The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (φ, A), despite being confined to a region in which both the magnetic field B and electric field E are zero. The underlying mechanism is the coupling of the electromagnetic potential with the complex phase of a charged particle's wave function, and the Aharonov–Bohm effect is accordingly illustrated by interference experiments.So the effect depends on the potential, and not just the fields.
The most commonly described case, sometimes called the Aharonov–Bohm solenoid effect, takes place when the wave function of a charged particle passing around a long solenoid experiences a phase shift as a result of the enclosed magnetic field, despite the magnetic field being negligible in the region through which the particle passes and the particle's wavefunction being negligible inside the solenoid. This phase shift has been observed experimentally.
The potential and fields are all locally defined, so what is the problem?
The problem is that only the fields are directly observable, and there is considerable discretion in defining the potential. Sometimes the potential is defined to satisfy a distant condition. This is allowed, because gauge symmetry means it has the same physical effect.
From the viewpoint of differential geometry, the potential is a connection on a complex line bundle, and is a purely local object. It is more fundamental than the fields.
The paradox is that an electron can interfere with itself after going around a non-null-homotopic loop with a flat complex line bundle. Arguably there is something nonlocal about that. I don't think so. It is not like action-at-a-distance at all.