Platonic hyperbolic monopoles

TL;DR

This paper constructs explicit hyperbolic monopole examples with various charges and symmetries, using circle-invariant instanton data, and analyzes their properties including Higgs field zeros and energy densities.

Contribution

It introduces new constraints on ADHM instanton data ensuring circle invariance and provides explicit rational formulae for hyperbolic monopoles with symmetry.

Findings

Explicit hyperbolic monopole solutions with platonic symmetry.

Higgs field zeros can exceed monopole charge in number.

Energy density and monopole scattering behaviors are characterized.

Abstract

We construct a number of explicit examples of hyperbolic monopoles, with various charges and often with some platonic symmetry. The fields are obtained from instanton data in four-dimensional Euclidean space that are invariant under a circle action, and in most cases the monopole charge is equal to the instanton charge. A key ingredient is the identification of a new set of constraints on ADHM instanton data that are sufficient to ensure the circle invariance. Unlike for Euclidean monopoles, the formulae for the squared Higgs field magnitude in the examples we construct are rational functions of the coordinates. Using these formulae, we compute and illustrate the energy density of the monopoles. We also prove, for particular monopoles, that the number of zeros of the Higgs field is greater than the monopole charge, confirming numerical results established earlier for Euclidean monopoles. We also present some one-parameter families of monopoles analogous to known scattering events for Euclidean monopoles within the geodesic approximation.