The Geometry of Self-dual 2-forms

Abstract

We show that self-dual 2-forms in 2n dimensional spaces determine a $n^2-n+1$ dimensional manifold ${\mathcal{S}}_{2n}$ and the dimension of the maximal linear subspaces of ${\mathcal{S}}_{2n}$ 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 ${\mathcal{S}}_{2n}$ 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 ${\mathcal{S}}_8$ 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 $\mathcal{C}{l}_7$ given by octonionic multiplication. We discuss the relation of the linear subspaces with the representations of Clifford algebras.

Latex, 15 pages

References in corpus (1)

• Self-Dual Yang-Mills Fields in Eight Dimensions

Cited by in corpus (2)

• Analytic Loops and Gauge Fields
• The Geometry of Self-Dual Gauge Fields