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 hyperkähler 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.

17 pages, 4 figures

References in corpus (1)

• Periodic Monopoles from Spectral Curves

Cited by in corpus (7)

• Non-relativistic twistor theory and Newton--Cartan geometry
• Dynamics of monopole walls
• Periodic Monopoles from Spectral Curves
• Geometry of Solutions of Hitchin Equations on R^2
• Hyperkahler Metrics from Monopole Walls
• Higher charge periodic monopoles
• Scaling limits of periodic monopoles