Yang-Mills-Higgs connections on Calabi-Yau manifolds

TL;DR

This paper investigates Yang-Mills-Higgs connections on Calabi-Yau manifolds, showing that certain Hermitian structures satisfying the Yang-Mills-Higgs equation also satisfy it when the Higgs field is zero, revealing structural properties of these connections.

Contribution

It proves that Hermitian structures satisfying the Yang-Mills-Higgs equation on Ricci-flat K"ahler manifolds also satisfy the equation with zero Higgs field, extending understanding of these connections.

Findings

Hermitian structures satisfy the Yang-Mills-Higgs equation for zero Higgs field

If a semistable Higgs bundle exists with non-zero Higgs field, the manifold is Calabi-Yau

Results apply to principal Higgs bundles on Ricci-flat K"ahler manifolds

Abstract

Let $X$ be a compact connected K\"ahler--Einstein manifold with $c_1(TX)\, \geq\, 0$. If there is a semistable Higgs vector bundle $(E\,,\theta)$ on $X$ with $\theta\,\not=\,0$, then we show that $c_1(TX)=0$, any $X$ satisfying this condition is called a Calabi--Yau manifold, and it admits a Ricci--flat K\"ahler form \cite{Ya}. Let $(E\,,\theta)$ be a polystable Higgs vector bundle on a compact Ricci--flat K\"ahler manifold $X$. Let $h$ be an Hermitian structure on $E$ satisfying the Yang--Mills--Higgs equation for $(E\,,\theta)$. We prove that $h$ also satisfies the Yang--Mills--Higgs equation for $(E\,,0)$. A similar result is proved for Hermitian structures on principal Higgs bundles on $X$ satisfying the Yang--Mills--Higgs equation.