Bogomolny equations for vortices in the noncommutative torus

We derive Bogomolny-type equations for the Abelian Higgs model defined on the noncommutative torus and discuss its vortex like solutions. To this end, we carefully analyze how periodic boundary conditions have to be handled in noncommutative space and discussed how vortex solutions are constructed. We also consider the extension to an $U(2)\times U(1)$ model, a simplified prototype of the noncommutative standard model.