La théorie supersymmetrique N = 2 de Yang-Mills est considérée pour tous les groupes de jauge classiques (SU(N), SO(N) et Sp(N)). Les expressions formelles pour l'action Wilsonienne effective pour (presque) tous les modèles compatible avec la condition de la liberté asymptotique sont obtenues. Les équations qui déterminent les courbes de Seiberg et Witten sont proposées. Dans quelque cas elles sont résolues. Il est montre que pour tous les modèles considères les corrections à l' instant on qui viennent de ces équations sont en accord avec les calculs directs. Ainsi elles sont en accord avec les calculs bases sur les courbes de Seiberg et Witten qui viennent de la théorie M. Il est montre donc que pour une grande classe de modèles les prédictions de la théorie M coïncident avec les calculs directs. Ceci est fait pour toutes les modèles considères au niveau des calculs à l' instant on. Pour quelques modèles ceci est fait au niveau des courbes algébriques.

In this thesis, we set up a framework to define knot invariants for each choice of a symmetric space. In order to address this task, we start by defining appropriate notions of singular bundles and singular connections for a given symmetric space. We can associate a moduli space to any singular bundle defined over a compact 4-manifold with possibly non-empty boundary. We study these moduli spaces and show that they enjoy nice properties. For example, in the case of the symmetric space SU(n)/SO(n) the moduli space can be perturbed to an orientable manifold. Although this manifold is not necessarily compact, we introduce a comapctification of it. We then use this moduli space for singular bundles defined over 4-manifolds of the form YxR to define knot invariants. In another direction we mimic the construction of Donaldson invariants to define polynomial invariants for closed 4-manifolds equipped with smooth action of Z/2Z. / Mathematics

Orientador: Jayme Vaz Jr / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-02T06:58:26Z (GMT). No. of bitstreams: 1 Hoefel_EduardoOuteiralCorrea_M.pdf: 3091537 bytes, checksum: f00ee4b0eba7ea00e03a3ba084791085 (MD5) Previous issue date: 2002 / Resumo: Nesta dissertação apresentamos uma descrição do formalismo matemático das teorias de gauge introduzindo os conceitos de grupos e álgebras de Lie, fibrados principais, conexões e curvatura. Em seguida introduzimos as álgebras de Clifford e os spinors, tais conceitos são utilizados no capítulo final onde apresenta-se algllmas de suas aplicações em teorias de gauge. Uma aplicação é dada pelas formas diferenciais assumindo valores em uma álgebra de Clifford: mostra-se como as formas de conexão e curvatura são dadas por formas a valores em álegebras de bivetores, estas últimas são as álgebras de Lie dos grupos Spin. Outra aplicação consiste em mostrar, usando o Teorema de Periodicidade das álgebras de Clifford, como algumas transformações conformes do espaço-tempo são dadas pela ação do grupo $pin(2,4) sobre paravetores ]R + ]R4,1. Finalizamos mostrando a construção de monopolos e instantons através do teorema de inversão para spinors de Pauli e Dirac, vistos como elementos de sub-álgebras pares de álgebras de Clifford, e a estreita relação deste teorema com as fibrações de Hopf, ilustrando a relação existente entre Topologia e Física / Abstract: This dissertation begins with a description of the mathematical formulation of gauge theories, introducing the concepts of Lie groups and Lie algebras, principal bundles, connection and curvature. Then, Clifford algebras and spinors are introduced. The final chapter presents some applications of Clifford algebras in gauge theories. The first application is given by Clifford algebra valued differential forms: we shown how the connection and curvature 2-forms are given by bivector algebra valued forms, bivector algebras are the Lie algebras of spin groups. Another application consist of showing, through the Periodicity Theorem of Clifford algebras, how some conformal transformations of the space-time are given by the action of the $pin(2,4) group over the paravectors R+ R4,1. ln the last application, the construction of monopoles and instantons is presented through the lnversion Theorem for Pauli and Dirac spinors, considered as elements of the even sub-algebra of the Clifford algebra. The close relationship between this theorem and the Hopf fibrations is emphasized, ilustrating the link between Topology and Physics / Mestrado / Mestre em Matemática Aplicada

