# Manton's five vortex equations from self-duality

## TL;DR

This paper shows that Manton's five vortex equations are derived from anti-self-dual Yang--Mills equations via symmetry reductions, linking vortex solutions to gauge groups and constructing vortex moduli space metrics.

## Contribution

It establishes the connection between Manton's vortex equations and anti-self-dual Yang--Mills equations, and introduces a method to generate higher vortex numbers and associated moduli space metrics.

## Key findings

- Vortices correspond to specific gauge groups like SU(1,1) and SU(2).
- Higher vortex numbers can be obtained by superposing different vortex equations.
- The vortex moduli space metric derived from Yang--Mills energy can be indefinite for non-compact gauge groups.

## Abstract

We demonstrate that the five vortex equations recently introduced by Manton ariseas symmetry reductions of the anti-self-dual Yang--Mills equations in four dimensions. In particular the Jackiw--Pi vortex and the Ambj\o rn--Olesen vortex correspond to the gauge group $SU(1, 1)$, and respectively the Euclidean or the $SU(2)$ symmetry groups acting with two-dimensional orbits. We show how to obtain vortices with higher vortex numbers, by superposing vortex equations of different types. Finally we use the kinetic energy of the Yang--Mills theory in 4+1 dimensions to construct a metric on vortex moduli spaces. This metric is not positive-definite in cases of non-compact gauge groups.