On monopole systems with weak axial symmetry

Abstract

Let (Φ,) be an SO(3) Yank-Mills-Higgs system which is a real-analytic, static, finite-energy solution of the Bogomolny field equation Φ. We show that the zero-set of the current is of dimension at most one. Using this property of we obtain the curious result that if the system is axially symmetric, in the weak sense that all local scalar gauge-invariants are axially symmetric, the topological charges must be located on the axis of symmetry and must be of equal magnitude and alternate sign. In particular, if the charges are of uniform sign they must be concentrated at a single point. The fact that the charges of spherically symmetric monopoles are bounded by unity is obtained as a corollary. It is also shown that a master-potential for the invariant fields that was found earlier to exist for systems with additional symmetry, exists as a direct consequence of weak axial symmetry alone.