融合范畴上的多模范畴。我们证明了所得的管代数自然地配备了$C^*$弱Hopf代数结构。我们表明局域在域墙上的拓扑激发由相应域墙管代数的表示分类，在双模范畴的函子范畴可以以保持融合的方式嵌入到域墙管代数的表示范畴中的意义上。我们进一步在此背景下建立了折叠技巧和Morita理论。然后，最关键的是，我们使用一个斜配对，从弱Hopf边界管代数严格构造出Drinfeld量子双代数，并建立了域墙管代数与边界管代数的Drinfeld量子双代数之间的同构。受域墙管代数与边界管代数的量子双代数之间对应关系的启发，我们引入了$N$-元代数的概念，并展示它如何在多模域墙设置中出现。最后，我们考虑两个域墙之间的缺陷，表明这样的缺陷可以通过域墙缺陷管代数的表示来表征。我们简要概述了如何在这一表示理论框架内系统地处理这些缺陷。 显示英文摘要

摘要： We investigate domain walls between 2d gapped phases of Turaev-Viro type topological quantum field theories (TQFTs) by constructing domain wall tube algebras. We begin by analyzing the domain wall tube algebra associated with bimodule categories, and then extend the construction to multimodule categories over $N$ base fusion categories. We prove that the resulting tube algebra is naturally equipped with a $C^*$ weak Hopf algebra structure. We show that topological excitations localized on domain walls are classified by representations of the corresponding domain wall tube algebra, in the sense that the functor category of bimodules admits a fusion-preserving embedding into the representation category of the domain wall tube algebra. We further establish the folding trick and Morita theory in this context. Then, most crucially, we provide a rigorous construction of the Drinfeld quantum double from weak Hopf boundary tube algebras using a skew-pairing, and establish an isomorphism between domain wall tube algebra and Drinfeld quantum double of boundary tube algebras. Motivated by the correspondence between the domain wall tube algebra and the quantum double of the boundary tube algebras, we introduce the notion of an $N$-tuple algebra and demonstrate how it arises in the multimodule domain wall setting. Finally, we consider defects between two domain walls, showing that such defects can be characterized by representations of a domain wall defect tube algebra. We briefly outline how these defects can be systematically treated within this representation-theoretic framework.