Generalized Cardy conditions of topological defect lines

TL;DR

This paper introduces a systematic method to determine topological defect lines in minimal models using modular invariants, establishing a correspondence with primary fields and confirming results with topological field theory.

Contribution

It provides a new procedure to classify TDLs in minimal models based solely on the modular invariant partition function, including explicit examples like the 3-state Potts model.

Findings

Bijective correspondence between TDLs and primary fields in diagonal models

Explicit classification of TDLs in the 3-state Potts model

Results consistent with 3D topological field theory methods

Abstract

We propose a systematic procedure to work out systems of topological defect lines (TDLs) in minimal models. The only input of this method is the modular invariant partition function. For diagonal and permutation diagonal models, we prove there is a bijection between simple TDLs and primary fields preserving fusion rules. For block-diagonal models, we work out simple TDLs in the $3$-state Potts model as an example. The results agree with those in $3D$ topological field theory methods.