Stable Yang-Mills connections on Special Holonomy Manifolds

TL;DR

This paper proves that energy-minimizing Yang-Mills connections with full holonomy on special holonomy manifolds are instantons or holomorphic, under certain curvature conditions, advancing understanding of gauge theories on these geometries.

Contribution

It establishes conditions under which energy-minimizing Yang-Mills connections on G2 and Calabi-Yau manifolds are instantons or holomorphic, linking curvature properties to geometric structures.

Findings

Energy-minimizing G2 Yang-Mills connections with full holonomy are G2-instantons.

Energy-minimizing connections on Calabi-Yau 3-folds with full holonomy are holomorphic.

Additional curvature conditions are necessary for these characterizations.

Abstract

We prove that energy minimizing Yang-Mills connections on a compact $G_{2}$-manifold has holonomy equal to $G_{2}$ are $G_{2}$-instantons, subject to an extra condition on the curvature. Furthermore, we show that energy minimizing connections on a compact Calabi-Yau $3$-fold has holonomy equal to $SU(3)$ subject to a similar condition are holomorphic.