In this thesis, the implications of the existence of Killing spinors in a spacetime are investigated. In particular, it is shown that in vacuum and electrovacuum spacetimes a Killing spinor, along with some assumptions on the associated Killing vector in an asymptotic region, guarantees that the spacetime is locally isometric to a member of the Kerr or Kerr-Newman family. It is shown that the characterisation of these spacetimes in terms of Killing spinors is an alternative expression of characterisation results of Mars (Kerr) and Wong (Kerr-Newman) involving restrictions on the Weyl curvature and matter content. In the next section, the construction of a geometric invariant characterising initial data for the Kerr-Newman spacetime is described. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr-Newman initial data, and so characterises this type of data. First, the characterisation of the Kerr-Newman spacetime in terms of Killing spinors is illustrated. The space spinor formalism is then used to obtain a set of four independent conditions on an initial Cauchy hypersurface that guarantee the existence of a Killing spinor on the development of the initial data. Following a similar analysis in the vacuum case, the properties of solutions to the approximate Killing spinor equation are studied, and used to construct the geometric invariant. Finally, the problem of Killing spinor initial data in the characteristic problem is investigated. It is shown that data need only be speci ed on the bifurcation surface of the two intersecting null hypersurfaces in order to guarantee the existence of a Killing spinor in a neighbourhood of the bifurcation surface. This characterises the class of spacetimes known as distorted black holes, which include but is strictly larger than the Kerr family of spacetimes.

We give a review of the construction and application of spinor fields in general relativity and an account of the spinor-based Geroch-Held-Penrose (GHP) formalism. Specifically, we discuss using the GHP formalism to integrate Einstein's equations as suggested by Held  and developed by Edgar and Ludwig and discuss the similaritites with the Cartan-Karlhede classification of spacetimes. We use this integration method to find a one-parameter subclass and a degenerate case, for which the Cartan-Karlhede algorithm terminates at second order, of the Petrov type III, vacuum Robinson-Trautman metrics. We use the GHP formalism to find the Killing vectors, using theorems by Edgar and Ludwig. The one-parameter family admits exactly two Killing fields, whereas the degenerate case admits three and is Bianchi type VI. Finally we use the Cartan-Karlhede algorithm to show that our class, including the degenerate case, is equivalent to a subclass found by Collinson and French. Our degenerate case corresponds to an example metric given by Robinson and Trautman and is known to be the unique algebraically special vacuum spacetime with diverging rays and a three-dimensional isometry group.

In this thesis we extend Lerman’s cutting construction to spin-c structures. Every spin-c structure on an even-dimensional Riemannian manifold gives rise to a Dirac operator D+ acting on sections of the associated spinor bundle. The spin-c quantization of a spin-c manifold is defined to be ker(D+)−coker(D+). It is a virtual vector space, and in the presence of a Lie group action, it is a virtual representation. In 2004, Guillemin et al defined signature quantization and showed that it is additive under cutting. We prove that the spin-c quantization of an S^1-manifold is also additive under cutting. Our proof uses the method of localization, i.e., we express the spin-c quantization of a manifold in terms of local data near connected components of the fixed point set. For a symplectic manifold (M,ω), a spin-c prequantization is a spin-c structure together with a connection compatible with ω. We explain how one can cut a spin-c prequantization and show that the choice of a spin-c structure on the complex plane (which is part of the cutting process) must be compatible with the moment level set along which the cutting is performed. Finally, we prove that the spin-c and metaplectic-c groups satisfy a universal property: Every structure that makes the construction of a spinor bundle possible must factor uniquely through a spin-c structure in the Riemannian case, or through a metaplectic-c structure in the symplectic case.

In this thesis we extend Lerman’s cutting construction to spin-c structures. Every spin-c structure on an even-dimensional Riemannian manifold gives rise to a Dirac operator D+ acting on sections of the associated spinor bundle. The spin-c quantization of a spin-c manifold is defined to be ker(D+)−coker(D+). It is a virtual vector space, and in the presence of a Lie group action, it is a virtual representation. In 2004, Guillemin et al defined signature quantization and showed that it is additive under cutting. We prove that the spin-c quantization of an S^1-manifold is also additive under cutting. Our proof uses the method of localization, i.e., we express the spin-c quantization of a manifold in terms of local data near connected components of the fixed point set. For a symplectic manifold (M,ω), a spin-c prequantization is a spin-c structure together with a connection compatible with ω. We explain how one can cut a spin-c prequantization and show that the choice of a spin-c structure on the complex plane (which is part of the cutting process) must be compatible with the moment level set along which the cutting is performed. Finally, we prove that the spin-c and metaplectic-c groups satisfy a universal property: Every structure that makes the construction of a spinor bundle possible must factor uniquely through a spin-c structure in the Riemannian case, or through a metaplectic-c structure in the symplectic case.

Apresentamos uma revisão dos elementos básicos do estudo da teoria clássica das supercordas em backgrounds planos e curvos, dando ênfase ao caso importante em que o background e a variedade AdS5 S5. Nós inclumos um estudo da corda bosonica para revisarmos alguns conceitos básicos da teoria de campos conforme em duas dimensões. Em seguida estudamos a teoria das supercordas em um espaço plano onde apresentamos uma introdução pedagógica ao formalismo de espinores puros. A ultima parte e dedicada a generalização da ação de Green-Schwarz para o caso de AdS5 S5 e uma apresentação do modelo sigma do formalismo de espinores puros no mesmo background. / We present a review of the basic elements of the study of classical superstring theory in at and curved backgrounds, giving emphasis to the very important case of the AdS5S5 background. We include a study of the bosonic string to review some basic concepts of two dimensional conformal eld theory. We then move on to the superstring in at space where we present a pedagogical introduction to the pure spinor formalism of superstrings. The last part is devoted to the generalization of the Green-Schwarz action to AdS5 S5 and a presentation of the pure spinor sigma model in the same background.

