标题： 规范或扩展体和边界共形场理论：在拓扑物质中的体和域壁问题及其由（假）模不变量描述的应用 显示英文标题

标题： Gauging or extending bulk and boundary conformal field theories: Application to bulk and domain wall problem in topological matter and their descriptions by (mock) modular covariant

摘要： 我们研究在（模糊）边界共形场理论（BCFT）和体共形场理论中的规范操作（或群扩展），以及它们在拓扑有序系统中各种现象的应用。 我们将所得理论应用于共形场理论（CFT）的重整化群（RG）流与可测试的一般类别的配分函数中拓扑量子场理论分类之间的对应关系。 可以得到对应于 $1+1$ 维系统的质量RG流或模糊BCFT的 $2+1$ 维拓扑有序相的体拓扑性质。 我们提出了一种类似于非可逆对称性的Lieb-Shultz-Mattis定理的模糊BCFT的质量凝聚障碍。 与 $2+1$ 维的体拓扑简并和 $1+1$ 维的量子相相关，我们构建了一组新的BCFT。 我们还研究了 $1+1$ 维CFT的质量无间隙RG流到 $2+1$ 维拓扑秩序的含义，这与L. Kong和H. Zheng在[Nucl. Phys. B 966 (2021), 115384], arXiv:1912.01760中的早期提议密切相关，与Schellekens和Gato-Rivera的整数自旋简单电流有关。 我们研究了由两种质量无间隙流连接的两个CFT的乘积的性质。 分析中出现的（假）模不变量似乎包含新的内容。 通过对耦合模型应用折叠技巧，我们提供了一种解决有隙和带电域墙的一般方法。 可以获得任何子通过域墙传输的一般现象学。 我们的工作为基于共形场理论或模不变量分类的已建立数据的拓扑相未来的理论和数值研究提供了统一的方向。 显示英文摘要

摘要： We study gauging operations (or group extensions) in (smeared) boundary conformal field theories (BCFTs) and bulk conformal field theories, and their applications to various phenomena in topologically ordered systems. We apply the resultant theories to the correspondence between the renormalization group (RG) flow of CFTs and the classification of topological quantum field theories in the testable information of general classes of partition functions. One can obtain the bulk topological properties of $2+1$ dimensional topological ordered phase corresponding to the massive RG flow of $1+1$ dimensional systems, or smeared BCFT. We present an obstruction of mass condensation for smeared BCFT analogous to the Lieb-Shultz-Mattis theorem for noninvertible symmetry. Related to the bulk topological degeneracies in $2+1$ dimensions and quantum phases in $1+1$ dimensions, we construct a new series of BCFT. We also investigate the implications of the massless RG flow of $1+1$ dimensional CFT to $2+1$ dimensional topological order, which corresponds to the earlier proposal by L. Kong and H. Zheng in [Nucl. Phys. B 966 (2021), 115384], arXiv:1912.01760, closely related to the integer-spin simple current by Schellekens and Gato-Rivera. We study the properties of the product of two CFTs connected by the two kinds of massless flows. The (mock) modular covariants appearing in the analysis seem to contain new ones. By applying the folding trick to the coupled model, we provide a general method to solve the gapped and charged domain wall. One can obtain the general phenomenology of the transportation of anyons through the domain wall. Our work gives a unified direction for the future theoretical and numerical studies of the topological phase based on the established data of classifications of conformal field theories or modular invariants.