Non-existence of Higgs fields on Calabi-Yau Manifolds

TL;DR

This paper proves that on compact Calabi-Yau manifolds, semistable Higgs bundles with vanishing Chern classes must have trivial Higgs fields, implying such bundles are semistable without non-trivial Higgs structures.

Contribution

It establishes the non-existence of non-trivial Higgs fields on semistable Higgs bundles with vanishing Chern classes over Calabi-Yau manifolds using Yang-Mills-Higgs flow.

Findings

Higgs fields are trivial for semistable Higgs bundles with vanishing Chern classes on Calabi-Yau manifolds

Such bundles are necessarily semistable

Yang-Mills-Higgs flow is a key tool in the proof

Abstract

In this article, we study the Higgs vector bundles $(E,\theta)$ over a compact Calabi-Yau manifolds $X$. We use Yang-Mills-Higgs flow to prove that if a semistable Higgs bundle with vanishing Chern classes over a compact connected Calabi-Yau manifold, then the Higgs field $\theta$ is trivial. In particular, the vector bundle $E$ must be semistable.