The Geometry of Self-dual 2-forms

TL;DR

This paper explores the geometric structure of self-dual 2-forms in even-dimensional spaces, revealing their manifold properties, maximal linear subspaces, and connections to Clifford algebras and octonions.

Contribution

It characterizes the manifold of self-dual 2-forms, determines the dimensions of their maximal linear subspaces, and links these structures to Clifford algebra representations and octonionic multiplication.

Findings

The manifold ${\

The maximal linear subspaces have dimensions related to the Radon-Hurwitz number.

Seven-dimensional subspaces include forms related to octonions and Clifford algebra representations.

Abstract

We show that self-dual 2-forms in 2n dimensional spaces determine a $n^2-n+1$ dimensional manifold ${\cal S}_{2n}$ and the dimension of the maximal linear subspaces of ${\cal S}_{2n}$ is equal to the (Radon-Hurwitz) number of linearly independent vector fields on the sphere $S^{2n-1}$. We provide a direct proof that for $n$ odd ${\cal S}_{2n}$ has only one-dimensional linear submanifolds. We exhibit $2^c-1$ dimensional subspaces in dimensions which are multiples of $2^c$, for $c=1,2,3$. In particular, we demonstrate that the seven dimensional linear subspaces of ${\cal S}_{8}$ also include among many other interesting classes of self-dual 2-forms, the self-dual 2-forms of Corrigan, Devchand, Fairlie and Nuyts and a representation of ${\cal C}l_7$ given by octonionic multiplication. We discuss the relation of the linear subspaces with the representations of Clifford algebras.