On the constraints defining BPS monopoles

TL;DR

This paper refines the mathematical constraints that define spectral curves of SU(2) BPS monopoles, linking different conditions and analyzing symmetric monopole examples using twistor methods.

Contribution

It provides an improved formulation of the Ercolani-Sinha constraints and relates them to Corrigan-Goddard conditions, with applications to symmetric monopole spectral curves.

Findings

Derived an improved version of the Ercolani-Sinha constraints.

Showed Corrigan-Goddard conditions as a special case of these constraints.

Analyzed the spectral curve of the tetrahedrally symmetric 3-monopole.

Abstract

We discuss the explicit formulation of the transcendental constraints defining spectral curves of SU(2) BPS monopoles in the twistor approach of Hitchin, following Ercolani and Sinha. We obtain an improved version of the Ercolani-Sinha constraints, and show that the Corrigan-Goddard conditions for constructing monopoles of arbitrary charge can be regarded as a special case of these. As an application, we study the spectral curve of the tetrahedrally symmetric 3-monopole, an example where the Corrigan-Goddard conditions need to be modified. A particular 1-cycle on the spectral curve plays an important role in our analysis.