Non-invertible Symmetries and Higher Representation Theory I

TL;DR

This paper explores the structure of non-invertible symmetries in three-dimensional gauge theories through higher representation theory, revealing new topological defects and their fusion properties.

Contribution

It provides a detailed description of symmetry categories arising from gauging finite higher groups, focusing on 2-representations and their fusion in three dimensions.

Findings

Topological surface defects labeled by 2-representations

Fusion 2-category of 2-representations describes symmetry

Determination of symmetry categories for gauge theories with disconnected groups

Abstract

The purpose of this paper is to investigate the global categorical symmetries that arise when gauging finite higher groups in three or more dimensions. The motivation is to provide a common perspective on constructions of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. This paper focusses on gauging finite groups and split 2-groups in three dimensions. In addition to topological Wilson lines, we show that this generates a rich spectrum of topological surface defects labelled by 2-representations and explain their connection to condensation defects for Wilson lines. We derive various properties of the topological defects and show that the associated symmetry category is the fusion 2-category of 2-representations. This allows us to determine the full symmetry categories of certain gauge theories with disconnected gauge groups. A subsequent paper will examine gauging more general higher groups in higher dimensions.