Finite 2-group gauge theory and its 3+1D lattice realization

TL;DR

This paper develops a 3+1D lattice gauge theory for finite 2-groups using Tannaka-Krein reconstruction, generalizing Kitaev's model, and explores the algebraic structure of topological defects.

Contribution

It introduces a novel 3+1D lattice model for finite 2-group gauge theories based on the Dijkgraaf-Witten TQFT, extending Kitaev's 2+1D quantum double framework.

Findings

String-like operators form the quantum double of the 2-group.

Topological defects are modules over the quantum double.

Model generalizes known 2+1D topological phases.

Abstract

In this work, we employ the Tannaka-Krein reconstruction to compute the quantum double $\mathcal D(\mathcal G)$ of a finite 2-group $\mathcal G$ as a Hopf monoidal category. We also construct a 3+1D lattice model from the Dijkgraaf-Witten TQFT functor for the 2-group $\mathcal G$, generalizing Kitaev's 2+1D quantum double model. Notably, the string-like local operators in this lattice model are shown to form $\mathcal D(\mathcal G)$. Specializing to $\mathcal G = \mathbb{Z}_2$, we demonstrate that the topological defects in the 3+1D toric code model are modules over $\mathcal D(\mathbb{Z}_2)$.