Classification of (2+1)D invertible fermionic topological phases with symmetry

Understand

We provide a classification of invertible topological phases of interacting fermions with symmetry in two spatial dimensions for general fermionic symmetry groups $G_f$ and general values of the chiral central charge $c_-$.

• Here $G_f$ is a central extension of a bosonic symmetry group $G_b$ by fermion parity, $(-1)^F$, specified by a second cohomology class $[\omega_2] \in \mathcal{H}^2(G_b, \mathbb{Z}_2)$.
• Our approach proceeds by gauging fermion parity and classifying the resulting $G_b$ symmetry-enriched topological orders while keeping track of certain additional data and constraints.
• We perform this analysis through two perspectives, using $G$-crossed braided tensor categories and Spin$(2c_-)_1$ Chern-Simons theory coupled to a background $G$ gauge field.