Types of quantum field thories
The Aharonov-Bohm effect is a configuration of the electromagnetic field which has vanishing electric/magnetic field strength (vanishing Faraday tensor ) and but is nevertheless non-trivial, in that the vector potential is non-trivial. Since the vector potential affects the quantum mechanical phase on the wavefunction of electrons moving in an electromagnetic field, in such a configuration classical physics sees no effect, but the phase of quantum particles, which may be observed as a interference pattern on some screen, does.
More technically, a configuration of the electromagnetic field is generally given by a circle-principal connection and an Aharonov-Bohm configuration is one coming from a flat connection, whose curvature/field strength hence vanishes, but which is itself globally non-trivial. This is only possible on spaces (spacetimes) which have a non-trivial fundamental group, hence for instance it does never happen on Minkowski spacetime.
Let be the plane with the origin removed, and consider the space (thought of as 3d Cartesian space with the z-axis removed) and spacetime (thought of as the previous configuration statically moving in time).
For the following argument only the topological structure of the space matters, and nothing needs to explicitly depend on the -coordinate and the time-coordinate, so for notational simplicity we may suppress these and consider just .
On this space minus the x-axis consider the polar coordinates with
Accordingly we have the differential 1-forms
Here the expression on the right extends smoothly also to the -axis and this extension we call
From the way this is constructed it is clear that is a closed differential form
the field strength vanishes ;
but there is no gauge transformation relating to the trivial field configuration.