Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nesta tese estudamos uma extensão supersimétrica do mecanismo de Gribov no caso N = 1 em supercampos. Abordamos as teorias de super Yang-Mills em D = 4 e super Yang-Mills-Chern-Simons em D = 3. E verificamos como nestes casos o princípio de calibre leva ao cenário de confinamento de Gribov. / In this thesis we study a supersymmetric extension of the Gribov mechanism in the case of N = 1 with superfields. We approached the theories of super Yang-Mills in D = 4 and super Yang-Mills-Chern-Simons in D = 3. And we verify how in these cases the gauge principle leads to Gribov confinement scenario.

Consideramos modelos não-comutativos de campos escalares e fermiônicos acoplados com um campo de Chern-Simons em 2+ 1 dimensões e mostramos que, pelo menos em um laço, o modelo contendo somente um campo fermiônico, na representação fundamental, minimalmente acoplado ao campo de Chern-Simons, é consistente no sentido que não há divergências infravermelhas não-integráveis presentes no modelo. Contrariamente, divergências infravermelhas perigosas ocorrem se o campo fermiônico pertence à representação adjunta ou se consideramos o acoplamento com a matéria escalar. A formulação do modelo de Chern-Simons supersimétrico em termos de supercampos também é analisada, sendo livre de singularidades infravermelhas não integráveis e, na verdade, finito no caso em que o campo de matéria pertence à representação fundamental. No caso da representação adjunta, isso ocorre somente para uma particular escolha de calibre. Analisando a parte de paridade ímpar das funções de vértice de dois e três pontos do campo de calibre, calculamos, em um laço, as correções ao coeficiente do termo de Chern-Simons no modelo de Higgs-Chern-Simons não comutativo no caso de temperatura zero e no limite de altas temperaturas. A altas temperaturas, mostramos que o limite estático desta correção é proporcional a T mas a primeira correção devida à não-comutatividade aumenta como T log T. Nossos resultados são funções analíticas do parâmetro não-comutativo. / We consider 2+ 1 dimensional noncommutative models of scalar and fermionic fields coupled to the Chern-Simons field. We show that, at least up to one loop, the model containing only a fermionic field in the fundamental representation minimally coupled to the Chern-Simons field is consistent in the sense that there are no nonintegrable infrared divergences. By contrast, dangerous infrared divergences occur if the fermion field belongs to the adjoint representation or if the coupling of scalar matter is considered instead. The superfield formulation of the supersymmetric Chern-Simons model is also analyzed and shown to be free of nonintegrable infrared singularities and actually finite if the matter field belongs to the fundamental representation of the supergauge group. In the case of the adjoint representation this only happens in a particular gauge. By analyzing the odd parity part of the gauge field two and three point vertex functions, the one-loop radiative correction to the Chern-Simons coefficient is computed in noncommutative Chern-Simons-Higgs model at zero and at high temperature. At high temperature, we show that the static limit of this correction is proportional to T but the first noncommutative correction increases as T log T. Our results are analytic functions of the noncommutative parameter.

nÃo hà / Neste trabalho, apresenta-se o formalismo canÃnico de quantizaÃÃoo da gravidade, tanto em sua formulaÃÃo original, para a qual a mÃtrica à a variÃvel canÃnica, quanto na de Ashtekar, onde a conexÃo autodual assume o papel de variÃvel canÃnica. Nesta Ãltima formulaÃÃo, as equaÃÃes de vÃnculo do formalismo sÃo drasticamente simplificadas, e, fazendo uso da teoria de Chern-Simons, constrÃi-se um estado que satisfaz estas equaÃÃes no vÃcuo, constituindo uma importante soluÃÃo para a equaÃÃo de Wheeler-DeWitt. O estado de Chern-Simons tambÃm tem uma representaÃÃo em loops, que recebe este nome por ser formulada em termos dos loops de Wilson.

