On the existence charge three SU(2) monopole: algebro-geometric approach

Abstract:
Charge three SU(2) monopole solution to the Bogomolny equation in the BPS limit is considered within Hitchin-Nahm algebro-geometric construction. The Hitchin constraints are analyzed for the genus four monopole curve with Z₃ symmetry. It is shown that the moduli space of the curve covers Humbert surface H₄. That leads to decomposition of the initial theta-functions to Jacobian theta's. Using this mashinary Hitchin conditions are explicitly resolved. The analysis undertaken leads to the conculision that for the given class of monopole curves only monopole with tetrahedral symmetry exists whilst there are no monopole solutions beyond this symmetry.