## 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.

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