On Charge-3 Cyclic Monopoles

TL;DR

This paper determines the spectral curve of charge 3 cyclic monopoles, using symmetry and algebraic geometry tools to relate genus 4 and genus 2 curves, providing explicit monopole data.

Contribution

It introduces a method to compute the spectral curve of charge 3 cyclic monopoles by leveraging symmetry and Richelot correspondence, connecting genus 4 and genus 2 curves.

Findings

Explicit spectral curve for charge 3 monopoles with cyclic symmetry

Use of Richelot correspondence to relate genus 4 and genus 2 curves

Comparison of different approaches to monopole spectral data

Abstract

We determine the spectral curve of charge 3 BPS su(2) monopoles with C_3 cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a (Toda) spectral curve of genus 2. A well adapted homology basis is presented enabling the theta functions and monopole data of the genus 4 curve to be given in terms of genus 2 data. The Richelot correspondence, a generalization of the arithmetic mean, is used to solve for this genus 2 curve. Results of other approaches are compared.