On Charge 3 Symmetric Monopoles

TL;DR

This paper investigates the existence of symmetric charge 3 monopoles in SU(2) gauge theory by analyzing their algebraic-geometric spectral curves and solving the associated Ercolani-Sinha constraints.

Contribution

It establishes the existence of cyclic charge 3 monopoles by solving the Ercolani-Sinha constraints for their genus 4 spectral curves.

Findings

Existence of cyclic charge 3 monopoles confirmed.

Spectral curves satisfy Ercolani-Sinha constraints.

Method provides a framework for analyzing symmetric monopoles.

Abstract

Monopoles are solutions of an SU(2) gauge theory in $\mathbb{R}^{3}$ satisfying a lower bound for energy and certain asymptotic conditions, which translate as topological properties encoded in their charge. Using methods from integrable systems, monopoles can be described in algebraic-geometric terms via their {spectral curve}, i.e. an algebraic curve, given as a polynomial P in two complex variables, satisfying certain constraints. In this thesis we focus on the Ercolani-Sinha formulation, where the coefficients of P have to satisfy the Ercolani-Sinha constraints, given as relations amongst periods. A particular class of such monopoles is studied, namely charge 3 monopoles with a symmetry by $C_{3}$, the cyclic group of order 3. This class of cyclic 3-monopoles is described by the genus 4 spectral curve $\hat{X}$, subject to the Ercolani-Sinha constraints: the aim of the present work is to establish the existence of such monopoles, which translates into solving the Ercolani-Sinha constraints for $\hat{X}$.