Curved twistor spaces - Ward, R; Ward, R. S. (1977)
﻿ This thesis is concerned with the problem of "coding" the information of various zero-rest-mass fields into the complex structure of "curved twistor spaces". Chapter 2 is devoted to various preliminaries: a brief outline of twistor theory; an introduction to vector bundles and sheaf cohomology and some of their applications in twistor theory; and a discussion of potentials for electromagnetic fields. Chapter 3 deals with left-handed (i.e. anti-self-dual) electromagnetic fields and describes in some detail the associated curved twistor spaces. It is shown how holomorphic functions on the curved spaces give rise to "charged" zero-rest-mass fields on space-time. The first section of Chapter 4 gives the corresponding results for left-handed gravitational fields, using Penrose's "nonlinear graviton" construction. The rest of Chapter 4 is devoted to the concept of twistors relative to a hypersurface in a general curved space-time. In §4.2 the hypersurface is taken to be spacelike; the hypersurface twistors are described and the problem of using holomorphic hypersurface twistor functions to generate fields on the hypersurface and in space-time is discussed. Next the hypersurface is taken to be null. The structure of the associated hypersurface twistor space and and#x210b;-space are described in some detail. The twistor space has a natural inner product and, if the hypersurface is shear-free, then it has a "fibred" structure as well. In §4.4 the hypersurface twistor language is used to show that the propagation of twistors through an analytic pp-wave is given by the unfolding of a canonical transformation. Chapter 5 extends the "electromagnetic" construction of Chapter 3 to non-Abelian gauge theories; left-handed gauge fields are described in terms of complex vector bundles over protective twistor space.