Sheaf cohomology in twistor diagrams
One of the earlier achievements of twistor theory was the description of free zero rest mass fields on complexified Minkowski space in terms of holomorphic functions on twistor space. Interactions between these fields are given by certain spacetime integrals (represented by Feynmann diagrams), and some of these integrals have been translated into contour integrals in products of twistor spaces (represented by twistor diagrams). The principal advantage of the twistor diagram formalism is that it is necessarily finite. The main purpose of this thesis is to explore the uses of two mathematical techniques in twistor diagrams. The first is the "blowing up" process familiar to algebraic geometers. It arises naturally in the translation from the massless scalar ϕ4(vertex to the corresponding twistor diagram (called the "box" diagram). A detailed study of this translation reveals that there are three contours over which the box diagram can be integrated, one for each of the channels in the ϕ4 interaction. The second technique is sheaf cohomology theory, vhich vas introduced to make rigorous the twistor description of zero rest mass fields by replacing twistor functions by elements of sheaf cohomology groups. We show how to interpret fragments of twistor diagrams - which normally represent twistor functions - as these sheaf cohomology elements. Chapter 1 introduces, briefly, the basic ideas of twistor geometry, the twistor description of fields, and twistor diagrams. In chapter 2 we demonstrate the existence of contours for part of the Möller scattering diagram using singular homology theory, while chapter 3 gives the details of the translation to the box diagram (already referred to) and compares it with the scalar product diagram. The last two chapters (4 and 5) deal with the sheaf cohomology of tree diagrams and the scalar product diagram respectively.