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By L. P. Hughston, R. S. Ward

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I " """ t/ Fi gure A A new viewpoint concerning twistor functions has been 9radual1y emerging which makes good mathematical sense of all this: a twistor function ;s really to be viewed as a representative function (or element of a sheaf cohomology group. coc~le) defining an Now , the twistor theorist. ",hen attacked by a pu r ist fo r shoddiness in the domains, can counter-attack ar med with his sheaf! Thumbnail sketch of (relevant) sheaf cohomology theo r'y First let us recall how ordinary (tech) cohomology ~Iorks .

J. ' (t~l: in9 '/. In this case we ~et the sheaf , denoted D, of ... on X. S119h:ly more consider "twisted" holomorphic functions . e . functiOns whose values are not j ust ordinary complex nulTbers , but takel) in some complex line bund le 29 "spin - ~Ieighted" over X (think of functions, for example). eous degree n in the twistor variable. (and holomorphic) of some fixed For each open set U L X He take G to u consist of all such tNisted functions on ll , and the resulting sheaf, denoted t~listed O(n), is called the "sheaf of ge)"ms of holomorphic functions ~lore (on X).

Where x and are real and is timelike future-pointing. l l A posi tive frequency field 't'A ' . 0' with n primed indices is said to have helicity~ , whereas a positive frequency field ¢lA ... O with n unprimed indices ha s helicity -~n . (In many of the sections of this Chapter, however , the terms helicity and spin are used even when referring to fields that are not pos itive frequency . ) The points of Cit correspond to lines which lie entirely in the 'top half' PT+ or projective twistor space (Penrose & l~acCallum 1972) .

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