Non Abelian Vortices as Instantons on Noncommutative Discrete Space

There seems to be close relationship between the moduli space of vortices and the moduli space of instantons, which is not yet clearly understood from a standpoint of the field theory. We clarify the reasons why many similarities are found in the methods for constructing the moduli of instanton and vortex, viewed in the light of the notion of the self-duality. We show that the non-Abelian vortex is nothing but the instanton in $R^{2} \times Z_{2}$ from a viewpoint of the noncommutative differential geometry and the gauge theory in discrete space. The action for pure Yang-Mills theory in $R^{2} \times Z_{2}$ is equivalent to that for Yang-Mills-Higgs theory in $R^{2}$.