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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.

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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.

Made available in DSpace on 2014-08-27T14:36:41Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-04-11Bitstream added on 2014-08-27T15:57:11Z : No. of bitstreams: 1 000777505.pdf: 1234881 bytes, checksum: 67337304770a8c89e87d9c2186ca05cc (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Escrevemos uma forma simplética para teorias com spin diferente de zero utilizando a descrição de twistors. Nossa abordagem segue das equações escritas por Penrose, e nossos resultados sugerem uma ação alternativa a teoria de Fronsdal. Diferenças entre as duas abordagens aparecem, muito embora elas sejam, essencialmente, equivalentes / We write a symplectic form on higher spin theories using twistor description. Our approach follows from Penrose’s equations, and our results seem to suggest an action di?erent from Fronsdal theory. Di?erences between the two approaches also appear, although they are, essentially, equivalent / CNPq: 130458/2012-0

Orientador: Andrey Yuryevich Mikhaylov / Banca:Vladimir Perchine / Banca: Eduardo Pontón Bayona / Resumo: Escrevemos uma forma simplética para teorias com spin diferente de zero utilizando a descrição de twistors. Nossa abordagem segue das equações escritas por Penrose, e nossos resultados sugerem uma ação alternativa a teoria de Fronsdal. Diferenças entre as duas abordagens aparecem, muito embora elas sejam, essencialmente, equivalentes / Abstract: We write a symplectic form on higher spin theories using twistor description. Our approach follows from Penrose's equations, and our results seem to suggest an action different from Fronsdal theory. Differences between the two approaches also appear, although they are, essentially, equivalent / Mestre

In this thesis, we are concerning about the Twistor theory, field originally motivated purely physically, although these days fully developed into the many fields of mathem- atics and physics. With its complexion Twistor theory influences algebraic geometry, Clifford analysis as well as the String theory or Theory of quantum gravity. In the thesis we describe the origin of twistors projective or not. Mathematical background to the twistor theory is covered in the first chapter, where we study Clifford algebras and their representations. In the first part of the second chapter we are describing non-projective twistors as representation elements of certain Spin-group, and we find the connection with the standard definition of non-projective twistors as a kernel of the twistor operator. In the last part of the second chapter, we create a space of pro- jective twistors and show its certain properties, especially its correspondence with the complexified compactified Minkowski spacetime.

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