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

Abstract

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.

Initially, this article has been written as full paper version of the letter version of arXiv:2405.05178, but this article contains many fundamental generalizations with concrete examples. Typos are corrected and references are added (v2). Typos are corrected (v3)

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Cited by in corpus (3)

• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
• Boundary critical phenomena in the quantum Ashkin-Teller model