Vortex condensation in Chern–Simons model with a general Higgs potential on a torus

We consider the self-dual equations in a relativistic U ( 1 ) × U ( 1 ) Chern–Simons model with a general Higgs potential. We prove the existence of maximal vortex condensates for the system of equations on a torus, and derive some properties of vortex condensates.