Octonion-valued forms and the canonical 8-form on Riemannian manifolds with a $Spin(9)$-structure

TL;DR

This paper introduces a new explicit algebraic formula for the unique $Spin(9)$-invariant 8-form on the octonionic plane, generalizing the K"{a}hler 2-form expression and applicable to Riemannian manifolds with $Spin(9)$-structure.

Contribution

It provides a novel explicit algebraic formula for the $Spin(9)$-invariant 8-form using octonion-valued coordinate 1-forms, extending previous descriptions.

Findings

New explicit algebraic formula for the $Spin(9)$-invariant 8-form

Generalization of the K"{a}hler 2-form expression

Expressions for Kraines, Cayley, and associative calibrations

Abstract

It is well known that there is a unique $Spin(9)$-invariant 8-form on the octonionic plane that naturally yields a canonical differential 8-form on any Riemannian manifold with a weak $Spin(9)$-structure. Over the decades, this invariant has been studied extensively and described in several equivalent ways. In the present article, a new explicit algebraic formula for the $Spin(9)$-invariant 8-form is given. The approach we use generalizes the standard expression of the K\"{a}hler 2-form. Namely, the invariant 8-form is constructed only from the two octonion-valued coordinate 1-forms on the octonionic plane. For completeness, analogous expressions for the Kraines form, the Cayley calibration and the associative calibration are also presented.