A Simons type condition for instability of $F$-Yang-Mills connections

TL;DR

This paper extends Simons' theorem to $F$-Yang-Mills connections, providing a criterion for their instability on convex hypersurfaces, generalizing classical results for Yang-Mills connections.

Contribution

It introduces a new instability criterion for non-flat $F$-Yang-Mills connections, generalizing existing theorems to a broader class of gauge theories.

Findings

Non-flat $F$-Yang-Mills connections are unstable under certain conditions.

The instability criterion involves the dimension and the differential of $F$.

Results extend classical Simons theorem to $F$-Yang-Mills setting.

Abstract

$F$-Yang-Mills connections are critical points of $F$-Yang Mills functional on the space of connections of a principal fiber bundle, which is a generalization of Yang-Mills connections, $p$-Yang-Mills connections and exponential Yang-Mills connections and so on. Here, $F$ is a strictly increasing $C^{2}$-function. In this paper, we extend Simons theorem for an instability of Yang-Mills connections to $F$-Yang-Mills connections. We derive a sufficient condition that any non-flat, $F$-Yang-Mills connection over convex hypersurfaces in a Euclidean space is instable. In the sphere case, this condition is expressed by an inequality with respect to its dimension and a degree of the differential of the function $F$. The proofs of the results are given by extending Kobayashi-Ohnita-Takeuchi's calculation to $F$-Yang-Mills connections.