Apresentamos uma revisão dos elementos básicos do estudo da teoria clássica das supercordas em backgrounds planos e curvos, dando ênfase ao caso importante em que o background e a variedade AdS5 S5. Nós inclumos um estudo da corda bosonica para revisarmos alguns conceitos básicos da teoria de campos conforme em duas dimensões. Em seguida estudamos a teoria das supercordas em um espaço plano onde apresentamos uma introdução pedagógica ao formalismo de espinores puros. A ultima parte e dedicada a generalização da ação de Green-Schwarz para o caso de AdS5 S5 e uma apresentação do modelo sigma do formalismo de espinores puros no mesmo background. / We present a review of the basic elements of the study of classical superstring theory in at and curved backgrounds, giving emphasis to the very important case of the AdS5S5 background. We include a study of the bosonic string to review some basic concepts of two dimensional conformal eld theory. We then move on to the superstring in at space where we present a pedagogical introduction to the pure spinor formalism of superstrings. The last part is devoted to the generalization of the Green-Schwarz action to AdS5 S5 and a presentation of the pure spinor sigma model in the same background.

Orientador: Julio Marny Hoff da Silva / Resumo: Espinores exóticos surgem quando a topologia da variedade $M$ tomada como sendo o espaço-tempo é suposta ser não-trivial, no sentindo que seu grupo fundamental é não-trivial: $\pi_1(M) \neq 0$. Assim, um novo termo exótico $\partial_\mu \theta$ surge na equação dinâmica destes espinores, e novas propriedades se apresentam. A não-trivialidade de $\pi_1(M)$ pode ser diretamente ligada a própria existência de buracos negros. Assim, estudamos, nesta tese, relações entre estruturas espinoriais exóticas e a taxa de emissão de radiação Hawking por buracos negros assintoticamente \textit{flat} em Relatividade Geral, encontrando equações diferenciais para o termo exótico, o que dá a possibilidade de inferir uma forma explícita para $\theta$. Também, tratamos aqui dos chamados espinores RIM, que são espinores que respeitam uma equação dinâmica não-linear chamada de equação não-linear de Heisenberg. Apresentamos dois lemas relativos a estes espinores: um deles encontrando restrições para ocorrer a decomposição de espinores de Dirac em termos de espinores RIM, e outro que nega a existência de espinores RIM exóticos, ou seja, relaciona a existência de espinores RIM a própria estrutura topológica do espaço-tempo. Ainda, encontramos um método de classificarmos os espinores RIM nas classes de Lounesto. Por fim, apresentamos, na forma de dois teoremas, maneiras de deformar homotopicamente tais espinores no que chamamos de \textit{spinor-plane}. / Abstract: Exotic spinors emerge when the topology associatd to the manifold $M$, which is token as being the spacetime, is suppose to be non-trivial, in the sense that its fundamental group is non-trivial: $\pi_1(M) \neq 0$. Thus, a new exotic term $\partial_\mu \theta$ rises from the dynamical equation related to these spinors, and new properties are in order. The non-triviality of $\pi_1(M)$ may be directly linked to the very existence of black holes. In this vein, we study some relations between exotic spinorial structures and the Hawking radiation emission rate by asymptotically flat black holes solutions of General Relativity, finding an equation from which an explicity form for the exotic term could be inferred. Moreover, we work on the so-called RIM spinors, which are spinor fields satisfying a non-linear dynamical equation known as Heiseing non-linear equation. We present two \textit{lemmata} related to these spinors: one of them gives us restrictions for the decompostion of Dirac fields in terms of RIM spinors to occur, while the other deny the existence of exotic RIM spinors, i.e., it relates the very existence of RIM spinors to the spacetime topological structure. Besides, we develop a classifying method for RIM spinors into the Lounesto classes. Finally, we present, in the form of two theorems, ways to homotopically deform such spinors in what we call the spinor-plane. / Doutor

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This thesis reviews the method of Mafra, Schlotterer, and Stieberger (2011) for computing the full colour ordered N-point open superstring amplitude using the Pure Spinor formalism. We introduce relevant elements of super Yang-Mills theory and examine the basics of the Pure Spinor formalism, with a focus on tools for amplitude computation. We then define a series of objects with increasingly useful BRST transformation properties, which greatly simplify the calculations, and show how these properties can be determined using a diagrammatic method. Finally, we use the explicit four- and five-point amplitude computations as stepping stones to compute the general N-point amplitude, which factors into a set of kinematic integrals multiplying SYM subamplitudes.

Superspace is a fundamental tool in the study of supersymmetry, one that while often used is seldom defined with a proper amount of mathematical rigor. This paper examines superspace and presents three different constructions of it; the original by Abdus Salam and J. Strathdee as well as two modern methods by Alice Rogers and Buchbinder-Kuzenko.Though the structures arrived at are the same the two modern constructions differ in methods, elucidating different important aspects of super-space. Rogers focuses on the underlying structure through the study of supermanifolds, and Buchbinder-Kuzenko the direct correlation with the Poincare superalgebra, and the parametrisation in terms of exponents.