On charge-3 cyclic monopoles

Abstract

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.

Geodesic Equations and Algebro-Geometric Methods

“…Concerning the numerical calculation of the analytical expressions, the complete elliptic integrals can very efficiently be computed by using the arithmetic geometric mean. This can be generalized to genus two hyperelliptic integrals, see [16,17].…”

Spectral curves are transcendental

“…Although the examples 1, 2 (for n ≥ 3), 6, 7 are for reducible curves and so outwith the theorem, they too are transcendental. The final curve has C 3 symmetry and quotients over a genus 2 curve [6]. The transcendence of the periods here requires Theorem (3.1); a genus 2-variant of the AGM due to Richelot may be used for their computation.…”

Victor Enolski (1945–2019)

“…By using Humbert theory and results of O. Bolza we showed that only the tetrahedrally symmetric monopole in this family satisfied Hitchin's third constraint. By enlarging the class of curves to the general 3 symmetric charge-three curve, Victor, Antonella D'Avanzo (one of my students) and I were able to construct those curves associated with monopoles and provided a new family of examples [7]. Unfortunately the curve is described in terms of a vanishing period implicitly relating coefficients of the quotient curve.…”