Bogomolny equations for vortices in the noncommutative torus

TL;DR

This paper derives Bogomolny equations for vortices in the noncommutative torus within the Abelian Higgs model, analyzing boundary conditions and constructing vortex solutions, with extensions to a noncommutative standard model prototype.

Contribution

It introduces Bogomolny equations for vortices on the noncommutative torus and details the construction of solutions, including boundary condition handling and model extensions.

Findings

Derived Bogomolny equations for noncommutative torus

Constructed vortex solutions respecting noncommutative boundary conditions

Extended analysis to a noncommutative $U(2)\times U(1)$ model

Abstract

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.