On the Stability of Yang-Mills Bundles over $S^4$

TL;DR

This paper examines the topological stability of Yang-Mills bundles over the 4-sphere, showing that standard U(1) gauge theory on this manifold lacks stable critical points due to topological constraints.

Contribution

It introduces a topological criterion for the existence of critical points in Yang-Mills bundles and demonstrates the instability of U(1) gauge theory on $S^4$ based on this criterion.

Findings

No critical points for U(1) gauge theory on $S^4$

Stability requires a self-consistent 2-manifold substructure

Results align with general dynamical systems stability theory

Abstract

The stability of Yang-Mills bundles over the usual $S^4$ space-time manifold is investigated according to the topological methods. The necessary gauge- and topological invaraint criterion for the exsitence of the related critical points is defined. It is shown that according to this criterion there exists no critical point even for the action functional of the standard U(1) gauge theory of electrodynamics on a $S^4$ manifold in view of its topological structure and therefore such a theory can not be stable. We will discuss also a general consequence of this result according to which for a stable U(1) Yang-Mills theory over a compact 4-manifold, this manifold should possess some self consistent compact 2-manifold substructure. These results are also in agreement with the known very general result for the {\it structural stability} of dynamical systems.