Topologically stratified energy minimizers in a product Abelian field theory

We study a recently developed product Abelian gauge field theory by Tong and Wong hosting magnetic impurities. We first obtain a necessary and sufficient condition for the existence of a unique solution realizing such impurities in the form of multiple vortices. We next reformulate the theory into an extended model that allows the coexistence of vortices and anti-vortices. The two Abelian gauge fields in the model induce two species of magnetic vortex-lines resulting from N_s vortices and P_s anti-vortices (s=1, 2) realized as the zeros and poles of two complex-valued Higgs fields, respectively. An existence theorem is established for the governing equations over a compact Riemann surface S which states that a solution with prescribed N₁, N₂ vortices and P₁, P₂ anti-vortices of two designated species exists if and only if the inequalities|N1+N2-(P1+P2)|<|S|π,|N1+2N2-(P1+2P2)|<|S|π, hold simultaneously, which give bounds for the 'differences' of the vortex and anti-vortex numbers in terms of the total surface area of S. The minimum energy of these solutions is shown to assume the explicit valueE=4π(N1+N2+P1+P2), given in terms of several topological invariants, measuring the total tension of the vortex-lines.