Dihedrally Symmetric Monopoles and Affine Toda Equations

TL;DR

This paper establishes a correspondence between certain symmetric monopoles in gauge theory and solutions to affine Toda equations of specific types, revealing a deep link between geometric symmetry and integrable systems.

Contribution

It demonstrates that $SU(2)$ BPS monopoles with dihedral symmetry correspond to Nahm data derived from affine Toda equations of particular types, depending on the monopole charge.

Findings

Monopoles with dihedral symmetry relate to affine Toda equations of $C_l^{(1)}$ and $A_{2(l-1)}^{(2)}$ types.

The correspondence holds for monopoles of charge $k=2l$ and $k=2l-1$ respectively.

Provides a classification of symmetric monopoles via integrable affine Toda systems.

Abstract

We show that any $SU(2)$ BPS monopole of charge $k$ with rotational spatial dihedral symmetry is gauge equivalent to the Nahm data obtained from affine Toda equations of $C_l^{(1)}$ type when $k=2l$ or $A_{2(l-1)}^{(2)}$ type when $k=2l-1$.