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In this thesis, we discuss a geometric construction analogous to the Ward correspondence for the KP equations. We propose a Dirac operator based on the inverse scattering transform for the KP-II equation and discuss the similarities and differences to the Ward correspondence. We also consider the KP-I equation, describing a geometric construction for a certain class of solutions. We also discuss the general inverse scattering of the equation, how this is related to the KP-II equation and the problems with describing a single geometric construction that incorporates both equations. We also consider the Davey-Stewartson equations, which have a similar behaviour. We demonstrate explicitly the problems of localising the theory with generic boundary conditions. We also present a reformulation of the Dirac operator and demonstrate a duality between the Dirac operator and the first Lax operator for the DS-II equations. We then proceed to generalise the Dirac operator construction to generate other integrable systems. These include the mKP and Ishimori equations, and an extension to the KP and mKP hierarchies.

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.

In this thesis, I study the relativistic effects of matter inhomogeneities on the accelerating expansion of the Universe. The acceleration is often taken to be caused by the presence of an exotic fluid called Dark Energy, or else a non-zero 'cosmological constant' term in the field equations of General Relativity. I consider whether this result could instead be an artefact caused by using an incorrect model to interpret observations. The standard 'concordance' cosmological model assumes the Cosmological Principle, which states that the matter distribution on large scales is homogeneous. One possibility is that correction terms appear in the field equations when small-scale inhomogeneities are smoothed over to produce this homogeneous model. These 'backreaction' effects could affect the dynamics of the spacetime, causing an apparent acceleration. I clarify the relationship between acceleration of the averaged spacetime and acceleration inferred from observable quantities, and show that they are closely related in statistically-homogeneous spacetimes. Another possibility is that the Universe could be inhomogeneous on large scales. If there was a large ‘void’, with us at the centre, the lensing of light by the void could reproduce the observations that imply cosmic acceleration. I show that a popular class of void models, based on spherically-symmetric Lemaitre-Tolman-Bondi spacetimes, are unable to simultaneously fit a selection of observational data, thus effectively ruling-out this possibility. These data include the Kinematic Sunyaev-Zel'dovich (KSZ) effect, which is a distortion/shift of the Cosmic Microwave Background (CMB) frequency spectrum caused by the Compton scattering of photons by hot gas in galaxy clusters. This, and other distortions of the CMB frequency spectrum, are sensitive to the degree of anisotropy in the CMB about a scattering cluster. I suggest tests involving these observables that exploit the strong link between isotropy and homogeneity to (a) distinguish between different causes of a deviation from spatial flatness on the horizon scale, and (b) potentially confirm the Cosmological Principle using observations. Finally, I describe a novel Bayesian CMB component separation method for extracting the Sunyaev-Zel'dovich signal of clusters from CMB sky maps.

In the first part of this thesis, we discuss some aspects of the four-dimensional N = 2 superconformal index of theories of class S. We first consider a generalized index on S¹ × S³/Z_r, and prove S-duality in a particular fugacity slice. We then go on to study the (round) superconformal index in the presence of surface defects. We develop a systematic prescription to compute surface defects labeled by arbitrary irreducible representations of the gauge group and subject those defects to various tests in several different limits. Each of these limits is interesting in its own right, and we go on to explore them in some depth. In the second part of this thesis, we construct the gravity duals of large N supersymmetric gauge theories defined on squashed five-spheres with SU(3) × U(1) symmetry. The gravity duals are constructed in Euclidean Romans F(4) gauged supergravity in six- dimensions, and uplift to massive type IIA supergravity. We compute the partition function and Wilson loop in the large N limit of the gauge theory and compare them to their corresponding supergravity dual quantities. As expected from AdS/CFT, both sides agree perfectly. Based on these results, we conjecture a general formula for the partition function and Wilson loop on any five-sphere background, which for fixed gauge theory depends only on a certain supersymmetric Killing vector. We then go on to construct rigid supersymmetric gauge theories on more general Riemannian five-manifolds. We follow a holographic approach, realizing the manifold as the conformal boundary of the six-dimensional bulk supergravity solution. This leads to a systematic classification of five-dimensional supersymmetric backgrounds with gravity duals.

