Non-Invertible Symmetries as Condensation Defects in Finite-Group Gauge Theories

TL;DR

This paper develops a framework to realize and analyze non-invertible symmetries in finite-group gauge theories as condensation defects, providing explicit constructions, fusion rules, and applications to both abelian and non-abelian cases.

Contribution

It introduces a method to represent non-invertible symmetries as condensation defects in finite-group gauge theories, extending previous work to general spacetime dimensions and non-abelian groups.

Findings

Derived the correspondence between domain walls and condensation defects in $ ext{Z}_N$ gauge theory.

Provided explicit fusion rules and symmetry actions for these defects.

Applied the framework to non-abelian groups like the Dihedral group $ ext{D}_4$.

Abstract

In recent work, we developed a method to construct invertible and non-invertible symmetries of finite-group gauge theories as topological domain walls on the lattice. In the present work, we consider abelian and non-abelian finite-group gauge theories in general spacetime dimension, and demonstrate how to realize these symmetries as condensation defects, i.e., as suitable insertions of lower dimensional topological operators. We then compute the fusion rules and action of these symmetries using their condensation expression and the algebraic properties of the lower dimensional objects that make them. We illustrate the discussion in $\mathbb{Z}_N$ gauge theory, where we derive the correspondence between domain walls, labeled by subgroups and actions for the doubled gauge group, and higher gauging condensation defects, labeled by subalgebras of the global symmetry. As a primary application, we obtain the condensation expression for the invertible symmetries of abelian gauge theories defined by outer automorphisms of the gauge group. We also show how to use these ideas to derive the action for certain non-abelian groups. For instance, one can obtain the action for the Dihedral group $\mathbb{D}_4$ by gauging a swap symmetry of $\mathbb{Z}_2\times\mathbb{Z}_2$ gauge theory.