Gauge Fields with Generic Singularities

Let Mbe an n-dimensional manifold equipped with an Abelian Yang–Mills field with connection form . We consider an external potential function Vand examine the existence and regularity of the vortex lines of the form +Vdtwhich define the motion of a particle weakly coupled to the Yang–Mills field on M. These curves are smooth unless the curvature form d is singular and in this paper we treat this singular case from a generic aspect. The problem reduces to the division properties for smooth functions and differential forms, the development of which constitutes the main part of the work presented here.