Symmetric Monopoles

TL;DR

This paper explores the spectral curves and rational maps of $SU(2)$ monopoles, analyzing their symmetries, moduli space structure, and geodesic scattering, with conjectures on monopoles with polyhedral symmetries.

Contribution

It introduces the concept of strongly centred monopoles, studies their symmetry-invariant submanifolds, and proposes new conjectures on monopoles with regular solid symmetries.

Findings

Strongly centred monopoles form a geodesic submanifold.

Invariant monopoles under cyclic rotations form surfaces of revolution.

Proposed spectral curves for monopoles with polyhedral symmetries.

Abstract

We discuss the spectral curves and rational maps associated with $SU(2)$ Bogomolny monopoles of arbitrary charge $k$. We describe the effect on the rational maps of inverting monopoles in the plane with respect to which the rational maps are defined, and discuss the monopoles invariant under such inversion. We define the strongly centred monopoles, and show they form a geodesic submanifold of the $k$-monopole moduli space. The space of strongly centred $k$-monopoles invariant under the cyclic group of rotations about a fixed axis, $C_k$, is shown to consist of several surfaces of revolution, generalizing the two surfaces obtained by Atiyah and Hitchin in the 2-monopole case. Geodesics on these surfaces give a novel type of $k$-monopole scattering. We present a number of curves in $TP_1$ which we conjecture are the spectral curves of monopoles with the symmetries of a regular solid. These conjectures are based on analogies with Skyrmions.