Yang-Mills connections valued on the octonionic algebra

TL;DR

This paper investigates Yang-Mills theories with gauge fields valued in octonionic algebra and finds that only standard (2) or (1) gauge groups are possible under certain conditions.

Contribution

It demonstrates that Yang-Mills theories with octonionic algebra-valued gauge fields reduce to familiar (2) or (1) gauge theories, revealing limitations of octonionic formulations.

Findings

Only (2) and (1) gauge groups are possible.

Octonionic algebra does not lead to new gauge theories.

Standard gauge theories are the only consistent under mild assumptions.

Abstract

We consider a formulation of Yang-Mills theory where the gauge field is valued on an octonionic algebra and the gauge transformation is the group of automorphisms of it. We show, under mild assumptions, that the only possible gauge formulations are the usual $\mathfrak{su}(2)$ or $\mathfrak{u}(1)$ Yang-Mills theories.