Well, let`s put the question in this form: any vector A with a zero curl ∇×A=0 can be expressed "locally" as a gradient A = ∇Ω for some function Ω. But its global version, .i.e. all along the topological space, is valid for path connected spaces only. For this reason the integral ∫A.dr is not...
Due to its form, gauge transformations for the typical electrodynamics potentials are "local" in nature. That`s: they exists for path connected topological spaces. So, there exists global gauge transformations or are all of them local in nature?. If the answer is "yes", i.e. if there are global...