Topological defect lines in bosonized Parafermionic CFTs

Topological defect lines (TDLs) are extended line operators which Fact on the Hilbert space of two-dimenional 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[Z(kappa)]-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 generalizing several previously used methods by introducing the notion of an extended S matrix. We apply the method to the cases k = 2 5 to extract corresponding TDL fusion algebras.