Parabolic Higgs bundles and cyclic monopole chains

We formulate a correspondence between SU(2) monopole chains and “spectral data”, consisting of curves in CP¹ x CP¹ 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 Z₂k. We present images of these symmetric monopole chains that were constructed using a numerical Nahm transform.