{'fraction_non_alphanumeric': 0.043559427504667084, 'fraction_numerical': 0.006222775357809583, 'mean_word_length': 4.4324324324324325, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Elementary states, supergeometry and twistor theory

﻿ It is shown that H p-1 (P + , 0 (-m-p)) is a Frechet space, and its dual is H q-1 (P - , 0 (m-q)), where P + and P - are the projectivizations of subsets of generalized twistor space (≌ ℂ p-q ) on which the hermitian form (of signature (p,q)) is positive and negative definite respectively, and 0 (-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 p-1 (P + , 0 (-m-p)). 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 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.