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Kaluza-Klein theories generally in $(4+D)$ and more specifically in five dimensions are reviewed. The magnetic monopole solutions found in the Kaluza-Klein theories are generally reviewed and their generalizations to Anti-de Sitter spacetimes are discussed.

In this thesis, we study various aspects of physics in higher-dimensional manifolds involving a single extra dimension. After giving some historical perspective on the motivation for studying higher-dimensional theories of physics, we describe classical tests for a non-compact extra dimension utilizing test particles and pointlike gyroscopes. We then turn our attention to the problem of embedding any given <i>n</i>-dimensional spacetime within an (<i>n</i>+1)-dimensional manifold, paying special attention to how any structure from the extra dimension modifies the standard <i>n</i>-dimensional Einstein equations. Using results derived from this investigation and the formalism derived for test particles and gyroscopes, we systematically introduce three specific higher-dimensional models and classify their properties; including the Space-Time-Matter and two types of braneworld models. The remainder of the thesis concentrates on specific higher-dimensional cosmological models drawn from the above mentioned scenarios; including an analysis of the embedding of Friedmann-Lemaitre-Robertson-Walker submanifolds in 5-dimensional Minkowski and topological Schwarzschild spaces, and an investigation of the dynamics of a <i>d</i>-brane that takes the form of a thin shell encircling a (<i>d</i>+2)-dimensional topological black hole in anti-deSitter space. The latter is derived from a finite-dimensional action principle, which allows us to consider the canonical quantization of the model and the solutions of the resulting Wheeler-DeWitt equation.

In this thesis, we study various aspects of physics in higher-dimensional manifolds involving a single extra dimension. After giving some historical perspective on the motivation for studying higher-dimensional theories of physics, we describe classical tests for a non-compact extra dimension utilizing test particles and pointlike gyroscopes. We then turn our attention to the problem of embedding any given <i>n</i>-dimensional spacetime within an (<i>n</i>+1)-dimensional manifold, paying special attention to how any structure from the extra dimension modifies the standard <i>n</i>-dimensional Einstein equations. Using results derived from this investigation and the formalism derived for test particles and gyroscopes, we systematically introduce three specific higher-dimensional models and classify their properties; including the Space-Time-Matter and two types of braneworld models. The remainder of the thesis concentrates on specific higher-dimensional cosmological models drawn from the above mentioned scenarios; including an analysis of the embedding of Friedmann-Lemaitre-Robertson-Walker submanifolds in 5-dimensional Minkowski and topological Schwarzschild spaces, and an investigation of the dynamics of a <i>d</i>-brane that takes the form of a thin shell encircling a (<i>d</i>+2)-dimensional topological black hole in anti-deSitter space. The latter is derived from a finite-dimensional action principle, which allows us to consider the canonical quantization of the model and the solutions of the resulting Wheeler-DeWitt equation.

In Part I we are dealing with effective description of Yang-Mills theories based on gauge-invarint variables. For pure Yang-Mills we study the spin-charge separation varibles. The dynamics in these variables resembles the Skyrme-Faddeev model. Thus the spin-charge separation is an important intermediate step between the fundamental Yang-Mills theory and the low-energy effective models, used to model the low-energy dynamics of gluons. Similar methods may be useful for describing the Electroweak sector of the Standard Model in terms of gauge-invariant field variables called supercurrents. We study the geometric structure of spin-charge separation in 4D Euclidean space (paper III) and elaborate onconnection with gravity toy model. Such reinterpretation gives a way to see how effective flat background metric is created in toy gravity model by studying the appearance of dimension-2 condensate in the Yang-Mills (paper IV). For Electroweak theory we derive the effective gauge-invariant Lagrangian by doing the Kaluza-Klein reduction of higher-dimensional gravity with 3-brane, thus making explicit the geometric interpretation for gauge-invariant supercurrents. The analogy is then made more precise in the framework of exact supergravity solutions. Thus, we interpret the Higgs effect as spontaneous breaking of Kaluza-Klein gauge symmetry and this leads to interpretation of Higgs field as a dilaton (papers I and II). In Part II of the thesis we study rather simple field theories, called “geometric” or “instantonic”. Their defining property is exact localization on finite-dimensional spaces – the moduli spaces of instantons. These theories allow to account exactly for non-linearity of space of fields, in this respect they go beyond the standard Gaussian perturbation theory. In paper V we show how to construct a geometric theory of chiral boson by embedding it into the geometric field theory. In Paper VI we elaborate on the simplest geometric field theory – the supersymmetric Quantum Mechanics and construct new non-perturbative topological observables that have a transparent meaning both in geometric and in the Hamiltonian formalisms. In Paper VII we are motivated by making perturbations away from the simple instantonic limit. For that we need to carefully define the observables that are quadratic in momenta and develop the way to compute them in geometric framework. These correspond geometrically to bivector fields (or, in general, the polyvector fields). We investigate the local limit of polyvector fields and compare the geometric calculation with free-field approach.

