Vortices as Instantons in Noncommutative Discrete Space: Use of $Z_{2}$ Coordinates

TL;DR

This paper demonstrates that vortices in a Yang-Mills-Higgs model on two-dimensional space can be interpreted as instantons in a noncommutative space extended by a discrete $Z_{2}$ component, linking vortex solutions to instanton solutions via noncommutative geometry.

Contribution

It constructs a noncommutative $Z_{2}$ space and shows how vortex BPS equations can be viewed as self-dual equations, proposing a new perspective on vortex-instanton correspondence.

Findings

Vortices are equivalent to instantons in $R^{2} imes Z_{2}$ space.

BPS vortex equations can be reformulated as self-dual equations.

Potential to express vortex BPS equations as ADHM equations.

Abstract

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.