Stable solutions to the abelian Yang--Mills--Higgs equations on $S^2$ and $T^2$

TL;DR

This paper proves that stable solutions to abelian Yang--Mills--Higgs equations on the 2-sphere and 2-torus are actually solutions to the simpler vortex equations, extending known results to these surfaces.

Contribution

It establishes that under natural conditions, stable solutions on $S^2$ and $T^2$ satisfy vortex equations, linking second-order and first-order formulations in these geometries.

Findings

Stable solutions on $S^2$ satisfy vortex equations.

Stable solutions on $T^2$ satisfy vortex equations.

Method uses Bourguignon--Lawson's approach for stable connections.

Abstract

We show under natural assumptions that stable solutions to the abelian Yang--Mills--Higgs equations on Hermitian line bundles over the round $2$-sphere actually satisfy the vortex equations, which are a first-order reduction of the (second-order) abelian Yang--Mills--Higgs equations. We also obtain a similar result for stable solutions on a flat $2$-torus. Our method of proof comes from the work of Bourguignon--Lawson concerning stable $SU(2)$ Yang--Mills connections on compact homogeneous $4$-manifolds.