The Geometry of Self-Dual Gauge Fields
Abstract
Self-dual 2-forms in D=2n dimensions are characterised by an eigenvalue criterion. The equivalence of various definitions of self-duality is proven. We show that the self-dual 2-forms determine a n2-n+1 dimensional manifold S2n and the dimension of the maximal linear subspaces of S2n$ is equal to the Radon-Hurwitz number of linearly independent vector fields on the sphere S2n-1. The relation between the maximal linear subspaces and the representations of Clifford algebras is noted. A general procedure based on this relation for the explicit construction of linearly self-dual 2-forms is given. The construction of the octonionic instanton solution in D=8 dimensions is discussed.
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