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Phys."],"published-print":{"date-parts":[[2020,5]]},"abstract":"<jats:p> 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. <\/jats:p><jats:p> 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. <\/jats:p>","DOI":"10.1142\/s0129055x20500117","type":"journal-article","created":{"date-parts":[[2019,10,10]],"date-time":"2019-10-10T03:57:37Z","timestamp":1570679857000},"page":"2050011","source":"Crossref","is-referenced-by-count":17,"title":["Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry"],"prefix":"10.1142","volume":"32","author":[{"given":"Alex","family":"Bullivant","sequence":"first","affiliation":[{"name":"School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK"},{"name":"Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK"}]},{"given":"Marcos","family":"Cal\u00e7ada","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK"},{"name":"Departamento de Matem\u00e1tica e Estat\u00edstica, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR, Brazil"}]},{"given":"Zolt\u00e1n","family":"K\u00e1d\u00e1r","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK"}]},{"given":"Jo\u00e3o Faria","family":"Martins","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK"}]},{"given":"Paul","family":"Martin","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK"}]}],"member":"219","published-online":{"date-parts":[[2019,11,4]]},"reference":[{"key":"S0129055X20500117BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/BF00398315"},{"key":"S0129055X20500117BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-46552-9_2"},{"key":"S0129055X20500117BIB003","doi-asserted-by":"publisher","DOI":"10.1007\/s10714-010-1070-9"},{"key":"S0129055X20500117BIB004","first-page":"423","volume":"12","author":"Baez J. 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