Parabolic Higgs bundles and cyclic monopole chains

TL;DR

This paper establishes a new correspondence between monopole chains and spectral data involving curves and parabolic line bundles, utilizing the Nahm transform, and classifies symmetric monopole chains with numerical examples.

Contribution

It introduces a spectral data framework for monopole chains analogous to rational maps and classifies symmetric solutions using the Nahm transform.

Findings

Spectral data correspondence for monopole chains

Classification of symmetric monopole chains

Numerical images of monopole chains

Abstract

We formulate a correspondence between SU(2) monopole chains and ``spectral data'', consisting of curves in $\mathbb{CP}^1\times\mathbb{CP}^1$ equipped with parabolic line bundles. This is the analogue for monopole chains of Donaldson's association of monopoles with rational maps. The construction is based on the Nahm transform, which relates monopole chains to Higgs bundles on the cylinder. As an application, we classify charge $k$ monopole chains which are invariant under actions of $\mathbb{Z}_{2k}$. We present images of these symmetric monopole chains that were constructed using a numerical Nahm transform.