The geometry of self-dual two-forms


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Bilge A., Dereli T., Kocak S.

JOURNAL OF MATHEMATICAL PHYSICS, vol.38, no.9, pp.4804-4814, 1997 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 38 Issue: 9
  • Publication Date: 1997
  • Doi Number: 10.1063/1.532125
  • Title of Journal : JOURNAL OF MATHEMATICAL PHYSICS
  • Page Numbers: pp.4804-4814

Abstract

We show that self-dual two-forms in 2n-dimensional spaces determine a n(2)-n+1-dimensional manifold S-2n and the dimension of the maximal linear subspaces of S-2n is equal To the (Radon-Hurwitz) number of linearly independent vector fields on the sphere S2n-1. We provide a direct proof that for n odd 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 S-8 also include among many other interesting classes of self-dual two-forms, the self-dual two-forms of Corrigan, Devchand, Fairlie, and Nuyts [Nucl. Phys. B 214, 452 (1983)] and a representation of Cl-7 given by octonionic multiplication. We discuss the relation of the Linear subspaces with the representations of Clifford algebras. (C) 1997 American Institute of Physics.