Applications of Sheaf Cohomology in Twistor Theory

Richard Jozsa

Applications of Sheaf Cohomology in Twistor Theory

A dissertation submitted in partial fulfilment of the requirement for the degree of Master of Science, 1976

Introduction
Complex Manifolds, Vector Bundles and Twistor Space
Sheaves, Sheaf Cohomology and Twistor Functions
Deformations of Complex Structure and the Nonlinear Graviton
References

CHAPTER 1: INTRODUCTION

The past 75 years in theoretical physics has seen the introduction of a wide variety of new mathematical constructions for the description of physical phenomena in regions beyond our immediate experience. It is a curious and interesting fact that all of classical physics (and indeed much of quantum mechanics and especially relativity) is based on the real number system and formulated in terms of differential equations, a local description. Why is it that the real number system, just one of an infinite variety of mathematical structures (and a rather complicated one) has secured such a totally dominating position in physics? The use of real numbers has become so ingrained that it is difficult even to conceive of a measurement which does not produce the measured quantity as a numerical value. The construction of the real number system out of the finite ordinals via the rationals was motivated a long time ago by the need to solve certain kinds of equations. Perhaps with our increased knowledge of the microstructure of spacetime some of these steps may be reconsidered to produce a mathematical structure more suited to the needs of modern physics.

One of the most interesting trends in recent physics has been the gradual recognition that complex numbers are in some sense more fundamental than real numbers. It may seem that the move from real to complex numbers is a small step but this is not the case, as has often been emphasised by Penrose (e.g. Penrose 1975, Introduction). The extra mathematical structure in the complex number field, giving rise to theorems like Cauchy's integral formula, represents an interplay between local and global properties, totally absent in the real number system, which has been exploited to great effect in twistor theory. Also we have the local isomorphisms (Penrose 1974)

O(1,3)  <---  SL(2,C)
C(1,3)  <---  SU(2,2)
  O(3)  <---  SU(2)

showing that the structure of spacetime can be neatly reformulated in terms of complex spaces and the basic role played by complex numbers in quantum mechanics is well known. Twistor theory, perhaps more than any other physical theory, is built around complex structures. For example, in the representation of zero rest mass fields on spacetime, by means of twistor functions, the field equations are essentially replaced merely by a condition of holomorphicity (Penrose 1975, 1969) and in the nonlinear graviton (Penrose 1976) curved vacuum spacetimes are generated by deforming the complex structure of flat twistor space.

One of the natural paths along which complex analysis and contour integral techniques can be developed leads to sheaf theory and sheaf cohomology which, as a result, takes up a fundamental role in the mathematical apparatus of twistor theory. In the following work, among other results it will be seen how zero rest mass fields on spacetime, originally described as real spinor fields satisfying field equations, later by arbitrary holomorphic functions of one twistor, are now interpreted as elements of a sheaf cohomology group. Also, It will be seen how deformations of complex structures, described in terms of sheaf cohomology, can be used to generate the general half flat solution of Einstein's vacuum equations. Perhaps these results depending essentially on the structure of complex numbers give some clues towards an under- standing of the fundamental role of complex structures in physics but a full understanding, which perhaps incorporates the union of quantum mechanics and relativity, is yet to emerge.