Submitted by ADRIANA VICTORIA ARAUJO SALCEDO null (adrianaaraujo@ift.unesp.br) on 2017-09-27T12:24:25Z No. of bitstreams: 1 dissertacao.pdf: 919386 bytes, checksum: 9947da9ce880a14adcc3e1ac56f121e6 (MD5) / Rejected by Monique Sasaki (sayumi_sasaki@hotmail.com), reason: Solicitamos que realize uma nova submissão seguindo as orientações abaixo: No campo “Versão a ser disponibilizada online imediatamente” foi informado que seria disponibilizado o texto completo porém no campo “Data para a disponibilização do texto completo” foi informado que o texto completo deverá ser disponibilizado apenas 6 meses após a defesa. Caso opte pela disponibilização do texto completo apenas 6 meses após a defesa selecione no campo “Versão a ser disponibilizada online imediatamente” a opção “Texto parcial”. Esta opção é utilizada caso você tenha planos de publicar seu trabalho em periódicos científicos ou em formato de livro, por exemplo e fará com que apenas as páginas pré-textuais, introdução, considerações e referências sejam disponibilizadas. Se optar por disponibilizar o texto completo de seu trabalho imediatamente selecione no campo “Data para a disponibilização do texto completo” a opção “Não se aplica (texto completo)”. Isso fará com que seu trabalho seja disponibilizado na íntegra no Repositório Institucional UNESP. Por favor, corrija esta informação realizando uma nova submissão. Agradecemos a compreensão. on 2017-09-28T13:58:32Z (GMT) / Submitted by ADRIANA VICTORIA ARAUJO SALCEDO null (adrianaaraujo@ift.unesp.br) on 2017-09-28T20:52:46Z No. of bitstreams: 1 dissertacao.pdf: 919386 bytes, checksum: 9947da9ce880a14adcc3e1ac56f121e6 (MD5) / Approved for entry into archive by Monique Sasaki (sayumi_sasaki@hotmail.com) on 2017-09-29T18:18:48Z (GMT) No. of bitstreams: 1 salcedo_ava_dr_ift.pdf: 919386 bytes, checksum: 9947da9ce880a14adcc3e1ac56f121e6 (MD5) / Made available in DSpace on 2017-09-29T18:18:48Z (GMT). No. of bitstreams: 1 salcedo_ava_dr_ift.pdf: 919386 bytes, checksum: 9947da9ce880a14adcc3e1ac56f121e6 (MD5) Previous issue date: 2017-08-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Due to the existence of an invariant length at the Planck scale, Einstein special relativity breaks down at that scale. A possible solution to this problem is arguably to replace the Poincare-invariant Einstein special relativity by a de Sitter invariant special relativity. Such replacement produces concomitant changes in all relativistic theories, including of course general relativity, which changes to what is called de Sitter modified general relativity, whose gravitational field equation is the de Sitter modified Einstein equation. A crucial property of this theory is that both the background de Sitter curvature and the gravitational dynamical curvature turns out to be included in the same curvature tensor. This means that the cosmological term Λ no longer explicitly appears in Einstein equation, and is consequently not restricted to be constant. In the first part of the thesis, a new definition for black hole entropy is defined. This new notion of entropy is strongly attached to the local symmetry, given the fact to be composed of two parts: the usual translational-related entropy plus an additional piece related to the proper conformal transformation. Also, it is obtained the de Sitter modified Schwarzschild solution, and using this solution we explore the consequences for the definition of entropy, as well as for the thermodynamics of the Schwarzschild-de Sitter system. In the second part the Newtonian limit of the de Sitter modified Einstein equation is obtained, and the ensuing Newtonian Friedmann equations are show to provide a good account of the dark energy content of the present-day universe. Finally, by using the same Newtonian limit, the circular velocity of stars around the galactic center is studied. It is shown that the de Sitter modified Newtonian force, which becomes effective only in the Keplerian region of the galaxy, could possibly explain the flat rotation curve of galaxies without necessity of supposing the existence of dark matter. / Devido a existência de um comprimento invariante na escala de Planck, a relatividade especial de Einstein deixa de ser valida naquela escala. Uma solução possível para esse problema é trocar a relatividade especial de Einstein, a qual tem o grupo de Poincaré como grupo de simetria, por uma relatividade especial invariante sob o grupo de de Sitter. Essa troca ira produzir mudanças concomitantes em todas as teorias relativísticas, incluindo naturalmente a teoria da relatividade geral. Essa teoria da origem ao que denominamos de Sitter modified general relativity, cuja equação para o campo gravitacional foi chamada de de Sitter modified Einstein equation. Uma propriedade crucial dessa teoria é que tanto a curvatura de fundo de de Sitter como a curvatura dinâmica da gravitação estão ambas incluídas no mesmo tensor de curvatura. Isso significa que o termo cosmológico Λ não aparece explicitamente na equação de Einstein, e consequentemente não é restrito a ser uma constante. Trabalhando no contexto da de Sitter modified general relativity, na primeira parte da tese, uma nova definição de entropia para buraco negro é definido. Esta nova noção de entropia está fortemente ligado à simetria local, dado o fato de ser composto por duas partes: uma associada as translação e uma parte adicional relacionada com a transformação conformal. Assim mesmo, nós obtemos a solução de Schwarzschild modificada por de Sitter. Usando essa solução exploramos as consequências para a definição de entropia, bem como para a termodinâmica do sistema de Schwarzschild-de Sitter. Na segunda parte da tese obtemos o limite Newtoniano da de Sitter modified Einstein equation, e usamos as correspondentes equações de Friedmann Newtonianas para estudar o problema da energia escura. Mostramos que essas equações fornecem uma solução bastante razoável para a existência de energia escura do universo atual. Finalmente, usamos o mesmo limite Newtoniano para estudar a velocidade circular de estrelas ao redor do núcleo galáctico. Mostramos que a força Newtoniana modificada por de Sitter, a qual torna-se ativa apenas na região Kepleriana da galáxia, pode explicar as curvas de rotação planas sem necessidade de supor a existência de matéria escura. / CAPES: 33015015001P7

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.