Non-invertible symmetries in finite-group gauge theory

TL;DR

This paper explores the structure and properties of invertible and non-invertible symmetries in finite-group gauge theories across various dimensions, revealing novel dualities, fusion rules, and topological defects.

Contribution

It introduces new examples of non-invertible symmetries, including Fibonacci fusion rules and generalized Cheshire string defects, expanding understanding of topological gauge theories.

Findings

Discovery of non-invertible electric-magnetic duality in 3+1d $Z_2$ gauge theory

Identification of Fibonacci fusion rules in 2+1d dihedral gauge theories

Generalization of Cheshire string defects to various codimensions and gauge groups

Abstract

We investigate the invertible and non-invertible symmetries of topological finite-group gauge theories in general spacetime dimensions, where the gauge group can be abelian or non-abelian. We focus in particular on the 0-form symmetry. The gapped domain walls that generate these symmetries are specified by boundary conditions for the gauge fields on either side of the wall. We investigate the fusion rules of these symmetries and their action on other topological defects including the Wilson lines, magnetic fluxes, and gapped boundaries. We illustrate these constructions with various novel examples, including non-invertible electric-magnetic duality symmetry in 3+1d $\mathbb{Z}_2$ gauge theory, and non-invertible analogs of electric-magnetic duality symmetry in non-abelian finite-group gauge theories. In particular, we discover topological domain walls that obey Fibonacci fusion rules in 2+1d gauge theory with dihedral gauge group of order 8. We also generalize the Cheshire string defect to analogous defects of general codimensions and gauge groups and show that they form a closed fusion algebra.