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

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, eta(3)+alpha eta zeta(2)+beta zeta(6)+gamma zeta(3)-beta = 0, with real parameters alpha, beta and gamma. In the case alpha = 0 we prove that the only suitable values of gamma/beta are +/- 5 root 2 (beta is given below) which corresponds to the tetrahedrally symmetric solution. We then extend this result by continuity to non-zero values of the parameter alpha and find finally a new one-parameter family of monopole curves with C-3 symmetry.