For a smooth manifold $M$, possibly with boundary and corners, and a Lie group $G$, we consider a suitable description of gauge fields in terms of parallel transport, as groupoid homomorphisms from a certain path groupoid in $M$ to $G$. Using a cotriangulation $\mathscr{C}$ of $M$, and collections of finite-dimensional families of paths relative to $\mathscr{C}$, we define a homotopical equivalence relation of parallel transport maps, leading to the concept of an extended lattice gauge (ELG) field. A lattice gauge field, as used in Lattice Gauge Theory, is part of the data contained in an ELG field, but the latter contains further local topological information sufficient to reconstruct a principal $G$-bundle on $M$ up to equivalence. The space of ELG fields of a given pair $(M,\mathscr{C})$ is a covering for the space of fields in Lattice Gauge Theory, whose connected components parametrize equivalence classes of principal $G$-bundles on $M$. We give a criterion to determine when ELG fields over different cotriangulations define equivalent bundles.

中文翻译：

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规范场和格的同伦类

对于光滑流形 $M$，可能有边界和角，以及李群 $G$，我们考虑在平行传输方面对规范场的合适描述，作为从 $M$ 中的某个路径 groupoid 到 $$ 的 groupoid 同态G$。使用 $M$ 的协三角 $\mathscr{C}$ 以及相对于 $\mathscr{C}$ 的有限维路径族的集合，我们定义了平行运输图的同伦等价关系，从而得出了扩展晶格规范 (ELG) 场。在格子规范理论中使用的格规范场是包含在 ELG 场中的数据的一部分，但后者包含更多的局部拓扑信息，足以在 $M$ 上重建主 $G$-bundle 直到等价。给定对 $(M, \mathscr{C})$ 是格子规范理论中域空间的覆盖，其连通分量参数化了 $M$ 上的主 $G$-bundles 的等价类。我们给出了一个标准来确定不同协三角上的 ELG 场何时定义等效丛。