Yang-Mills connections on $G_{2}$-manifolds and Calabi-Yau 3-folds

TL;DR

This paper investigates Yang-Mills energy minimizers on $G_{2}$-manifolds and Calabi-Yau 3-folds, showing that such connections are special geometric structures: $G_{2}$-instantons and holomorphic bundles respectively.

Contribution

It establishes that energy-minimizing Yang-Mills connections on these manifolds are necessarily $G_{2}$-instantons and holomorphic bundles, revealing their geometric nature.

Findings

Yang-Mills energy minimizers are $G_{2}$-instantons on $G_{2}$-manifolds.

On Calabi-Yau 3-folds, minimizers correspond to holomorphic bundles.

Connections satisfy stability conditions leading to these geometric structures.

Abstract

We consider the minimum Yang-Mills energy on the complete $G_{2}$-manifolds and Calabi-Yau 3-folds,the connection $A$ is a stability Yang-Mills connection on the $G$-bundle $E$.We prove that the connection must be a $G_{2}$-instanton on $G_{2}$-manifold and the bundle is holomorphic on Calabi-Yau 3-fold with holonomy $SU(3)$.