| 總結: | <p>It is shown that H<sup>p-1</sup> (P<sup>+</sup>, <em>0</em> (-m-p)) is a Fréchet space, and its dual is H<sup>q-1</sup>(P<sup>-</sup>, <em>0</em> (m-q)), where P<sup>+</sup> and P<sup>-</sup> are the projectivizations of subsets of generalized twistor space (≌ ℂ<sup>p-q</sup>) on which the hermitian form (of signature (p,q)) is positive and negative definite respectively, and <em>0</em>(-m-p) denotes the sheaf of germs of holomorphic functions homogeneous of degree -m-p. It is then proven, for p = 2 and q = 2, that the subspace consisting of all twistor elementary states is dense in H<sup>p-1</sup>(P<sup>+</sup>, <em>0</em>(-m-p)).</p> <p>A supermanifold is a ringed space consisting of an underlying classical manifold and an augmented sheaf of <strong>Z</strong><sub>2</sub>-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 Yang-Mills 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 Yang-Mills 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 Yang-Mills 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.</p>
|
|---|