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Diese Doktorarbeit behandelt die bosonische Einbettung der geeichten N = 8 Supergravitation in elf Dimensionen. Die höher dimensionalen Felder müssen zuerst nichtlinear umdefiniert werden, sodass ihre supersymmetrischen Transformationen mit denen der vierdimensionalen Felder verglichen werden können. So wurden in der Literatur nichtlineare Beziehungen zwischen den neu definierten elfdimensionalen Feldern und den Feldern der N = 8 Supergravitation gefunden. Darauf basierend können nun direkte Ansätze gefunden werden, die eine vierdimensionale in eine elfdimensionale Lösung der Supergravitation einbetten. Die Arbeit präsentiert alle Ansätze für die skalaren internen Felder. Zuerst werden die schon bekannten Einbettungsformeln für die inverse Metrik, das Dreiform-Potential mit gemischter Indexstruktur sowie das Sechsform-Potential zusammengefasst. Danach werden neue Ansätze für die explizite interne Metrik, das vollständige Dreiform-Potential, den Warp Faktor, die Vierform Feldstärke sowie den Freund-Rubin Faktor gefunden. Die Einbettung der Vektorbosonen hängt dann nur von den skalaren Feldern ab. Der zweite Teil der Arbeit benutzt die gefundenen Einbettungsformeln, um gruppeninvariante Lösungen der elfdimensionalen Supergravitation zu finden. In solchen Fällen hängen die höherdimensionalen Felder ausschließlich von speziellen gruppeninvarianten Tensoren ab, die auf die jeweilige interne Geometrie angepasst sind. Als Beispiel wird zuerst die schon bekannte Einbettung der G2 invarianten Supergravitation zusammengefasst. Dann wird eine neue SO(3)×SO(3) invariante Löung der elfdimensionalen Supergravitation gefunden. Schließlich wird die Konsistenz der gefundenen Lösungen für eine maximal symmetrische Raumzeit überprüft. Die Ergebnisse können auf andere Kompaktifizierungen verallgemeinert werden, z.B. auf die nichtkompakten CSO(p,q,r) Eichungen oder auf die Reduzierung der Typ IIB Supergravitation zu fünf Dimensionen. / This thesis presents the complete embedding of the bosonic section of gauged N = 8 supergravity into 11 dimensions. The fields of 11-dimensional supergravity are reformulated in a non-linear way, such that their supersymmetry transformations can be compared to the four-dimensional ones. In this way, non-linear relations between the redefined higher-dimensional fields and the fields of N = 8 supergravity were already found in the literature. This is the basis for finding direct uplift Ansätze for the bosonic fields of 11-dimensional supergravity in terms of the four-dimensional ones. This work gives the scalar Ans¨atze for the internal fields. First, the well known uplift formulae for the inverse metric, the three-form potential with mixed index structure and the six-form potential are summarized. Secondly, new embedding formulae for the explicit internal metric, the full three-form potential and the warp factor are presented. Additionally, two subsequent non-linear Ansätze for the full internal four-form field strength and the Freund-Rubin term are found. Finally, the vector uplift can simply be found in terms of the obtained scalar fields. The second part of this thesis uses the obtained embedding formulae in order to construct group invariant solutions of 11-dimensional supergravity. In such cases, the higher-dimensional fields can be written solely in terms of certain group invariant tensors that are adapted to the particular geometry of the internal space. Two such examples are discussed in detail. The first one is the well-known uplift of G2 gauged supergravity. Furthermore, a new SO(3)×SO(3) invariant solution of 11-dimensional supergravity is found. In particular, the consistency of both solutions is explicitly checked for a maximally symmetric spacetime. The results may be generalized to other compactifications, e.g. the non-compact CSO(p, q, r) gaugings or the reduction from type IIB supergravity to five dimensions.