Un des problèmes fondamentaux qui survient dans la formulation de la théorie quantique des champs dans l’espace de Sitter est la transition possible vers l’espace Anti-de Sitter. Par un calcul semi-classique, on peut calculer le taux de transition, qui est non-nul. Or, une comparaison des degrés de liberté suggèrent qu’ils sont incompatibles et que la transition ne devrait pas avoir lieu. Une analyse plus approfondie d’un modèle de stabilité du vide met en lumière deux conditions qui pourraient résoudre ce paradoxe. En appliquant ces conditions au modèle standard, on obtient une borne sur la masse du boson de Higgs ainsi que sur la constante cosmologique. Bien qu’elles n’offrent pas une résolution complète du problème de la hiérarchie et de la constante cosmologique, ces contraintes pourraient jouer un rôle dans la formulation d’une théorie de la gravité quantique. / A fundamental issue regarding the formulation of a consistent quantum de Sitter space theory is the possible transition to Anti-de Sitter space. Indeed, a semiclassical computation gives a nonzero estimate for the transition probability, while a counting of the degrees of freedom for both spaces shows their incompatilibity, leading to the expectation that the transition is impossible. Through a deeper analysis of quantum tunneling in a semiclassical theory including gravity, one can outline two consistency conditions that could alleviate this seemingly paradoxical disparity. Applying these consistency conditions to the Standard Model shed a different light on the hierarchy problem and the cosmological constant problem, although it does not solve them altogether. Nevertheless, they could play a role in the formulation of a consistent theory of quantum gravity.

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Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-22T22:29:57Z (GMT). No. of bitstreams: 1 Branco_LucasMagalhaesPereiraCastello_M.pdf: 1981391 bytes, checksum: 7ecd7674514f634b8bb527c0bcab1a06 (MD5) Previous issue date: 2013 / Resumo: Neste trabalho os aspectos básicos da teoria de calibre são abordados, incluindo as noções de conexão e curvatura em fibrados principais e vetoriais, considerações sobre o grupo de transformações de calibre e o espaço de moduli de soluções para a equação anti-auto-dual em dimensão quatro (o espaço de moduli de instantons). Posteriormente, mapas momento e redução são introduzidos. Primeiramente, no contexto clássico de geometria simplética e depois no contexto de geometria hyperkähler. Por fim, são apresentadas aplicações da teoria de mapas momento e redução em teoria de calibre. As equações ADHM são introduzidas e mostra se que estas podem ser dadas como o conjunto de zeros de um mapa momento hyperkähler. Além disso, considerações são feitas acerca da construção ADHM de instantons, que relaciona soluções dessas equações com as soluções da equação de anti-auto-dualidade. O espaço de moduli de conexões planas é também abordado. Neste caso, a curvatura é vista como um mapa momento e os cálculos podem ser generalizados para o espaço de moduli de conexões planas sobre variedades Kähler de dimensões mais altas e para o espaço de moduli de instantons sobre variedades hyperkähler de dimensão quatro / Abstract: In this work it is developed the basic concepts of gauge theory, including the notions of connections and curvature on principal bundles and vector bundles, considerations on the group of gauge transformations and the moduli space of anti-self-dual connections in dimension four (the instanton moduli space). After, moment maps and reduction are introduced. First in the classical context of symplectic geometry, then in hyperkähler geometry. At last, applications to the theory of moment maps and reduction in gauge theory are given. The ADHM equations are introduced and it is shown that solutions to these equations can be given by the zeros of a hyperkähler moment map. Furthermore, the ADHM construction, that relates the ADHM equations to instanton solutions, is discussed. The moduli space of flat connections over a Riemann surface is also treated. In this case, the curvature is seen as a moment map and the calculations can be generalized to flat connections over higher-dimensional Kähler manifolds and to the instanton moduli space over four dimensional hyperkähler manifolds / Mestrado / Matematica / Mestre em Matemática

