Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization

TL;DR

This paper introduces a framework for understanding both invertible and noninvertible gauge symmetries in 2+1D topological orders using string-net models, revealing the first noninvertible categorical gauge symmetry in such systems.

Contribution

It develops a systematic approach to classify and realize gauge symmetries, including noninvertible ones, in 2+1D topological phases through string-net models and duality transformations.

Findings

First noninvertible categorical gauge symmetry in 2+1D topological order

Duality between string-net models with Morita equivalent fusion categories

Construction of symmetry transformations via duality and isomorphisms

Abstract

We develop a systematic framework for understanding symmetries in topological phases in 2+1 dimensions using the string-net model, encompassing both gauge symmetries that preserve anyon species and global symmetries permuting anyon species, including both invertible symmetries describable by groups and noninvertible symmetries described by categories. As an archetypal example, we reveal the first noninvertible categorical gauge symmetry of topological orders in 2+1 dimensions: the Fibonacci gauge symmetry of the doubled Fibonacci topological order, described by the Fibonacci fusion 2-category. Our approach involves two steps: first, establishing duality between different string-net models with Morita equivalent input fusion categories that describe the same topological order; and second, constructing symmetry transformations within the same string-net model when the dual models have isomorphic input fusion categories, achieved by composing duality maps with isomorphisms of degrees of freedom between the dual models.