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From the Introduction, p. 1. During the past decade, the theory of twistors has been introduced and developed, primarily by Professor Roger Penrose, as part of a long-term program aimed at resolving certain difficulties in present-day physical theory. These difficulties include, firstly, the problem of combining quantum mechanics and general relativity, and, secondly, the question of whether the concept of a continuum is at all relevant to physics. Most models of space-time used in general relativity employ the idea of a manifold consisting of a continuum of points. This feature of the models has often been criticised, on the grounds that physical observations are essentially discrete in nature; for reasons that are mathematical, rather than physical, the gaps between these observations are filled in a continuous fashion (see, for example, Schrodinger (I), pp.26-31). Although analysis (in its generally accepted form) demands that quantities should take on a continuous range of values, physics, as such,does not make such a demand. The situation in quantum mechanics is not all that much better since, although some quantities such as angular momentum can only take on certain discrete values, one still has to deal with the complex continuum of probability amplitudes. From this point of view it would be desirable to have all physical laws expressed in terms of combinatorial mathematics, rather than in terms of (standard) analysis.

It is shown that H^p-1 (P⁺, 0 (-m-p)) is a Fréchet space, and its dual is H^q-1(P^-, 0 (m-q)), where P⁺ and P^- are the projectivizations of subsets of generalized twistor space (≌ ℂ^p-q) on which the hermitian form (of signature (p,q)) is positive and negative definite respectively, and 0(-m-p) denotes the sheaf of germs of holomorphic functions homogeneous of degree -m-p. It is then proven, for p = 2 and q = 2, that the subspace consisting of all twistor elementary states is dense in H^p-1(P⁺, 0(-m-p)). A supermanifold is a ringed space consisting of an underlying classical manifold and an augmented sheaf of <strong>Z</strong>₂-graded algebras locally isomorphic to an exterior algebra. The subcategory of the category of ringed spaces generated by such supermanifolds is referred to as the super category. A mathematical framework suitable for describing the generalization of Yang-Mills theory to the super category is given. This includes explicit examples of supercoordinate changes, superline bundles, and superconnections. Within this framework, a definition of the full super Yang-Mills equations is given and the simplest case is studied in detail. A comprehensive account of the generalization of twistor theory to the super category is presented, and it is used in an attempt to formulate a complete description of the super Yang-Mills equations. New concepts are introduced, and several ideas which have previously appeared in the literature at the level of formal calculations are expanded and explained within a consistent framework.

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.

This thesis is concerned with an investigation of twistorial structures present in curved Lorentzian space-times. Chapter 1 introduces the basic definitions and some theorems that will be used later in the text. Chapter 2 investigates generalised connections that arise in twister theory. First the Cartan con-formal connection is studied, and some of the geometry underlying it is shown to be that used by Fefferman and Graham C133. Also a condition that a space-time is conformal to vacuum is given. Secondly the theory of the Chern connection associated to a C.R. manifold is developed in such a way as to make the calculation of the connection associated to a twistor C.R. manifold straight forward. A new proof of the Chern theorem of existence and uniqueness is given. The Chern connection of a twistor C.R. manifold is then calculated, and discussed. In particular S-dimensionai C.R. manifolds arising as twistor C.R. manifolds are characterised. Canonical structures peculiar to the twister case are discussed. Applications of C.R. manifold theory to algebraically special space-times are suggested. Chapter three analyses how various twistorial structures behave in linearised general relativity. First, deformations of the space of complex null geodesies corresponding to variations of the conformal structure of space-time are shown to be generated by hami1tonians. Those that correspond to variations in the metric satisfying the field equations are given, along with hamiltonians corresponding to different fields and field equations. Beneralisations to nonlinear equations are discussed. These ideas are applied to hypersurface twisters in linearised theory, using fiat hypersurfaces and Cech cohoeology. Expressions are obtained for the deformation of the complex structure of the spaces and their evolution. The results are generalised to non flat hypersurfaces using Dolbeault cohomolcgy. It is shown that certain canonically defined forms on the spin bundle are preferred Dolbeault representatives for derivatives of the twister cohomology classes corresponding to the linearised field. In chapter four I generalise the results of chapter three to curved space using the Chern connection. In particular twistorial formulations of the constraint equations are given, and a formula for the evolution that satisfies the the vacuum evolution equations is given in terms of an "infinity" twistor and a "time" twister. This is then discussed. In chapter five I make some comments on the interpretation of a three form on the spin bundle discovered by B.A.J. Sparling as the gravitational hami1tonian. I then use this to show that one can give an interpretation of Penrose's quasi-local angular momentum twistor in terms of the canonical formalism.

