I'm not the person you're replying to, but yes A definitely is the electromagnetic vector potential (from classical electrodynamics). In gauge theory A tells you how to relate the phase of Ψ at nearby points in space/time.

Now that may purely a choice of convention for Ψ at different points in space/time (a choice of "gauge" in the jargon), but where it gets interesting is if your successive nearby points in space/time trace out a closed loop. If your A is such that the phase of Ψ ends up different as a result of going round the loop, you have an electromagnetic field!

Actually since it's a spacetime potential, it's a 4-vector valued. So it's related to both the (three-)vector potential A and the scalar potential V.

Related question: is it settled which one is "real"? A or B?

https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect#P...

The more correct technical term is "electromagnetic four potential", or "four vector".

A is quantum, the electromagnetic potential is classical, so they are not really the same thing.