G2-structures and octonion bundles

TL;DR

This paper explores the geometric and algebraic structures of G2-structures on 7-manifolds, introducing an octonion bundle, a compatible covariant derivative, and analyzing torsion and curvature in this context.

Contribution

It develops a novel framework linking G2-structures with octonion bundles, defining an octonionic covariant derivative, and relating torsion to octonionic connections and gauge transformations.

Findings

Torsion acts as an octonionic connection with curvature in the G2 representation.

Change of G2-structure gauge transforms torsion as an octonion-valued 1-form.

Critical points of an energy functional correspond to divergence-free torsion.

Abstract

We use a G2-structure on a 7-dimensional Riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonion covariant derivative on this bundle that is compatible with the octonion product. The torsion of the G2-structure is then shown to be an octonionic connection for this covariant derivative with curvature given by the component of the Riemann curvature that lies in the 7-dimensional representation of G2. We also interpret the choice of a particular G2-structure within the same metric class as a choice of gauge and show that under a change of this gauge, the torsion does transform as an octonion-valued connection 1-form. Finally, we also show an explicit relationship between the octonion bundle and the spinor bundle, define an octonionic Dirac operator and explore an energy functional for octonion sections. We then prove that critical points correspond to divergence-free torsion, which is shown to be an octonionic analog of the Coulomb gauge.