Recall that the Aharonov-Bohm effect can be interpreted mathematically as follows.
Consider an electromagnetic field A on some smooth manifold M, i.e., A is an element in the first differential cohomology of M, or in other words, an isomorphism class of complex hermitian line bundles with a metric connection over M.
The field strength of A is defined to be its curvature. It can happen that the field strength is zero, i.e., a charged particle traveling through M has no forces acting on it, yet we can still observe a nontrivial change in phase if the particle travels around a loop with a nontrivial holonomy. This change of phase can then be observed in a physical experiment.
In other words, a line bundle with connection can be flat without being trivial, in particular, it can have nontrivial holonomy.
Recently I learned that there is a dual (in a certain physical sense) effect to the Aharonov-Bohm effect, namely, the Aharonov-Casher effect, in which a charged particle is replaced by a neutral particle with a magnetic moment.
Can we interpret the Aharonov-Casher effect mathematically in a similar way to the Aharonov-Bohm effect?