Résumé anglais: <p><p>This thesis consists of an introductory text followed by two separate parts which may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in the vicinity of spacelike singularities (the BKL-limit). In this limit, spatial points decouple and the dynamics exhibits ultralocal behaviour which may be mapped to an auxiliary problem given in terms of a (possibly chaotic) hyperbolic billiard. In all supergravities arising as low-energy limits of string theory or M-theory, the billiard dynamics takes place within the fundamental Weyl chambers of certain hyperbolic Kac-Moody algebras, suggesting that these algebras generate hidden infinite-dimensional symmetries of gravity. We investigate the modification of the billiard dynamics when the original gravitational theory is formulated on a compact spatial manifold of arbitrary topology, revealing fascinating mathematical structures known as galleries. We further use the conjectured hyperbolic symmetry E10 to generate and classify certain cosmological (S-brane) solutions in eleven-dimensional supergravity. Finally, we show in detail that eleven-dimensional supergravity and massive type IIA supergravity are dynamically unified within the framework of a geodesic sigma model for a particle moving on the infinite-dimensional coset space E10/K(E10). <p><p>Part II of the thesis is devoted to a study of how (U-)dualities in string theory provide powerful constraints on perturbative and non-perturbative quantum corrections. These dualities are typically given by certain arithmetic groups G(Z) which are conjectured to be preserved in the effective action. The exact couplings are given by moduli-dependent functions which are manifestly invariant under G(Z), known as automorphic forms. We discuss in detail various methods of constructing automorphic forms, with particular emphasis on a special class of functions known as (non-holomorphic) Eisenstein series. We provide detailed examples for the physically relevant cases of SL(2,Z) and SL(3,Z), for which we construct their respective Eisenstein series and compute their (non-abelian) Fourier expansions. We also discuss the possibility that certain generalized Eisenstein series, which are covariant under the maximal compact subgroup K(G), could play a role in determining the exact effective action for toroidally compactified higher derivative corrections. Finally, we propose that in the case of rigid Calabi-Yau compactifications in type IIA string theory, the exact universal hypermultiplet moduli space exhibits a quantum duality group given by the emph{Picard modular group} SU(2,1;Z[i]). To verify this proposal we construct an SU(2,1;Z[i])-invariant Eisenstein series, and we present preliminary results for its Fourier expansion which reveals the expected contributions from D2-brane and NS5-brane instantons. <p><p>/<p><p>Résumé francais: <p><p>Cette thèse est composée d'une introduction suivie de deux parties qui peuvent être lues indépendemment. Dans la première partie, nous analysons des structures hyperboliques apparaissant dans l'étude de la gravité au voisinage d'une singularité de type espace (la limite BKL). Dans cette limite, les points spatiaux se découplent et la dynamique suit un comportement ultralocal qui peut être reformulé en termes d'un billiard hyperbolique (qui peut être chaotique). Dans toutes les supergravités qui sont des limites de basse énergie de théories de cordes ou de la théorie M, la dynamique du billiard prend place à l'intérieur des chambres de Weyl fondamentales de certaines algèbres de Kac-Moody hyperboliques, ce qui suggère que ces algèbres correspondent à des symétries cachées de dimension infinie de la gravité. Nous examinons comment la dynamique du billard est modifiée quand la théorie de gravité originale est formulée sur une variété spatiale compacte de topologie arbitraire, révélant ainsi de fascinantes structures mathématiques appelées galleries. De plus, dans le cadre de la supergravité à onze dimensions, nous utilisons la symétrie hyperbolique conjecturée E10 pour engendrer et classifier certaines solutions cosmologiques (S-branes). Finalement, nous montrons en détail que la supergravité à onze dimensions et la supergravité de type IIA massive sont dynamiquement unifiées dans le contexte d'un modèle sigma géodesique pour une particule se déplaçant sur l'espace quotient de dimension infinie E10/K(E10).