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Topological Defect Lines in bosonized Parafermionic CFTs

Babak Haghighat, Youran Sun

TL;DR

This work develops a unified framework to extract topological defect lines and their fusion algebras in bosonized Z_k parafermionic CFTs by introducing an extended S-matrix, $\hat{S}$, that allows a generalized Verlinde-like computation of TY category data. By applying the method to k = 2,3,4,5, it shows how duality defects and character splitting generate non-invertible TY lines, and it demonstrates that the extended S-matrix encapsulates the necessary information to fix defect fusion rules even when explicit character decompositions are not fully known. The results recover known TY subcategories for the Ising (k=2) and Potts-related (k=3) cases and reveal new TY structures for k=4,5, with k=5 providing a particularly explicit instance of the method, including a bootstrap-like determination of free parameters. Overall, the approach provides a concrete, scalable route to identify and characterize TDLs and their fusion in a broad class of RCFTs, with potential implications for non-invertible symmetries and condensed-matter applications.

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.

Topological Defect Lines in bosonized Parafermionic CFTs

TL;DR

This work develops a unified framework to extract topological defect lines and their fusion algebras in bosonized Z_k parafermionic CFTs by introducing an extended S-matrix, , that allows a generalized Verlinde-like computation of TY category data. By applying the method to k = 2,3,4,5, it shows how duality defects and character splitting generate non-invertible TY lines, and it demonstrates that the extended S-matrix encapsulates the necessary information to fix defect fusion rules even when explicit character decompositions are not fully known. The results recover known TY subcategories for the Ising (k=2) and Potts-related (k=3) cases and reveal new TY structures for k=4,5, with k=5 providing a particularly explicit instance of the method, including a bootstrap-like determination of free parameters. Overall, the approach provides a concrete, scalable route to identify and characterize TDLs and their fusion in a broad class of RCFTs, with potential implications for non-invertible symmetries and condensed-matter applications.

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[]-categories have been explicitly realized for the cases , , and 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 matrix. We apply the method to the cases to to extract corresponding TDL fusion algebras.
Paper Structure (16 sections, 75 equations, 3 figures, 5 tables)

This paper contains 16 sections, 75 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Topological Defect Lines
  3. Fusion Category
  4. Partition functions with defect lines
  5. The $\mathbb{Z}_k$ Parafermionic CFTs
  6. The relationship between $\mathbb{Z}_3$ Parafermions and the 3-state Potts model
  7. Duality Defects
  8. Finding Topological Defect Lines
  9. Basic Setup
  10. The Ising Model
  11. Three-State Potts Model
  12. $\mathbb{Z}_3$ Parafermion Theory
  13. $\mathbb{Z}_4$ Parafermion Theory
  14. $\mathbb{Z}_5$ Parafermion Theory
  15. Conclusion
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Illustration of fusion symbols' properties. The left figure is for Eq. \ref{['eq:NN=NN']}, while the right figure for Eq. \ref{['eq:Nsym']}.
  • Figure 2: Two ways to compute $Z_{a|b}$. On the left: periodic in space, between states $a$ and $b$; on the right: periodic in time, between boundary condition $a$ and $b$.
  • Figure 3: A torus with $\mathcal{L}_1$ inserted in the time direction and $\mathcal{L}_2$ inserted in the space direction