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This thesis presents an overview of recent developments in the applications of twistor theory to the study of the gravitational S-matrix in flat as well as curved space-times. We begin by introducing the novel geometric tool of twistor sigma models. These are two-dimensional chiral sigma models governing holomorphic maps from the Riemann sphere into twistor spaces of self-dual vacuum space-times. What follows is a concise list of their main highlights, in the order that the reader will encounter them: i) Solutions to the equations of motion of our sigma models provide the incidence relations of the twistor correspondence. ii) On-shell actions of our models compute Kahler potentials for hyperkahler metrics on the associated space-times. iii) Connected, tree-level correlators of our models give rise to tree-level, maximally helicity violating (MHV) graviton amplitudes of general relativity in flat space. iv) The chiral algebra of operators in our models enjoys an action of the loop algebra of the wedge subalgebra of w(1+infinity). This is associated with the soft sector of celestial holography. v) By coupling them to background twistor spaces of self-dual, vacuum space-times, our models can be used to derive tree-level MHV graviton amplitudes in self-dual, radiative space-times. The self-dual space-times mentioned in the last point provide an ideal laboratory for studying amplitudes in curved backgrounds. The corresponding formulae for MHV graviton amplitudes are built out of Hodges' determinants familiar from flat space, but also exhibit exciting new structures like gravitational wave tails.

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.

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.

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.

Building off senior and master thesis work, we present a class of topological plasma configurations characterized by their toroidal and poloidal winding numbers, $n_t$ and $n_p$, respectively. The case of $n_t=1$ and $n_p=1$ corresponds to the Kamchatnov-Hopf soliton, a magnetic field configuration everywhere tangent to the fibers of a Hopf fibration so that the field lines are circular, linked precisely once, and form the surfaces of nested tori. We show that for $n_t \in \mathbb{Z}^+$ and $n_p=1$, these configurations represent stable, localized solutions to the magnetohydrodynamic equations for an ideal incompressible fluid with infinite conductivity. Furthermore, we extend our stability analysis by considering a plasma with finite conductivity and estimate the soliton lifetime in such a medium as a function of the toroidal winding number. Torus knot topology is also inherent in electromagnetic and gravitational radiation. We show this by constructing spin-$N$ fields based on the elementary states of twistor theory. The twistor functions corresponding to the elementary states admit a parameterization in terms of the torus knots' poloidal and toroidal winding numbers, allowing one to choose the degree of linking or knotting of the associated field configuration. We describe the topology of the gravitational fields and their physical interpretation in terms of their tidal and frame drag phenomenology. Using the gravito-electromagnetic formalism, we show that the torus knot structure is exhibited in the tendex and vortex lines for the analogous linearized gravitational solutions. We extend the definition of hopfions to include a larger class of spin-$h$ fields and use this to classify the electromagnetic and gravitational hopfions of different algebraic types. The fields are constructed through the Penrose contour integral transform; thus, the singularities of the generating functions are directly related to the geometry of the resulting physical fields. We discuss this relationship and how the topological structure of the fields is related to the Robinson congruence. Since the topology appears in the lines of force for both electromagnetism and gravity, the gravito-electromagnetic formalism is used to analyze the gravitational hopfions and describe the time evolution of their tendex and vortex lines. We thus obtain similar configurations based on the same topological structure but varying spin-$h$ algebraic types. The null and type N fields propagate at the speed of light, while the non-null and type D fields radiate energy outward from the center. Finally, we discuss the type III gravitational hopfion, which has no direct electromagnetic analog, but find that it still exhibits some characteristic features common to the other hopfion fields.In the complete non-linear theory we present an exact solution of Einstein's equation that describes the gravitational shockwave of a massless particle on the horizon of a Kerr-Newman black hole—generalizing the Dray-'t~Hooft solution to the case of a rotating background. The back-reacted metric is of the generalized Kerr-Schild form and is Type II in the Petrov classification. We show that if the background tetrad is aligned with shear-free null geodesics, and the background Ricci tensor satisfies a simple condition, all nonlinearities in the perturbation will drop out of the curvature scalars. We reformulate Einstein-Hilbert gravity in the first-order formalism of Elie Cartan and find that the Riemann curvature tensor is the Yang-Mills gauge curvature of a local Lorentz group gauge connection. We derive and make heavy use of the method of spin coefficients (the Newman-Penrose formalism) in its compacted form (the Geroch-Held-Penrose formalism).

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.

It is shown that Hp-1 (P+, 0 (-m-p)) is a Fréchet space, and its dual is Hq-1(P-, 0 (m-q)), where P+ and P- are the projectivizations of subsets of generalized twistor space (≌ ℂp-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 Hp-1(P+, 0(-m-p)). A supermanifold is a ringed space consisting of an underlying classical manifold and an augmented sheaf of Z2-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 comprises three sections. In the first, real space-times admitting a solution to the two-index twistor (Killing spinor) equation are constructed and the separability properties of solutions to zero rest-mass field equations within these space-times are derived. In addition conditions for differential relations between components of opposite helicity are derived. In the second section class of half algebraically-special spaces is investigated, a class which may be characterised as those admitting a certain partial two-index twistor (Killing spinor). A formalism based on their structure as a two-dimensional family of flat null two-planes is developed and used to provide an explicit integration of the curvature equations, to discuss Hertz-type potentials for zero rest-mass fields and to examine a class of Einstein-Maxwell spaces and their perturbations. In the third section we examine the relation between the space of solutions to the twistor equation restricted to a general space--like two-surface of spherical topology (two-surface twistor space) and its dual. This is then used to discuss the nature of the local embedding of the two-surface in a space conformally related to complex Minkowski space.