2017/02/28 by Alex Bullivant, Marcos Calçada, Marcos Calcada +3
Mathematics · Physics and Astronomy · #BRST quantization #Gauge theory #Hamiltonian (control theory) #Hamiltonian lattice gauge theory #Holonomy #Homotopy and Cohomology in Algebraic Topology #Lattice field theory #Lattice gauge theory #Quantum many-body systems #Topological Materials and Phenomena #Topology (electrical circuits) #cond-mat.str-el #hep-th #math-ph #math.MP
paper · pdf · doi:10.1142/s0129055x20500117
published as Reviews in Mathematical Physics, Vol. 32, No. 04, 2050011 (2020) · V3: Major revision. Clarification of the action of vertex and edge operators. 52 pages, 18 figures. V4: Minor corrections. 53 pages, 18 figures
openalex created_date 2017/02/24 · arxiv created 2018/07/28 · openalex publication_date 2019/10/10 · arxiv updated 2020/06/05 · openalex updated_date 2026/08/05
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 study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in [Formula: see text] 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 combinatorialized 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 discretized 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.