Gauging Non-Invertible Symmetries in (2+1)d Topological Orders

TL;DR

This paper develops methods to gauge non-invertible symmetries in (2+1)d topological orders, generalizing invertible symmetry gauging and providing tools for constructing non-Abelian topological phases from Abelian ones.

Contribution

It introduces formal and practical techniques for gauging non-invertible symmetries, unifying approaches for different symmetry types and exploring their implications in topological quantum field theories.

Findings

Derived constraints on gaugeable symmetries and their duals.

Unified the gauging procedures for non-invertible 0-form and 1-form symmetries.

Provided a recipe for generating non-Abelian topological orders from Abelian ones.

Abstract

We present practical and formal methods for gauging non-invertible symmetries in (2+1)d topological quantum field theories. Along the way, we generalize various aspects of invertible 0-form gauging, including symmetry fractionalization, discrete torsion, and the fixed point theorem for symmetry action on lines. Our approach involves two complementary strands: the fusion of topological interfaces and Morita theory of fusion 2-categories. We use these methods to derive constraints on gaugeable symmetries and their duals while unifying the prescription for gauging non-invertible 0-form and 1-form symmetries and various higher structures. With a view toward recent advances in creating non-Abelian topological orders from Abelian ones, we give a simple recipe for non-invertible 0-form gauging that takes large classes of the latter to the former. We also describe conditions under which iterated gauging of invertible 0-form symmetries is equivalent to a single-step gauging of a non-invertible symmetry. We conclude with a set of concrete examples illustrating these various phenomena involving gauging symmetries of the infrared limit of the toric code.