Doubly Periodic Monopole Dynamics and Crystal Volumes

We determine the large-modulus dynamics of the U(N) doubly periodic BPS monopole in Yang-Mills-Higgs theory, called a monopole wall. We accomplish this by exploring its Higgs curve using the Newton polytope and amoeba. In particular, we prove that the monopole wall splits into subwalls when any of its moduli become large. The long-distance gauge and Higgs field interactions of these subwalls are abelian, allowing us to derive an asymptotic metric for the monopole wall moduli space via electromagnetic scattering. We carry out a generalized Legendre transform to determine complex coordinates and Kähler potential for the asymptotic metric. We prove that the Kähler potential is determined by the cut volume of a crystal associated with the Higgs curve, i.e. the volume of a region enclosed by a plane arrangement in R3