Higher lattice gauge theory from representations of 2-groups and 3+1D topological phases

TL;DR

The paper extends Kitaev's 2+1D lattice gauge theory to 3+1D by employing representations of 2-groups on a lattice described by a path 2-groupoid. It develops a full higher lattice gauge theory framework, including dressed lattices, path 2-groupoids, gauge configurations and holonomies, and a commuting projector Hamiltonian whose ground states are topological observables. The construction yields an exactly solvable model whose ground-state subspace corresponds to a Yetter-type 2-type TQFT, and it proves topological invariance under graph mutations, ensuring robustness to lattice details. Overall, the work provides a rigorous method to realize higher gauge theories in 3+1D and highlights potential implications for fault-tolerant quantum information processing and topological quantum computing.

Abstract

We construct a higher lattice gauge theory based on the representation of 2-groups described by a category of crossed modules on a lattice model described by path 2-groupoids. Using these lattice gauge representations, an exactly solvable Hamiltonian for topological phases in 3+1 dimensions is constructed. We show that the ground states of this model are topological observables.

Figures (4)

• Figure 1: Here we consider a left whiskering representation i.e., a left composition of a 1-morphism with a 2-morphism
• Figure 2: A configuration a standard tetrahedron $(v_0v_1v_2v_3).$
• Figure 5: Configuration graph: $v_\alpha$ is the plaquette base point, $v_i\,(i=1,2,3)$ are the vertices, $\Phi_{\gamma_j}(g_j)$ are the colored edges $\gamma_i$ and $\Phi_\alpha(u)$ is the colored plaquette $\alpha$ anti-clockwise rotation.
• Figure 10: A configurative graph

Theorems & Definitions (18)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 3.1
• Definition 3.2
• Definition 3.8
• Lemma 3.9
• proof
• Definition 3.10