U(1)×U(1)U(1)×U(1) Chern–Simons vortices in line bundles over a compact Riemann surface

The relativistic U(1)×U(1)U(1)×U(1) Chern–Simons model with two Higgs particles and two gauge fields has been studied in R2R2 and on a flat torus. We formulate a model for line bundles over a compact Riemann surface, and prove an existence theorem for the induced system of self-dual Chern–Simons vortex equations.