Topological Defect Lines in Two Dimensional Fermionic CFTs

TL;DR

This work develops a comprehensive framework for topological defect lines in two-dimensional fermionic CFTs, introducing fermionic defect operators and a novel q-type TDL carrying a 1d Majorana fermion. It formulates the spin-graded, fermionic F-moves and the super pentagon identity and proposes a rank-2 classification through super fusion categories, solving the universal data and constructing explicit F-symbols. The authors realize the full set of simple TDLs in standard fermionic minimal models and a substantial set in exceptional models, demonstrating how these categories encode anomaly and spin constraints on defect spectra. They also derive RG-flow constraints for flows preserving q-type TDLs, linking topological data to IR phase structure and providing a route to constrain 2d fermionic RG trajectories using category-theoretic tools.

Abstract

We consider topological defect lines (TDLs) in two-dimensional fermionic conformal field theories (CFTs). Besides inheriting all the properties of TDLs in bosonic CFTs, TDLs in fermionic CFTs could host fermionic defect operators at their endpoints and junctions. Furthermore, there is a new type of TDLs, called q-type TDLs, that have no analog in bosonic CFTs. Their distinguishing feature is an extra one-dimensional Majorana fermion living on the worldline of the TDLs. The properties of TDLs in fermionic CFTs are captured in the mathematical language of the super fusion category. We propose a classification of the rank-2 super fusion categories generalizing the $\mathbb Z_8$ classification for the anomalies of $\mathbb Z_2$ symmetry. We explicitly solve the F-moves for all the nontrivial categories, and derive the corresponding spin selection rules that constrain the spectrum of the defect operators. We find the full set of TDLs in the standard fermionic minimal models and a partial set of TDLs in the two exceptional models, which give CFT realizations to the rank-2 super fusion categories. Finally, we discuss a constraint on the renormalization group flow that preserves a q-type TDL.