Made available in DSpace on 2015-05-14T12:14:06Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1577961 bytes, checksum: c71d976d783495df566e0fa6baadf8ca (MD5) Previous issue date: 2009-08-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We start this work studying the appearance of geometric quantum phases as in the relativistic as in the non-relativistic quantum dynamics of a neutral particle with permanent magnetic and electric dipole moment which interacts with external electric and magnetic fields in the presence of linear topological defects. We describe the linear topological defects using the approach proposed by Katanaev and Volovich, where the topological defects in solids are described by line elements which are solutions of the Einstein's equations in the context of general relativity. We also analyze the in uence of non-inertial effects in the quantum dynamics of a neutral particle using two distinct reference frames for the observers: one is the Fermi-Walker reference frame and another is a rotating frame. As a result, we shall see that the difference between these two reference frames is in the presence/absence of dragging effects of the spacetime which makes its in uence on the phase shift of the wave function of the neutral particle. In the following, we shall use our study of geometric quantum phases to make an application on the Holonomic Quantum Computation, where we shall show a new approach to implement the Holonomic Quantum Computation via the interaction between the dipole moments of the neutral particle and external fields and the presence of linear topological defects. Another applications for the Holonomic Quantum Computation is based in the structure of the topological defects in graphene layers. In the presence of topological defects, a graphene layer shows two distinct phase shifts: one comes from the mix of Fermi points while the other phase shift comes from the topology of the defect. To provide a geometric description for each phase shift in the graphene layer, we use the Kaluza-Klein theory where we establish that the extra dimension describes the Fermi points in the graphene layer. Hence, we can implement the Holonomic Quantum Computation through the possibility to build cones and anticones of graphite in such way we can control the quantum uxes in graphene layers. In the last part of this work, we study the Landau quantization for neutral particles as in the relativistic dynamics and non-relativistic dynamics. In the non-relativistic dynamics, we study the Landau quantization in the presence of topological defects as in an inertial as in a non-inertial reference frame. In the relativistic quantum dynamics, we start our study with the Landau quantization in the Minkowisky considering two different gauge fields. At the end, we study the relativistic Landau quantization for neutral particles in the Cosmic Dislocation spacetime. / Neste trabalho estudamos inicialmente o surgimento de fases geometricas nas dinâmicas quânticas relativística e não-relativística de uma partícula neutra que possui momento de dipolo magnético e elétrico permanente interagindo com campos elétricos e magnéticos externos na presença de defeitos topológicos lineares. Para descrevermos defeitos topológicos lineares usamos a aproximação proposta por Katanaev e Volovich, onde defeitos lineares em sólidos são descritos por elementos de linha que são soluções das equações de Einstein no contexto da relatividade geral. Analisamos também a inuência de efeitos não-inerciais na dinâmica quântica de uma partícula neutra em dois tipos distintos de referenciais para os observadores: um é o referencial de Fermi-Walker e outro é um referencial girante. Vemos que a diferença entre dois referenciais está na presença/ausência de efeitos de arrasto do espaço-tempo que irá influenciar diretamente na mudança de fase na funçãao de onda da partícula neutra. Em seguida, usamos nosso estudo de fases geométricas para fazer aplicações na Computação Quântica Holonômica onde mostramos uma nova maneira de implementar a Computação Quântica Holonômica através da interação entre momentos de dipolo e campos externos e pela presença de defeitos topológicos lineares. Outra aplicação para a Computação Quântica Holonômica está baseada na estrutura de defeitos topológicos em um material chamado grafeno. Na presença de defeitos topológicos lineares, esse material apresenta duas fases quânticas de origens distintas: uma da mistura dos pontos de Fermi e outra da topologia do defeito. Para dar uma descrição geométrica para a origem de cada fase no grafeno usamos a Teoria de Kaluza-Klein, onde a dimensão extra sugerida por esta teoria descreve os pontos de Fermi no grafeno. Portanto, a implementação da Computação Quântica Holonômica no grafeno está baseada na possibilidade de construir cones e anticones de grafite de tal maneira que se possa controlar os fluxos quânticos no grafeno. Na última parte deste trabalho estudamos a quantização de Landau para partículas neutras tanto na dinâmica não-relativística quanto na dinâmica relativística. Na dinâmica não-relativítica, estudamos a quantização de Landau na presença de defeitos em um referecial inercial e, em seguida, em um referencial nãoo-inercial. Na dinâmica relativística, estudamos inicialmente a quantização de Landau no espaço-tempo plano em duas configurações de campos diferentes. Por fim, estudamos a quantização de Landau relativística para partículas neutras no espaço-tempo da deslocação cósmica.