Isolated Singularities of Yang-Mills-Higgs fields on surfaces

TL;DR

This paper investigates the behavior of Yang-Mills-Higgs fields near isolated singularities on surfaces, establishing precise decay estimates linked to holonomy, thus extending classical harmonic map results.

Contribution

It provides a sharp asymptotic decay estimate for Yang-Mills-Higgs fields near singularities, generalizing the classical removable singularity theorem for harmonic maps.

Findings

Decay rate determined by limit holonomy

Singularities may not be removable due to non-vanishing holonomy

Generalization of harmonic map singularity theorem

Abstract

We study isolated singularities of two dimensional Yang-Mills-Higgs fields defined on a fiber bundle, where the fiber space is a compact Riemannian manifold and the structure group is a compact connected Lie group. In general the singularity can not be removed due to possibly non-vanishing limit holonomy around the singular points. We establish a sharp asymptotic decay estimate of the Yang-Mills-Higgs field near a singular point, where the decay rate is precisely determined by the limit holonomy. Our result can be viewed as a generalization of the classical removable singularity theorem of two dimensional harmonic maps.