We generalize gauge theory on a graph so that the gauge group becomes a finite-dimensional ribbon Hopf algebra, the graph becomes a ribbon graph, and gauge-theoretic concepts such as connections, gauge transformations and observables are replaced by linearized analogs. Starting from physical considerations, we derive an axiomatic definition of Hopf algebra gauge theory, including locality conditions under which the theory for a general ribbon graph can be assembled from local data in the neighborhood of each vertex. For a vertex neighborhood with n incoming edge ends, the algebra of non-commutative ‘functions’ of connections is dual to a two-sided twist deformation of the n-fold tensor power of the gauge Hopf algebra. We show these algebras assemble to give an algebra of functions and gauge-invariant subalgebra of ‘observables’ that coincide with those obtained in the combinatorial quantization of Chern–Simons theory, thus providing an axiomatic derivation of the latter. We then discuss holonomy in a Hopf algebra gauge theory and show that for semisimple Hopf algebras this gives, for each path in the embedded graph, a map from connections into the gauge Hopf algebra, depending functorially on the path. Curvatures — holonomies around the faces canonically associated to the ribbon graph — then correspond to central elements of the algebra of observables, and define a set of commuting projectors onto the subalgebra of observables on flat connections. The algebras of observables for all connections or for flat connections are topological invariants, depending only on the topology, respectively, of the punctured or closed surface canonically obtained by gluing annuli or discs along edges of the ribbon graph.

中文翻译：

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带状图上的 Hopf 代数规范理论

我们在图上推广规范理论，使规范群变成有限维带状 Hopf 代数，图变成带状图，并且规范理论概念，如连接、规范变换和可观测量被线性化的类比取代。从物理考虑出发，我们推导出了 Hopf 代数规范理论的公理化定义，包括局部性条件，在该局部性条件下，一般带状图的理论可以从每个顶点附近的局部数据中组装起来。对于一个顶点邻域n输入边缘末端，连接的非交换“函数”的代数是对偶的n- 规范 Hopf 代数的折叠张量幂。我们展示了这些代数的组合，给出了一个函数代数和“可观测量”的规范不变子代数，这些代数与陈-西蒙斯理论的组合量化中获得的那些一致，从而提供了后者的公理推导。然后，我们讨论 Hopf 代数规范理论中的 holonomy，并表明对于半简单 Hopf 代数，对于嵌入图中的每条路径，这给出了从连接到规范 Hopf 代数的映射，具体取决于路径。曲率——典型地与带状图相关联的面周围的完整函数——然后对应于可观测量的代数的中心元素，并在平面连接上定义一组通勤投影到可观测量的子代数上。