Octahedral and Dodecahedral Monopoles

TL;DR

This paper constructs and visualizes specific symmetric monopoles with charge five and seven, revealing their geometric shapes and analyzing their scattering behavior using advanced numerical methods and rational maps.

Contribution

It introduces new symmetric monopole solutions with octahedral and icosahedral symmetry, employing a novel cluster decomposition formula and numerical ADHMN implementation.

Findings

Charge five monopole has octahedral symmetry and resembles an octahedron.

Charge seven monopole has icosahedral symmetry and resembles a dodecahedron.

Scattering geodesics are analyzed using rational maps and new formulas.

Abstract

It is shown that there exists a charge five monopole with octahedral symmetry and a charge seven monopole with icosahedral symmetry. A numerical implementation of the ADHMN construction is used to calculate the energy density of these monopoles and surfaces of constant energy density are displayed. The charge five and charge seven monopoles look like an octahedron and a dodecahedron respectively. A scattering geodesic for each of these monopoles is presented and discussed using rational maps. This is done with the aid of a new formula for the cluster decomposition of monopoles when the poles of the rational map are close together.