Geometry of Periodic Monopoles

Abstract

BPS monopoles on $\mathbb{R}^2\times S^1$ correspond, via the generalized Nahm transform, to certain solutions of the Hitchin equations on the cylinder $\mathbb{R}\times S^1$. The moduli space M of two monopoles with their centre-of-mass fixed is a 4-dimensional manifold with a natural hyperk\"ahler metric, and its geodesics correspond to slow-motion monopole scattering. The purpose of this paper is to study the geometry of M in terms of the Nahm/Hitchin data, i.e. in terms of structures on $\mathbb{R}\times S^1$. In particular, we identify the moduli, derive the asymptotic metric on M, and discuss several geodesic surfaces and geodesics on M. The latter include novel examples of monopole dynamics.