On charge 3 cyclic monopoles

Abstract

Monopoles are solutions of an SU(2) gauge theory in R3 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. In this thesis a particular class of such monopoles is studied, namely charge 3 monopoles with a symmetry by C3, the cyclic group of order 3. This class of cyclic 3-monopoles is described by the genus 4 spectral curve ^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 ^X . Exploiting the symmetry of the system,we manage to recast the problem entirely in terms of a genus 2 hyperelliptic curve X, the (unbranched) quotient of ^X by C3 . A crucial step to this aim involves finding a basis forH1(^X; Z), with particular symmetry properties according to a theorem of Fay. This gives a simple formfor the period matrix of ^X ; moreover, results by Fay and Accola are used to reduce the Ercolani-Sinha constraints to hyperelliptic ones on X. We solve these constraints onX numerically, by iteration using the tetrahedral monopole solution as starting point in the moduli space. We use the Arithmetic-GeometricMean method to find the periods onX: this method iswell understood for a genus 2 curve with real branchpoints; in this work we propose an extension to the situation where the branchpoints appear in complex conjugate pairs, which is the case for X. We are hence able to establish the existence of a curve of solutions corresponding to cyclic 3-monopoles.