Fermionic symmetry fractionalization in (2+1)D

Abstract: We develop a systematic theory of symmetry fractionalization for fermionic topological phases of matter in (2+1)D with a general fermionic symmetry group $G_f$. In general $G_f$ is a central extension of the bosonic symmetry group $G_b$ by fermion parity, $(-1)^F$, characterized by a non-trivial cohomology class $[\omega_2] \in \mathcal{H}^2(G_b, \mathbb{Z}_2)$. We show how the presence of local fermions places a number of constraints on the algebraic data that defines the action of the symmetry on the super-modular tensor category that characterizes the anyon content. We find two separate obstructions to defining symmetry fractionalization, which we refer to as the bosonic and fermionic symmetry localization obstructions. The former is valued in $\mathcal{H}^3(G_b, K(\mathcal{C}))$, while the latter is valued in either $\mathcal{H}^3(G_b,\mathcal{A}/{1,\psi})$ or $Z^2(G_b, \mathbb{Z}_2)$ depending on additional details of the theory. $K(\mathcal{C})$ is the Abelian group of functions from anyons to $\mathrm{U}(1)$ phases obeying the fusion rules, $\mathcal{A}$ is the Abelian group defined by fusion of Abelian anyons, and $\psi$ is the fermion. When these obstructions vanish, we show that distinct symmetry fractionalization patterns form a torsor over $\mathcal{H}^2(G_b, \mathcal{A}/{1,\psi})$. We study a number of examples in detail; in particular we provide a characterization of fermionic Kramers degeneracy arising in symmetry class DIII within this general framework, and we discuss fractional quantum Hall and $\mathbb{Z}_2$ quantum spin liquid states of electrons.