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!