Fermionic symmetry fractionalization in (2+1)D

We develop a systematic theory of symmetry fractionalization for fermionic topological phases of matter in (2+1)D with a general fermionic symmetry group Gf . In general Gf is a central extension of the bosonic symmetry group Gb by fermion parity, p ́1q F , characterized by a nontrivial cohomology class rω2s P H 2pGb,Z2q. 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 supermodular 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 H3pGb,KpCqq, while the latter is valued in either H 3pGb,A{t1, ψuq or Z 2pGb,Z2q depending on additional details of the theory. KpCq is the Abelian group of functions from anyons to Up1q phases obeying the fusion rules, A is the Abelian group defined by fusion of Abelian anyons, and ψ is the fermion. When these obstructions vanish, we show that distinct symmetry fractionalization patterns form a torsor over H2pGb,A{t1, ψuq. 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 Z2 quantum spin liquid states of electrons.