Monopoles, gauge fields and de Rham’s theorems

The topology assumed by most authors for a spacelike hypersurface in a spacetime containing a monopole is generally R3−{0}; save for the surface S2 isolating the monopole, this space is unbounded. For such a topology, a consistency relation of de Rham’s theorems shows that a single isolated monopole cannot exist. Monopoles, with charge ±m, if they exist at all, must occur in pairs having opposite magnetic charge. An extension of de Rham’s theorems to non-Abelian monopoles which are generalizations of Dirac monopoles (those characterized by π1(G), the fundamental group of the gauge group G) is made using the definition of an ordered integral of a path-dependent curvature over a surface. This integral is similar to that found in the non-Abelian Stokes theorem. The implications of de Rham’s theorems for non-Abelian monopoles are shown to be similar to the Abelian case.