Sites and googly twistor spaces. II. Yang–Mills fields

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 the previous paper, we analyzed the structure of the googly twistor space when no Yang–Mills field is present, and showed that it was 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.) In this paper, we treat the case where a gauge field is present. We show that the field is represented by a vector bundle over the site, and that massless fields minimally coupled to the Yang–Mills field are given by the elements of a sheaf cohomology group on the site. Also, this vector bundle is isomorphic to one over a leg-break twistor site. This provides a strong link between the googly and the leg-break spaces.