288 research outputs found
The Geometry of Self-dual 2-forms
We show that self-dual 2-forms in 2n dimensional spaces determine a
dimensional manifold and the dimension of the maximal linear
subspaces of is equal to the (Radon-Hurwitz) number of linearly
independent vector fields on the sphere . We provide a direct proof
that for odd has only one-dimensional linear submanifolds.
We exhibit dimensional subspaces in dimensions which are multiples of
, for . In particular, we demonstrate that the seven dimensional
linear subspaces of 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
given by octonionic multiplication. We discuss the relation of the linear
subspaces with the representations of Clifford algebras.Comment: Latex, 15 page
Self-dual Yang-Mills fields in eight dimensions
Strongly self-dual Yang-Mills fields in even dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fields F_{\mu \nu}. We derive a topological bound on {\bf R}^8, \int_{M} ( F,F )^2 \geq k \int_{M} p_1^2 where p_1 is the first Pontrjagin class of the SO(n) Yang-Mills bundle and k is a constant. Strongly self-dual Yang-Mills fields realise the lower bound
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