Non-abelian monopoles and vortices

TL;DR

This paper explores non-abelian generalizations of Seiberg-Witten equations on complex Kahler surfaces, comparing them with known vortex equations and discussing their differences.

Contribution

It introduces non-abelian versions of Seiberg-Witten equations inspired by vortex equations, expanding the theoretical framework for gauge theories on complex surfaces.

Findings

Non-abelian Seiberg-Witten equations differ from abelian cases.

Comparison between vortex equations and Seiberg-Witten equations.

Insights into the structure of gauge theories on Kahler surfaces.

Abstract

The Seiberg-Witten equations are defined on certain complex line bundles over smooth oriented four manifolds. When the base manifold is a complex Kahler surface, the Seiberg-Witten equations are essentially the Abelian vortex equations. Using known non-abelian generalizations of the vortex equations as a guide, we explore some non-abelian versions of the Seiberg-Witten equations. We also make some comments about the differences between the vortex equations that have previously appeared in the literature and those that emerge as Kahler versions of Seiberg-witten type equations.