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The Aharonov-Bohm effect is an observable property of the electromagnetic field which shows that the electromagnetic field strength differential 2-form $F$ alone does not characterize the physically distinguishable field configurations, but that only the “vector potential” of the field strength does, hence the circle bundle with connection of which which $F$ is the curvature 2-form.
Accoridngly, the Aharonov-Bohm effect is a certain interference pattern in the wavefunction of an electron that propagates in an electromagnetic background gauge field that is modeled by a circle bundle with connection which is a flat connection, hence has vanishing curvature/field strength just as the trivial electromagnetic field configuration does, but which is nevertheless a nontrivial connection. The observed interference comes from the holonomy of this non-trivial but flat connection around loops in spacetime which represent non-trivial elements of the fundamental group.