Classification of 2D Fermionic Systems with a $\mathbb Z_2$ Flavor Symmetry

TL;DR

This paper classifies superfusion categories for 2D fermionic systems with fermion-parity and a $bZ_2$ flavor symmetry, revealing 16 consistent categories distinguished by anomaly invariants.

Contribution

It introduces a classification scheme for superfusion categories with $bZ_2$ flavor symmetry, including explicit realizations and anomaly characterization.

Findings

Identifies three classes based on $W$ being m-type or q-type.

Derives 16 consistent superfusion categories from super-pentagon equations.

Provides explicit models using multiple Majorana fermions.

Abstract

We classify superfusion categories describing two-dimensional fermionic systems equipped with the universal fermion-parity symmetry, implemented by a topological defect line (TDL) $Z$, and an additional $\mathbb{Z}_2$ flavor symmetry generated by a $W$ TDL. Depending on whether $W$ is m-type or q-type, its fusion rules lead to three distinct classes, and solving the super-pentagon equations yields 16 consistent superfusion categories. These are labeled by invariants $(\nu_W,\nu_Z,\nu_{WZ})$, which determine the $\mathbb{Z}_8$ anomaly classes of the symmetries generated by $W$, $Z$, and $WZ$. We also provide explicit realizations using multiple Majorana fermions and comment on implications for fermionic CFTs and gapped phases.