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

Abstract

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. These results give a way to characterize and classify invertible fermionic topological phases in terms of a concrete set of data and consistency equations, which is more physically transparent and computationally simpler than the more abstract methods using cobordism theory and spectral sequences. Our results also generalize and provide a different approach to the recent classification of fermionic symmetry-protected topological phases by Wang and Gu, which have chiral central charge $c_- = 0$. We show how the 10-fold way classification of topological insulators and superconductors fits into our scheme, along with general non-perturbative constraints due to certain choices of $c_-$ and $G_f$. Mathematically, our results also suggest an explicit general parameterization of deformation classes of (2+1)D invertible topological quantum field theories with $G_f$ symmetry.