Non-existence of Higgs fields on Calabi-Yau Manifolds

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.