Tetrahedral and Cubic Monopoles

TL;DR

This paper numerically constructs and analyzes symmetric monopoles with tetrahedral and octahedral symmetries, revealing new scattering behaviors and energy density configurations in monopole dynamics.

Contribution

It introduces a numerical method for computing symmetric monopoles and constructs a family of charge four monopoles with tetrahedral symmetry, including octahedral as a special case.

Findings

Computed energy densities of symmetric monopoles.

Constructed a family of monopoles with tetrahedral symmetry.

Revealed novel monopole scattering behaviors.

Abstract

Using a numerical implementation of the ADHMN construction, we compute the fields and energy densities of a charge three monopole with tetrahedral symmetry and a charge four monopole with octahedral symmetry. We then construct a one parameter family of spectral curves and Nahm data which represent charge four monopoles with tetrahedral symmetry, which includes the monopole with octahedral symmetry as a special case. In the moduli space approximation, this family describes a novel kind of four monopole scattering and we use our numerical scheme to construct the energy density at various times during the motion.