Bogomol’nyi-Prasad-Sommerfield 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 $\mathcal{M}$ of two monopoles with their center of mass fixed is a four-dimensional manifold with a natural hyperkähler metric, and its geodesics correspond to slow-motion monopole scattering. The purpose of this paper is to study the geometry of $\mathcal{M}$ 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 $\mathcal{M}$, and discuss several geodesic surfaces and geodesics on $\mathcal{M}$. The latter include novel examples of monopole dynamics.

• Received 6 October 2013

DOI:https://doi.org/10.1103/PhysRevD.88.125013

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Published by the American Physical Society