Sites and googly twistor spaces. I. Vacuum

Abstract

(Anti-) self-dual Yang–Mills fields may be described by twistors of the same or opposite handedness as the fields. These are called the leg-break and googly descriptions, respectively. The leg-break twistor space is a complex manifold; the Yang–Mills field is given by a vector bundle over this manifold, and massless fields minimally coupled to the Yang–Mills field are given by elements of certain sheaf cohomology groups on the manifold. In this paper, the structure of the googly twistor space when no Yang–Mills field is present is elucidated. It is shown that the googly twistor space is a site. Sites are generalizations of topological spaces, in which the primitive concept is that of an open set rather than that of a point. The massless fields on space-time are given by the elements of a sheaf cohomology group on the site. Also, this site is isomorphic to a leg-break site, consisting of a family of open sets in the leg-break manifold. This provides a strong link between the googly and the leg-break spaces. The following paper treats the case where a Yang–Mills field is present.