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.

中文翻译：

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(3 + 1)D 中更高的格子、离散的二维完整和拓扑相位，具有更高的规范对称性

高规范理论是规范理论的高阶版本，它使得沿着嵌入存在规范 2 连接的流形中的表面定义二维完整成为可能。在本文中，我们研究了关于流形晶格分解的离散高规范理论的哈密顿模型。我们表明，更高晶格规范理论的构造是明确定义的，特别包括 [公式：参见文本] 维度中物质拓扑相的哈密顿量。我们的构建基于 Kitaev 量子双模型，用有限规范 2 组 2 连接代替有限规范连接。我们的哈密顿高格规范理论模型定义在任意维度的空间流形上，由略微组合的 CW 分解（2 格分解）呈现，其 1-cells 和 2-cells 携带离散的 1 维和 2 维完整数据。我们证明了哈密顿高格规范理论的基态退化是流形的拓扑不变量，与从流形到基础规范2群的分类空间的映射的同伦类的数量一致。我们的哈密顿算子的算子模型与离散二维完整算子密切相关，用于具有 2 格分解的流形上的离散 2 连接。因此，我们解决了嵌入在 2 晶格中的表面的离散二维完整的定义。提出了关于离散二维完整学的明确定义及其在组合和代数拓扑设置中的构造的几个结果。