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?
For example, if somebody asked me "Why are particles described as complex-valued functions in Quantum Mechanics?" then I would answer "Because experiments have shown that particles should not be thought as points, but behave like having a certain probability distribution over space, which moreover has been proven to interact like waves, i.e. having constructive and destructive phenomena, and therefore it seems as there is a phase information encoded in the particle together to the probability amplitude".
Or if somebody asked "And why to use spinorial bundles when describing electrons?" the I would answer "Well, particles behave like waves, i.e. functions, but if I accept there is some spin 'vector', it becomes altogether a 'vector-valued function', whose behavior is suitably described by means of this spinorial bundle".
Now suppose our ‘electromagnetic world’ is modelled by means of a complex line bundle with base the 4-dimensional spacetime. 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.