Classification of 2D Fermionic Systems with a ${\mathbb{Z}}_2$ Flavor Symmetry

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 $(\hat{I}{½}_W,\hat{I}{½}_Z,\hat{I}{½}_{WZ})$, 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.

17 pages+4 appendices