Excitations in strict 2-group higher gauge models of topological phases

TL;DR

This work constructs a solvable $(3+1)$-D topological model based on a finite strict $2$-group, realizing Yetter's homotopy $2$-type TQFT on a lattice and supporting loop- and point-like bulk excitations. Central to the analysis is a generalized tube algebra ${ m Tube}^ G[ ext{T}^2]$ whose simple modules are labeled by pairs $( ext{C},R)$, with $ ext{C}$ a boundary-colouring orbit and $R$ an irrep of the stabiliser; these classify elementary loop-like excitations. The authors introduce a canonical basis for the tube algebra, reveal its centre, and prove that the three-torus ground-state subspace is isomorphic to this centre, yielding a ground-state degeneracy equal to the number of elementary loop-like excitations. Limiting cases recover familiar DW and Crane–Yetter theories, offering physical interpretations of flux/charge components and confinement phenomena. The paper also discusses avenues for extending the formalism to higher-categorical inputs and exploring fusion/braiding structures via potential tube-algebra enhancements, with implications for understanding 3+1D topological order from higher gauge data.

Abstract

We consider an exactly solvable model for topological phases in (3+1)d whose input data is a strict 2-group. This model, which has a higher gauge theory interpretation, provides a lattice Hamiltonian realisation of the Yetter homotopy 2-type topological quantum field theory. The Hamiltonian yields bulk flux and charge composite excitations that are either point-like or loop-like. Applying a generalised tube algebra approach, we reveal the algebraic structure underlying these excitations and derive the irreducible modules of this algebra, which in turn classify the elementary excitations of the model. As a further application of the tube algebra approach, we demonstrate that the ground state subspace of the three-torus is described by the central subalgebra of the tube algebra for torus boundary, demonstrating the ground state degeneracy is given by the number of elementary loop-like excitations.