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 N-1, N-2 vortices and P-1, P-2 anti-vortices of two designated species exists if and only if the inequalities vertical bar N-1 + N-2 - (P-1 + P-2)vertical bar < vertical bar S vertical bar/pi, vertical bar N-1 + 2N(2) - (P-1 + 2P(2))vertical bar < vertical bar S vertical bar/pi, 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 value E = 4 pi(N-1 + N-2 + P-1 + P-2), given in terms of several topological invariants, measuring the total tension of the vortex-lines. (C) 2015 The Authors. Published by Elsevier B.V.