A cohomological description of connections and curvature over posets. (English) Zbl 1114.53019

Gauge theory can be understood as a principal bundle over a manifold. For applications to physics, the manifold is space-time, but in quantum field theory, it is natural to consider the base as a partially ordered set (poset). With this in mind, the authors adopt a cohomological approach. A principal bundle can be described in terms of transition functions, and they form a \(1\)-cocycle in Čech cohomology with values in a group \(G\). The authors regard a \(1\)-cohomology of a poset with values in \(G\) as a principal bundle over space-time. Then they introduce fundamental notions such as connection and curvature as the ones associated with \(1\)-cohomology. They establish a version of the Ambrose-Singer theorem. Finally, gauge transformations are discussed.


53C05 Connections (general theory)
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