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On Charge 3 Symmetric Monopoles

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 $\hat{X}$.