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

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 ω₂ in the group H²(G_b, Z₂). 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.  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.