Given that:

1) virtual photons mediate the electric and magnetic force fields

2) the magnetic field is the curl of the magnetic vector potential

3) the electric field is the negative gradient of the electric scalar potential

How do we understand the vector and scalar potentials in terms of virtual photons?

Specifically, I mean curl-free magnetic vector and gradient-free electric scalar potential fields, where no electromagnetic waves or physical forces are present.

For instance, what's actually happening outside of the solenoid in the Aharonov-Bohm experiment that alters the phase of the passing electron? Are virtual particles involved there? This question has really stumped me because I only have an undergrad level of understanding in physics.