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

TL;DR

<3-5 sentence high-level summary> The paper develops a comprehensive framework to gauge non-invertible symmetries in (2+1)d topological orders by unifying two main strands: fusion of topological interfaces and Morita theory of fusion 2-categories. It generalizes invertible 0-form gauging to non-invertible and higher-structure symmetries via algebras of surfaces, providing constraints, consistency conditions (including a generalized fixed-point theorem), and a practical recipe to convert Abelian topological orders into non-Abelian ones. The authors demonstrate the framework through detailed analyses and explicit gaugings in D(Z2) and related models, including realizations like D(D6) and Ising-related theories, and discuss when non-invertible gaugings can be decomposed into sequences of invertible gaugings. The work advances the construction and classification of non-Abelian topological orders from Abelian ancestors and clarifies dualities between 0-form and 1-form (and higher) symmetries in (2+1)d.

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.

Figures (27)

• Figure 1: Gauging $S_g$ identifies the lines $a$ and $\rho_g(a)$.
• Figure 2: Action of $\mathcal{I}$ on line operators.
• Figure 3: Fusion of a line $x$ in $\mathcal{B}_1$ from the left on the interface produces a line operator, $\mathcal{I}_L(x)$, on it. One can similarly consider the action of a line in $\mathcal{B}_2$ from the right.
• Figure 4: Fusion of a surface, $S\in \mathsf{Mod}(\mathcal{B}_1)$, with the interface $\mathcal{I}$ from the left produces a new interface, $\mathcal{I}'$.
• Figure 5: Unlike the case of fusion of general surfaces in $\mathsf{Mod}(\mathcal{B}_1)$ with $\mathcal{I}$, the fusion of $\mathcal{A}_S$ with the interface $\mathcal{I}$ leaves it unchanged.