Vortices as Instantons in Noncommutative Discrete Space: Use of $Z_2$ Coordinates

We show that vortices of Yang-Mills-Higgs model in $R^2$ space can be regarded as instantons of Yang-Mills model in $R^2\times Z_2$ space. For this, we construct the noncommutative $Z_2$ space by explicitly fixing the $Z_2$ coordinates and then show, by using the $Z_2$ coordinates, that BPS equation for the vortices can be considered as a self-dual equation. We also propose the possibility to rewrite the BPS equations for vortices as ADHM equations through the use of self-dual equation.