On the Existence of Non-Abelian Monopoles: the Algebro-Geometric Approach

Abstract

We develop the Atiyah-Drinfeld-Manin-Hitchin-Nahm construction to study SU (2) non-abelian charge 3 monopoles within the algebro-geometric method. The method starts with finding an algebraic curve, the monopole spectral curve, subject to Hitchin's constraints. We take as the monopole curve the genus four curve that admits a C 3 symmetry, η 3 + αηζ 2 + βζ 6 + γζ 3 − β = 0, with real parameters α, β and γ. In the case α = 0 we prove that the only suitable values of γ/β are ±5 √ 2 (β is given below) which corresponds to the tetrahedrally symmetric solution. We then extend this result by continuity to non-zero values of the parameter α and find finally a new one-parameter family of monopole curves with C 3 symmetry.2. The Atiyah-Drinfeld-Manin-Hitchin-Nahm construction. Although the Bogomolny equation is a first order partial differential equation in R 3 few explicit solutions 1 arXiv:1009.3837v1 [math-ph]