Higher symmetry and gapped phases of gauge theories

TL;DR

The paper develops a comprehensive framework for gapped phases of gauge theories described by 2-group symmetries, unifying Higgsed and confined sectors through a finite 2-group ${\mathbb G}=(\Pi_1,\Pi_2,\alpha,\beta)$. It provides both continuum and lattice formulations of 2-gauge theory, classifies topological actions in dimensions 2–4 via $H^d(B{\mathbb G},U(1))$, and details observables and dualities that connect to Dijkgraaf–Witten theories and higher-form symmetries. A central advance is the interpretation of higher-symmetry phases as symmetry-protected topological phases, with boundaries exhibiting anomalies and possible gapped realizations only through symmetry breaking or boundary topological order. The work also constructs explicit SPT examples for 2-groups, illustrating how Postnikov data and cocycles realize novel higher-symmetry protected phases and their lattice realizations, and connects to broader frameworks like Walker–Wang models. Overall, it extends topological classification from ordinary groups to 2-groups, enabling systematic study of new gapped phases protected by higher symmetries in various spacetime dimensions.

Abstract

We study topological field theory describing gapped phases of gauge theories where the gauge symmetry is partially Higgsed and partially confined. The TQFT can be formulated both in the continuum and on the lattice and generalizes Dijkgraaf-Witten theory by replacing a finite group by a finite 2-group. The basic field in this TQFT is a 2-connection on a principal 2-bundle. We classify topological actions for such theories as well as loop and surface observables. When the topological action is trivial, the TQFT is related to a Dijkgraaf-Witten theory by electric-magnetic duality, but in general it is distinct. We propose the existence of new phases of matter protected by higher symmetry.