Paper detail

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, $η^3+αηζ^2+βζ^6+γζ^3-β=0$, with real parameters $α$, $β$ and $γ$. In the case $α=0$ we prove that the only suitable values of $γ/β$ are $\pm 5\sqrt{2}$ ($β$ 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 {\em new} one-parameter family of monopole curves with $C_3$ symmetry.