Is there a mathematical explanation for the Aharonov-Casher effect? 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?
 A: This is the same mathematical effect (from reading their paper on it):  
The action-functional is effectively (ignoring kinetic term) $\bar{v}\cdot \bar{E}\times\bar{\mu}$. But this is equivalent to the precession of the magnetic moment in a magnetic field, $\bar{E}\times\bar{\mu}=e\bar{A}$ (a paper of Kan and Koh, 1992, actually explains this in great detail). From here the Lagrangian for the AC-effect is effectively the Lagrangian for the AB-effect, and is due to the vector potential... this is what the dual aspect is.
Clarification: In fact, their paper came across this effect by simply manipulating the view of the AB-effect in the case of a solenoid (the standard example). A solenoid can be represented as a bunch of magnetic moments lined up, and this is what they do to get the AC-effect. They explicitly attribute this to the vector potential, quote, "Is it possible to generate a situation in which a neutral particle exhibits the A-B effect? We will show that
this is indeed possible and is actually a necessary consequence of the physics described by Eq. (1)." Equation 1 here is the standard Lagrangian for particle motion, and involves the vector potential (that is how you get a potential term in the Lagrangian).
Aside: This is related to how us physics students learn [in Electrodynamics] that the only physical quantities are the E-field and B-field, and the vector potential $A$ and scalar potential $\phi$ are simply mathematical constructs to help computations... this is indeed true, as the electromagnetic field is described by virtual photons. Yet the AB-effect (and hence AC-effect) shows how through the vector potential we realize a topological condition on our fields!