Les isolants topologiques sont des isolants qui ne peuvent être différenciés des isolants atomiques que par une grandeur physique non locale appelée invariant topologique. L'effet Hall quantique et son équivalent sans champ magnétique l'isolant de Chern sont des exemples d'isolants topologiques. En présence d'interactions fortes, des excitations exotiques appelées anyons peuvent apparaître dans les isolants topologiques. L'effet Hall quantique fractionnaire (EHQF) est la seule réalisation expérimentale connue de ces phases. Dans ce manuscrit, nous étudions numériquement les conditions d'émergence de différents isolants topologiques fractionnaires. Nous nous concentrons d'abord sur l'étude de l'EHQF sur le tore. Nous introduisons une méthode de construction projective des états EHQF les plus exotiques complémentaire par rapport aux méthodes existantes. Nous étudions les excitations de basse énergie sur le tore de deux états EHQF, les états de Laughlin et de Moore-Read. Nous proposons des fonctions d'onde pour les décrire, et vérifions leur validité numériquement. Grâce à cette description, nous caractérisons les excitations de basse énergie de l'état de Laughlin dans les isolants de Chern. Nous démontrons également la stabilité d'autres états de l'EHQF dans les isolants de Chern, tels que les états de fermions composites, Halperin et NASS. Nous explorons ensuite des phases fractionnaires sans équivallent dans la physique de l'EHQF, d'abord en choisissant un modèle dont l'invariant topologique a une valeur plus élevée, puis en imposant au système la conservation de la symétrie par renversement du temps, ce qui modifie la nature de l'invariant topologique. / Topological insulators are band insulators which are fundamentally different from atomic insulators. Only a non-local quantity called topological invariant can distinguish these two phases. The quantum Hall effect is the first example of a topological insulator, but the same phase can arise in the absence of a magnetic field, and is called a Chern insulator. In the presence of strong interactions, topological insulators may host exotic excitations called anyons. The fractional quantum Hall effect is the only experimentally realized example of such phase. In this manuscript, we study the conditions of emergence of different types of fractional topological insulators, using numerical simulations. We first look at the fractional quantum Hall effect on the torus. We introduce a new projective construction of exotic quantum Hall states that complements the existing construction. We study the low energy excitations on the torus of two of the most emblematic quantum Hall states, the Laughlin and Moore-Read states. We propose and validate model wave functions to describe them. We apply this knowledge to characterize the excitations of the Laughlin state in Chern insulators. We show the stability of other fractional quantum Hall states in Chern insulators, the composite fermion, Halperin and NASS states. We explore the physics of fractional phases with no equivalent in a quantum Hall system, using two different strategies: first by choosing a model with a higher value of the topological invariant, second by adding time-reversal symmetry, which changes the nature of the topological invariant.

A passagem progressiva de estados de Landau bidimensional (2D) para estados tridimensional (3D) foi estudada em Poços Quânticos Parabólicos (PQW) largos (W = 1000 6000 Å). Utilizou-se como técnica de transporte medidas da magnetoresistência em campo magnético intenso (B = 0 15 T) e inclinado ( = 0 90°; perpendicular paralelo), a baixas temperaturas (T = 50 mK). Observou-se, através da dependência angular das oscilações de Shubnikov de Haas ( = 0 90°), em PQWs cheios, várias sub-bandas ocupadas (5 a 8), a coexistência de estados de Landau 2D e 3D, sendo o gás 3D formado pelo colapso das sub-bandas elevadas, e o gás 2D pertencendo à primeira sub-banda. Através de cálculos do alargamento dos níveis de Landau devido ao espalhamento elástico ( = /2 , onde é o tempo quântico) e de cálculos auto-consistentes da energia de separação entre sub-bandas do PQW (ij = Ej Ei; e 12=12/2), obtiveram-se as condições 2 j-1,j para as sub-bandas elevadas j = 3,4,..., corroborando com as observações experimentais da coexistência de estados de Landau 2D e 3D no poço. Em PQWs parcialmente cheios, com apenas 2 sub-bandas ocupadas, observou-se, através do efeito do anticruzamento de níveis de Landau, de medidas da dependência angular da energia de ativação no regime de efeito Hall quântico, e de comparações com resultados de cálculos da estrutura eletrônica de PQWs em campo magnético inclinado, a coexistência de estados de Landau 2D e 3D, ocorrendo somente em campos intensos e com inclinação acentuada ( = 80 90°). Esta coexistência é diferente da mencionada anteriormente, quando od estados de Landau 3D são observados já em campo perpendicular. / The gradual progress, or evolution, of the two-dimensional (2D) toward three-dimensional (3D) Landau states was studied in wide parabolic quantum Wells (W = 1000 6000 Å). As transport technique, we used measurements of the magnetoresistence in intense (B = 0 15 T) and tilted ( = 0 90°; perpendicular parallel) magnetic Field at low temperature (T = 50 Mk). We observed in PQWs with Five to eight sub-bands occupied full well the coexistence of the 2D and 3D Landau states, through the angular dependence of the Shubnikov de Hass oscillation ( = 0 90°), where the 2D states belong to the lowest sub-band and the 3D states are formed by overlap of the other sub-bands. We calculated the level broadening due to the elastic scattering rate ( = /2 , where is the quantum time), and the energy separation between sub-bands (ij = Ej Ei; e 12=12/2). We obtained 2 j-1,j to j=3,4,... . This confirms the experimental observations of the coexistence of the 2D and 3D states in the well. We also measured PQWs partially full 2 sub-bands occupied. Experiments revel anticrossing of the Landau level (LL) belonging to the lowest sub-band and the last LL belonging to the second sub-band. Such antisrossuing occurs due to a decrease of the energy of the LL with tilt angle. This observation was supported by measurements of the angular dependence of the activation energy in the quantum Hall regime. In these measurements, we also observed the coexistence of the 2D and 3D Landau states. However, the coexistence only occurs at large tilt angles ( = 80 90°). Thus, it is different from the coexistence above mentioned, when 3D Landau states are observed already in the perpendicular magnetic field.