<p><p><p>La deuxième partie de cette thèse est consacrée à étudier comment les dualités U en théorie des cordes fournissent des contraintes puissantes sur les corrections quantiques perturbatives et non perturbatives. Ces dualités sont typiquement données par des groupes arithmétiques G(Z) dont il est conjecturé qu'ils préservent l'action effective. Les couplages exacts sont donnés par des fonctions des moduli qui sont manifestement invariantes sous G(Z), et qu'on appelle des formes automorphiques. Nous discutons en détail différentes méthodes de construction de ces formes automorphiques, en insistant particulièrement sur une classe spéciale de fonctions appelées séries d'Eisenstein (non holomorphiques). Nous présentons comme exemples les cas de SL(2,Z) et SL(3,Z), qui sont physiquement pertinents. Nous construisons les séries d'Eisenstein correspondantes et leurs expansions de Fourier (non abéliennes). Nous discutons également la possibilité que certaines séries d'Eisenstein généralisées, qui sont covariantes sous le sous-groupe compact maximal, pourraient jouer un rôle dans la détermination des actions effectives exactes pour les théories incluant des corrections de dérivées supérieures compactifiées sur des tores.<p><p> / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T09:52:31Z (GMT). No. of bitstreams: 1 DeMartino_MarceloGoncalves_M.pdf: 1604556 bytes, checksum: be41ad6ca9fd66921624adce95bf0939 (MD5) Previous issue date: 2012 / Resumo: Este trabalho procurou apresentar os conhecimentos básicos necessários para trabalhar com a teoria de calibre em baixas dimensões e também mostrar algumas aplicações da mesma. Na parte básica da teoria, além de comentar aspectos da teoria de Hodge para variedades compactas, também se discute, com certo nível de detalhes, os conceitos de fibrados vetoriais e conexões, com ênfase dada para os cálculos locais com conexões e curvaturas. Duas aplicações mais concretas da teoria de calibre são apresentadas nesta dissertação. Primeiro, em dimensão quatro, discute-se a equação de Yang-Mills sobre 4-variedades e é apresentada uma solução para a equação anti-auto-dual, solução esta que é conhecida na literatura como ansatz de 't Hooft. Por fim, é apresentada a prova, baseado no artigo [DONALDSON, 1983], de um importante teorema devido a M. S. Narasimhan e C. S. Seshadri que relaciona os conceitos de estabilidade com o de existência de conexão unitária satisfazendo certa propriedade, em fibrados vetoriais complexos sobre superfícies de Riemann / Abstract: In this work it is developed the basic knowledge required to deal with gauge theory in low dimension and it is shown some applications of this theory. Regarding the basic knowledge, apart from discussing some aspects of Hodge theory over compact manifolds, it is also covered, with a certain deal of details, the concepts of vector bundles and connections, paying close attention to the local computations regarding connections and curvature. As for the applications of the theory, we start, in dimension four, by treating the Yang-Mills equation over 4-manifolds and it is showed a solution to the anti-self-dual Yang-Mills equation, solution that is known in the literature as the 't Hooft ansatz. At last, it is given a proof, following the paper [DONALDSON, 1983], of an important theorem due to M. S. Narasimhan and C. S. Seshadri that relates the algebro-geometric notion of stability to the differential-geometric notion of existence of unitary connection whose curvature satisfies a certain condition, on vector bundles over Riemann surfaces / Mestrado / Matematica / Mestre em Matemática

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Title: Instantons and Unitarily Inequivalent Quantum Vacua Author: Roman Derco Department: Institute of Particle and Nuclear Physics Supervisor: doc. Alfredo Iorio, Ph.D., Institute of Particle and Nuclear Physics Abstract: In the presented thesis we investigate the relationship between the topologically distinct instantonic vacua and the unitarily inequivalent vacua of the quantum field theory. We focus on quantum mechanical exam- ples, where instantons appear but the complications due to quantum gauge field theory are absent. A model for quantum dissipation and the theory of one particle escaping from a metastable minimum were compared, what led to some observations. A double well system was build from harmonic oscillators and an interaction term to get closer to the quantum dissipation model, where inequivalent representations are involved. We identified the particularly simple model of a quantum particle constrained on a circle to be the ideal toy model for spotting the relation among unitarily inequivalent vacua and topologically distinct vacua we were seeking for. 1