Title:

Elementary states, supergeometry and twistor theory

It is shown that H^{p1} (P^{+}, 0 (mp)) is a Fréchet space, and its dual is H^{q1}(P^{}, 0 (mq)), where P^{+} and P^{} are the projectivizations of subsets of generalized twistor space (≌ ℂ^{pq}) on which the hermitian form (of signature (p,q)) is positive and negative definite respectively, and 0(mp) denotes the sheaf of germs of holomorphic functions homogeneous of degree mp. It is then proven, for p = 2 and q = 2, that the subspace consisting of all twistor elementary states is dense in H^{p1}(P^{+}, 0(mp)). A supermanifold is a ringed space consisting of an underlying classical manifold and an augmented sheaf of Z_{2}graded algebras locally isomorphic to an exterior algebra. The subcategory of the category of ringed spaces generated by such supermanifolds is referred to as the super category. A mathematical framework suitable for describing the generalization of YangMills theory to the super category is given. This includes explicit examples of supercoordinate changes, superline bundles, and superconnections. Within this framework, a definition of the full super YangMills equations is given and the simplest case is studied in detail. A comprehensive account of the generalization of twistor theory to the super category is presented, and it is used in an attempt to formulate a complete description of the super YangMills equations. New concepts are introduced, and several ideas which have previously appeared in the literature at the level of formal calculations are expanded and explained within a consistent framework.