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In twistor theory the nonlinear graviton construction realises four-dimensional antiself- dual Einstein manifolds as Kodaira moduli spaces of rational curves in threedimensional complex manifolds. We establish a Newtonian analogue of this procedure, in which four-dimensional Newton-Cartan manifolds arise as Kodaira moduli spaces of rational curves with normal bundle O + O(2) in three-dimensional complex manifolds. The isomorphism class of the normal bundle is unstable with respect to general deformations of the complex structure, exhibiting a jump to the Gibbons- Hawking class of twistor spaces. We show how Newton-Cartan connections can be constructed on the moduli space by means of a splitting procedure augmented by an additional vector bundle on the twistor space which emerges when considering the Newtonian limit of Gibbons-Hawking manifolds. The Newtonian limit is thus established as a jumping phenomenon. Newtonian twistor theory is extended to dimensions three and five, where novel features emerge. In both cases we are able to construct Kodaira deformations of the flat models whose moduli spaces possess Galilean structures with torsion. In five dimensions we find that the canonical affine connection induced on the moduli space can possess anti-self-dual generalised Coriolis forces. We give examples of anti-self-dual Ricci-flat manifolds whose twistor spaces contain rational curves whose normal bundles suffer jumps to O(2 - k) + O(k) for arbitrarily large integers k, and we construct maps which portray these big-jumping twistor spaces as the resolutions of singular twistor spaces in canonical Gibbons-Hawking form. For k > 3 the moduli space itself is singular, arising as a variety in an ambient complex space. We explicitly construct Newtonian twistor spaces suffering similar jumps. Finally we prove several theorems relating the first-order and higher-order symmetry operators of the Schrödinger equation to tensors on Newton-Cartan backgrounds, defining a Schrödinger-Killing tensor for this purpose. We also explore the role of conformal symmetries in Newtonian twistor theory in three, four, and five dimensions.

We first consider four-dimensional gauge theory on twistor space, taking as a case study maximally supersymmetric Yang-Mills theory. Using a twistor action functional, we show that gauge theory scattering amplitudes are naturally computed on twistor space in a manner that is much more efficient than traditional space-time Lagrangian techniques at tree-level and beyond. In particular, by rigorously studying the Feynman rules of a gauge-fixed version of the twistor action, we arrive at the MHV formalism. This provides evidence for the naturality of computing scattering amplitudes in twistor space as well as an alternative proof of the MHV formalism itself. Next, we study other gauge theory observables in twistor space including gauge invariant local operators and Wilson loops, and discuss how to compute their expectation values with the twistor action. This enables us to provide proofs for the supersymmetric correlation function / Wilson loop correspondence as well as conjectures on mixed Wilson loop - local operator correlators at the level of the loop integrand. Furthermore, the twistorial formulation of such observables is naturally algebro-geometric; this leads to novel recursion relations for computing mixed correlators by performing BCFW-like deformations of the observables in twistor space. Finally, we apply these twistor actions to gravity. Using the on-shell equivalence between Einstein and conformal gravity in de Sitter space, we argue that the twistor action for conformal gravity should encode the tree-level graviton scattering amplitudes of Einstein's theory. We prove this in terms of generating functionals, and derive the flat space MHV amplitude as well as a recursive version of the MHV amplitude with cosmological constant. We also include some discussion of super-connections and Coulomb branch regularization on twistor space.

The correspondence of stationary, axisymmetric, asymptotically flat space-times and bundles over a reduced twistor space has been established in four dimensions. The main impediment for an application of this correspondence to examples in higher dimensions is the lack of a higher-dimensional equivalent of the Ernst poten- tial. This thesis will propose such a generalized Ernst potential, point out where the rod structure of the space-time can be found in the twistor picture and thereby provide a procedure for generating solutions to the Einstein field equations in higher dimensions from the rod structure, other asymptotic data, and the requirement of a regular axis. Examples in five dimensions are studied and necessary tools are developed, in particular rules for the transition between different adaptations of the patching matrix and rules for the elimination of conical singularities.

La relativité générale représente la description la plus précise de l'interaction gravitationnelle. Cependant, alors que la matière est régie par les lois de la mécanique quantique, la gravitation, elle, est une théorie fondamentalement classique. A l'échelle de Planck, c'est-à-dire à des distances d'environ 10E-35 mètres, les effets quantiques et ceux de la gravitation deviennent tous deux importants. A l'heure actuelle, un langage mathématique unifié et décrivant les effets physiques à cette échelle est toujours manquant. Il existe néanmoins plusieurs théories candidates à cette description, et l'une d'entre elles, la gravité quantique à boucles, est l'objet d'étude de cette thèse.Afin de tester si une théorie candidate peur fournir une description appropriée des propriétés quantiques du champ de gravitation, elle doit présenter une certaine cohérence interne du point de vue mathématique, et aussi être en accord avec les tests expérimentaux de la relativité générale. Le but de cette thèse est de développer certains outils mathématiques qui éclairent ces conditions de consistance interne, et qui permettent d'établir un lien entre différentes formulations de la théorie. / General relativity is the most precise theory of the gravitational interaction. It is a classical field theory. All matter, on the other hand, follows the rules of quantum theory. At the Planck scale, at about distances of the order of 10E-35 meters, both theories become equally important. Today, theoretical physics lacks a unifying language to explore what happens at this scale, but there are several candidate theories available. Loop quantum gravity is one them, and it is the main topic of this thesis. To see whether a particular proposal is a viable candidate for a quantum theory of the gravitational field it must be free of internal inconsistencies, and agree with all experimental tests of general relativity. This thesis develops mathematical tools to check these.