We formulate the $Z_N$ group gauging operations (or group extensions) in (smeared) boundary conformal field theories (BCFTs) and bulk conformal field theories, and study their applications to various phenomena in topologically ordered systems. We apply the resultant theories to analyze the correspondence between the renormalization group (RG) flow of CFTs and the classification of topological quantum field theories using the testable information of general classes of partition functions. One can obtain the bulk topological properties of the $2+1$ dimensional topological ordered phase corresponding to the massive RG flow of the $1+1$ dimensional systems equivalent to the smeared BCFT. We present an obstruction to mass condensation for the smeared BCFT analogous to the Lieb-Shultz-Mattis theorem for noninvertible symmetry. Corresponding to the bulk topological degeneracies in $2+1$ dimensions and gapped quantum phases in $1+1$ dimensions, we construct a new series of BCFT. We also study the implications of the massless RG flow of the $1+1$ dimensional CFTs corresponding to the domain wall or coupling problem in $2+1$ dimensional topological orders, which corresponds to the earlier proposal by L. Kong and H. Zheng [Nucl. Phys. B 966, 115384 (2021)] closely related to the integer-spin simple current by Schellekens and Gato-Rivera. We reveal that two distinct kinds of massless RG flow exist, coming from anomaly cancellation and anomaly matching related to Witt equivalence of bulk CFTs. The anomaly cancellation flows produce a new series of modular invariants as coupled models, and the anomaly matching flows produce nontrivial domain walls changing the chirality of anyons. Our paper gives a unified direction for future theoretical and numerical studies of the topological phase based on the established data of classifications of conformal field theories or modular invariants (and the corresponding category theories).

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• Anomalies
• Anyons
• Conformal field theory
• Edge states
• Topological order
• Symmetries in condensed matter
• Abstract algebra
• Renormalization group
• String theory techniques in condensed matter