# Stability and energy identity for Yang-Mills-Higgs pairs

## TL;DR

This paper investigates the properties and energy identities of Yang-Mills-Higgs pairs, establishing conditions for stability, curvature vanishing, and convergence behavior on high-dimensional and 4-dimensional manifolds.

## Contribution

It proves that stable Yang-Mills-Higgs pairs on spheres have Higgs fields of norm 1 and are Yang-Mills, with curvature vanishing in higher dimensions, and establishes an energy identity for sequences on 4-manifolds.

## Key findings

- Higgs field norm equals 1 for stable pairs on S^n, n > 3
- Curvature vanishes for n > 4
- Energy identity for sequences on 4-manifolds

## Abstract

In this paper, we study the properties of the critical points of Yang-Mills-Higgs functional, which are called Yang-Mills-Higgs pairs. We first consider the properties of weakly stable Yang-Mills-Higgs pairs on a vector bundle over S^n (n > 3). When n > 3, we prove that the norm of its Higgs field is 1 and the connection is actually Yang-Mills. More precisely, its curvature vanishes when n > 4. We also use the bubble-neck decomposition to prove the energy identity of a sequence of Yang-Mills-Higgs pairs over a 4-dimensional compact manifold with uniformly bounded energy. We show there is a subsequence converges smoothly to a Yang-Mills-Higgs pair up to gauge modulo finitely many 4-dimensional spheres with Yang-Mills connections.