Dans cette thèse nous nous intéressons aux flots de Monge-Ampère complexes, à leurs généralisations et à leurs applications géométriques sur les variétés hermitiennes compactes. Dans les deux premiers chapitres, nous prouvons qu'un flot de Monge-Ampère complexe sur une variété hermitienne compacte peut être exécuté à partir d'une condition initiale arbitraire avec un nombre Lelong nul en tous points. En utilisant cette propriété, nous con- firmons une conjecture de Tosatti-Weinkove: le flot de Chern-Ricci effectue une contraction chirurgicale canonique. Enfin, nous étudions une généralisation du flot de Chern-Ricci sur des variétés hermitiennes compactes, le flot de Chern-Ricci tordu. Cette partie a donné lieu à deux publications indépendantes. Dans le troisième chapitre, une notion de C -sous-solution parabolique est introduite pour les équations paraboliques, étendant la théorie des C -sous-solutions développée récem- ment par B. Guan et plus spécifiquement G. Székelyhidi pour les équations elliptiques. La théorie parabolique qui en résulte fournit une approche unifiée et pratique pour l'étude de nombreux flots géométriques. Il s'agit ici d'une collaboration avec Duong H. Phong (Université Columbia ) Dans le quatrième chapitre, une approche de viscosité est introduite pour le problème de Dirichlet associé aux équations complexes de type hessienne sur les domaines de Cn. Les arguments sont modélisés sur la théorie des solutions de viscosité pour les équations réelles de type hessienne développées par Trudinger. En conséquence, nous résolvons le problème de Dirichlet pour les équations de quotient de hessiennes et lagrangiennes spéciales. Nous établissons également des résultats de régularité de base pour les solutions. Il s'agit ici d'une collaboration avec Sl-awomir Dinew (Université Jagellonne) et Hoang-Son Do (Institut de Mathématiques de Hanoi). / In this thesis we study the complex Monge-Ampère flows, and their generalizations and geometric applications on compact Hermitian manifods. In the first two chapters, we prove that a general complex Monge-Ampère flow on a compact Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti- Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern-Ricci flow on compact Hermitian manifolds, namely the twisted Chern-Ricci flow. This part gave rise to two independent publications. In the third chapter, a notion of parabolic C -subsolution is introduced for parabolic non-linear equations, extending the theory of C -subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows. This part is a joint work with Duong H. Phong (Columbia University) In the fourth chapter, a viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in Cn. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by Trudinger. As consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions. This part is a joint work with Sl-awomir Dinew (Jagiellonian University) and Hoang-Son Do (Hanoi Institute of Mathematics).

Dado um C$^*$-sistema dinâmico $(A, G, \\alpha)$ define-se um homomorfismo, denominado de caráter de Chern-Connes, que leva elementos de $K_0(A) \\oplus K_1(A)$, grupos de K-teoria da C$^*$-álgebra $A$, em $H_{\\mathbb}^*(G)$, anel da cohomologia real de deRham do grupo de Lie $G$. Utilizando essa definição, nós calculamos explicitamente esse homomorfismo para os exemplos $(\\overline{\\Psi_^0(S^1)}, S^1, \\alpha)$ e $(\\overline{\\Psi_^0(S^2)}, SO(3), \\alpha)$, onde $\\overline{\\Psi_^0(M)}$ denota a C$^*$-álgebra gerada pelos operadores pseudodiferenciais clássicos de ordem zero da variedade $M$ e $\\alpha$ a ação de conjugação pela representação regular (translações). / Given a C$^*$-dynamical system $(A, G, \\alpha)$ one defines a homomorphism, called the Chern-Connes character, that take an element in $K_0(A) \\oplus K_1(A)$, the K-theory groups of the C$^*$-algebra $A$, and maps it into $H_{\\mathbb}^*(G)$, the real deRham cohomology ring of $G$. We explictly compute this homomorphism for the examples $(\\overline{\\Psi_^0(S^1)}, S^1, \\alpha)$ and $(\\overline{\\Psi_^0(S^2)}, SO(3), \\alpha)$, where $\\overline{\\Psi_^0(M)}$ denotes the C$^*$-álgebra gene\\-rated by the classical pseudodifferential operators of zero order in the manifold $M$ and $\\alpha$ the action of conjugation by the regular representation (translations).

