SU(N) Monopoles and Platonic Symmetry

TL;DR

This paper explores symmetric SU(N) monopoles, constructing families with Platonic symmetries, and analyzes their dynamics, revealing novel deformation behaviors and scattering processes in the moduli space.

Contribution

It introduces simplified methods for studying symmetric SU(N) monopoles and constructs explicit families with tetrahedral symmetry, analyzing their complex dynamics.

Findings

Constructed tetrahedrally symmetric SU(4) and SU(5) monopole families.

Discovered a monopole deformation where a tetrahedron transforms into its dual.

Numerically analyzed monopole scattering with octahedral symmetry.

Abstract

We discuss the ADHMN construction for SU(N) monopoles and show that a particular simplification arises in studying charge N-1 monopoles with minimal symmetry breaking. Using this we construct families of tetrahedrally symmetric SU(4) and SU(5) monopoles. In the moduli space approximation, the SU(4) one-parameter family describes a novel dynamics where the monopoles never separate, but rather, a tetrahedron deforms to its dual. We find a two-parameter family of SU(5) tetrahedral monopoles and compute some geodesics in this submanifold numerically. The dynamics is rich, with the monopoles scattering either once or twice through octahedrally symmetric configurations.