# Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry

## TL;DR

This paper develops a Hamiltonian model for higher lattice gauge theory in 3+1 dimensions, extending Kitaev's model with 2-group connections, and proves topological invariance of ground-state degeneracy.

## Contribution

It introduces a well-defined higher lattice gauge theory Hamiltonian using 2-group connections on CW-decompositions, generalizing Kitaev's model to higher dimensions.

## Key findings

- Ground-state degeneracy is a topological invariant.
- The model is well-defined on arbitrary CW-decompositions.
- Discrete 2-dimensional holonomy is rigorously constructed.

## Abstract

Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we will continue the study of Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. In particular, we show that a previously proposed construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in 3+1 dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly {\it combinatorialised} CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group. The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretised 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.