An operator algebraic approach to symmetry defects and fractionalization

TL;DR

This paper develops a rigorous operator algebraic framework for understanding symmetry defects and fractionalization in 2+1D topological orders, generalizing superselection theory to symmetry-enriched contexts.

Contribution

It introduces a $G$-crossed braided tensor category formalism for symmetry defects, extending superselection theory to symmetry-enriched topological phases.

Findings

Explicit computation of $G\mathsf{Sec}$ in spin systems

Recovery of skeletal data for symmetry fractionalization

Generalization of superselection theory to symmetry-enriched settings

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.