Monopoles, gauge fields and de Rham’s theorems

J. Phys. A: Math. Gen. 31 (1998) 7077-7094 (PDF)

The topology assumed by most authors for a spacelike hypersurface in a spacetime containing a monopole is generally Euclidean 3-space minus the origin; save for the spherical surface 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 the first homotopy group of 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.