Neste trabalho, estudamos diferentes construções e propriedades das classes características de variedades suaves e singulares. Para ilustrar a teoria, calculamos a obstrução de Euler de algumas superfícies singulares no espaço tridimensional e apresentamos uma fórmula do tipo Poincaré-Hopf para variedades singulares / In this work, we study different constructions and properties of the characteristics classes of smooth and singular manifolds. To ilustrate the theory, we compute the Euler obstructions of some singular surfaces in tridimensional space and state a Poincaré-Hopf´s formula for singular varieties

Dado um C$^*$-sistema dinâmico $(A, G, \\alpha)$ define-se um homomorfismo, denominado de caráter de Chern-Connes, que leva elementos de $K_0(A) \\oplus K_1(A)$, grupos de K-teoria da C$^*$-álgebra $A$, em $H_{\\mathbb}^*(G)$, anel da cohomologia real de deRham do grupo de Lie $G$. Utilizando essa definição, nós calculamos explicitamente esse homomorfismo para os exemplos $(\\overline{\\Psi_^0(S^1)}, S^1, \\alpha)$ e $(\\overline{\\Psi_^0(S^2)}, SO(3), \\alpha)$, onde $\\overline{\\Psi_^0(M)}$ denota a C$^*$-álgebra gerada pelos operadores pseudodiferenciais clássicos de ordem zero da variedade $M$ e $\\alpha$ a ação de conjugação pela representação regular (translações). / Given a C$^*$-dynamical system $(A, G, \\alpha)$ one defines a homomorphism, called the Chern-Connes character, that take an element in $K_0(A) \\oplus K_1(A)$, the K-theory groups of the C$^*$-algebra $A$, and maps it into $H_{\\mathbb}^*(G)$, the real deRham cohomology ring of $G$. We explictly compute this homomorphism for the examples $(\\overline{\\Psi_^0(S^1)}, S^1, \\alpha)$ and $(\\overline{\\Psi_^0(S^2)}, SO(3), \\alpha)$, where $\\overline{\\Psi_^0(M)}$ denotes the C$^*$-álgebra gene\\-rated by the classical pseudodifferential operators of zero order in the manifold $M$ and $\\alpha$ the action of conjugation by the regular representation (translations).

Η παρούσα διπλωματική εργασία πραγματοποιήθηκε στο τμήμα Σωματιδιακής Φυσικής του Πανεπιστημίου Santiago de Compostela της Ισπανίας και αποτελεί τη μελέτη της υπερσυμμετρίας στις τρεις χωροχρονικές διαστάσεις. Έμφαση δίνεται σε θεωρίες που περιέχουν τον όρο Chern-Simons που παιζεί συμαντικό ρόλο στους τομείς έρευνας της θεωρητικής φυσικής. Αρχικά, εισάγουμε τις εισαγωγικές ένοιες της υπερσυμμετρίας στις τρεις διαστάσεις και ακολούθως μελέτουμε την Ν=1 ελάχιστη θεωρία με διάφορες φυσικές ποσότες που περιέχουν τον όρο Chern-Simons. Στην συνέχεια, μελετάμε τις ABJM θεωρίες και αποδεικνύουμε ότι είναι αναλλοίωτες κάτω από μετασχηματισμούς βαθμίδας. Τέλος υπολογίζουμε τις κβαντικές διορθώσεις στην διαταρακτική θεωρία Chern-Simons. / The present thesis took part in Department of Particle Physics of University of Santiago de Compostela, Spain, and is the study of supersymmetry in three spacetime dimensions. Emphasis is given to theories containing the Chern-Simons term that plays an important role in the research areas of theoretical physics. First, we introduce the notion of supersymmetry in three dimensions and then we study the N = 1 minimal theory with various physical quantitative containing the term Chern-Simons. Then, we study the ABJM theories and prove that they are invariant under gauge transformations. Finally we calculate the quantum corrections to the perturbative Chern-Simons theory.