An operator algebraic approach to symmetry defects and fractionalization

Abstract

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 $G\mathsf{Sec}$. 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 $G{\mathsf{Sec}}_1$. This framework also gives us a completely rigorous understanding of symmetry fractionalization. To demonstrate the utility of our formalism, we compute $G\mathsf{Sec}$ explicitly in both short-range and long-range entangled spin systems with symmetry and recover the relevant skeletal data.

arXiv commentExtensively reworked the manuscript to improve precision, clarity, and rigor. Added discussion on $W^{\ast }$ property of $G\mathsf{Sec}$ and on $G$-crossed braided structure of $\mathsf{Hilb}(G,\nu )$. 94 pages, 22 figures. Comments welcome