Non Abelian Vortices as Instantons on Noncommutative Discrete Space

TL;DR

This paper demonstrates that non-Abelian vortices can be understood as instantons within a noncommutative discrete space framework, revealing deep connections between vortex and instanton moduli spaces.

Contribution

It shows that non-Abelian vortices are equivalent to instantons in a noncommutative $R^{2} imes Z_{2}$ space, bridging vortex and instanton theories through noncommutative geometry.

Findings

Vortices are equivalent to instantons in noncommutative discrete space.

The action in $R^{2} imes Z_{2}$ matches Yang-Mills-Higgs theory in $R^{2}$.

Method clarifies similarities in moduli space constructions.

Abstract

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} $.