As the article 'Electrodynamics in general spacetime' greatly explains, the U(1)-gauge theory is a good base for working in non-simply connected spaces. But I wonder whether there is a deep reason to relate electrodynamics and complex line bundles and connections (anything to do with quantum theory, perhaps?). I know Yang-Mills theory has to do with all this idea of studying bundles and connections, but I hope it has physical meaning and not just 'geometrical convenience'. Can anyone explain me intuitively this mathematics-physics interplay?
More concisely, suppose our ‘electromagnetic world’ is modelled by means of a complex line bundle with base the 4-dimensional spacetime.
-In Quantum Mechanics, a complex valued function over spacetime should be understood as a ‘phased’-distribution of the probability of a particle being in a certain place. How should be understood a section over our complex line bundle?
-And in this setting, which is the role of a connection?
As it may be difficult to answer this in a few words, it would be also helpful if somebody pointed out some small fragment of a book where this questions are studied. All ideas are welcomed.