A variational problem for a system of magnetic monopoles joined by Abrikosov vortices

An action functional, related to the Higgs model to field theory, depending on a complex scalar field and aU(1) connection is defined. The complex scalar field is a section of a line bundle associated to a principalU(1)-bundle with base space ³{x₁,...,x_n}. The pointsx₁,...,x_n are the positions ofn magnetic monopoles of magnetic chargesm₁,...,m_n, with. The existence of minimizers of the action functional is proven using direct methods of the calculus of variation. Regularity and decay properties of the minimizers are obtained. By constructing explicit comparison field configurations, we establish accurate upper and lower bounds for the action of the minimizers in a variety of special situations, e.g.n=2 andm₁=–m₂.