An operator algebraic approach to symmetry defects and fractionalization

We provide a superselection theory of symmetry defects in 2+1D symmetry enriched topological (SET) order in the infinite volume setting. For a finite symmetry group $G$ with a unitary on-site action, our formalism produces a $G$-crossed braided tensor category $G\mathsf{S}\mathsf{e}\mathsf{c}$. This superselection theory is a direct generalization of the usual superselection theory of anyons, and thus is consistent with this standard analysis in the trivially graded component $G{\mathsf{S}\mathsf{e}\mathsf{c}}_1$. This framework also gives us a completely rigorous understanding of symmetry fractionalization. To demonstrate the utility of our formalism, we compute $G\mathsf{S}\mathsf{e}\mathsf{c}$ explicitly in both short-range and long-range entangled spin systems with symmetry and recover the relevant skeletal data.