The article's description of the Aharonov–Bohm effect seems kinda misleading to me. It's not that particles are being affected by a field that's not there, it's that the particles are affected by the electromagnetic potential, which can be non-zero even though the field is zero (the two are related through some simple equations).
Some argue that it's neither a non-local interaction (the test particle being affected despite no field in its region) nor that the interaction is caused by the four-potential, which would then be physical and more fundamental than the field tensor. On the contrary, it may just be an artefact of the semi-classical treatment that is normally done: classical theory for the fields, quantum one for the test particle. See this, for example: https://arxiv.org/abs/1110.6169
So to be simple you're saying the field is the slope (or actually gradient) of the potential. So if it's constant (but high) there is no slope, but significant potential, like being on a mesa. And that difference in potential rather than slope affects the paired quantum particles?
Kinda. The magnetic field is the curl of the vector potential. If the vector potential is nonzero but curl-less then you get zero magnetic field, but the particles’ wavefunctions still feel the vector potential.
(Specifically, the phase of the wavefunction is affected; obviously nothing observable about a single wavefunction could be affected in the absence of a field, via the correspondence principle. But when you have two electrons, that phase difference does show up in their interference pattern.)
I think I understand the Aharonov–Bohm effect for magnetism, but I don't understand how you can have the branches at different gravitational potentials without the packets experiencing a field somewhere when they branch/join.
Right, but pairs of particles aren't relevant. The effect occurs for any particle that interacts electromagnetically. And the magnetic field is the curl of its potential.
