Topological Defect Lines in bosonized Parafermionic CFTs

TL;DR

This paper develops a unified framework to identify topological defect lines in bosonized Parafermionic conformal field theories, extending previous specific case studies to a broader range of models.

Contribution

It introduces an extended S matrix approach that combines multiple methods to systematically find TDLs in bosonized Parafermionic CFTs for k=2 to 5.

Findings

Unified method for TDLs in bosonized Parafermionic CFTs

Explicit TDL fusion algebras for k=2 to 5

Extension of previous case-specific results

Abstract

Topological defect lines (TDLs) are extended line operators which act on the Hilbert space of two-dimensional CFTs and satisfy non-trivial fusion algebras when forming junctions. Among the most interesting fusion algebras are the so-called Tambara-Yamagami (TY) fusion categories which are realized in (bosonized) Parafermionic CFTs. The corresponding TY[$\mathbb{Z}_k$]-categories have been explicitly realized for the cases $k=2$, $k=3$, and $k=4$ together with the action of the defect lines on the Hilbert space of the corresponding CFTs. For each of the cases, different methods have been used in the previous literature. In the current paper, we present a unified framework for finding the TDLs in bosonized Parafermionic CFTs. Our approach relies on combining several previously used methods and the definition of an extended $S$ matrix. We apply the method to the cases $k=2$ to $k=5$ to extract corresponding TDL fusion algebras.