I am struggling with understanding the magnetic (solenoidal) version of the Aharonov-Bohm effect. The magnetic vector potential, A, is defined as any vector field whose curl is the magnetic force field, B. This definition makes A not unique. The presence of the solenoid makes the region of A not simply-connected. Somehow, that prevents us from choosing (fixing the gauge) one unique A for the entire region and this causes a phase difference between electrons that travel one way around the solenoid versus those that travel the other way. I understand, in the electrostatic version of the AB effect why we cannot fix the gauge for the electrostatic potential (only simply-connected regions can be conservative), but why, in the above magnetic version can we not fix the gauge and why does that cause the phase difference/interference pattern? Thanks in advance.