Nesta tese, o limite não re1ativistico em teorias quânticas de campos em 2+1 dimensões é discutido1 perturbativamente, através da introdução de um corte intermediário que permite 0 cálculo de.expansão /P/m das amplitudes quânticas. especificando a origem, no espaço dos estados intermediários, de cada uma. das contribuições. Este procedimento é aplicado à teoria 4 e um esquema de redução a nível das amplitudes, que identifica a contribuição do setor de baixas energias com 0 resultado da teoria não re1ativistica, é proposto. Quando aplicado á teoria de Chern-Simons escalar, este procedimento sugere correlações relativísticas para o espalhamento Aharonov-Bohm. / n this thesis, the nonrelativistic limit of quantum field theories in 2 + 1 dimensions is discussed, perturbatively, through the introduction of an intermediate cutoff which generates the /p/m expansion of the quantum amplitudes and specifies the origin of each contribution in the space of the intermediary states. This scheme is applied to the theory 4 and a reduction procedure for the amplitudes that identify the low energy sector contribution with the results of the nonrelativistic theory is proposed. When applied to the scalar Chern-Simons theory, this procedure gives relativistic corrections to the Aharonov- Bohm scattering.

This thesis consists of two parts, which can be read separately. In the first part we study aspects of topological field theories. We show how to topologically twist three-dimensional N=2 supersymmetric Chern-Simons theory using a contact structure on the underlying manifold. This gives us a formulation of Chern-Simons theory together with a set of auxiliary fields and an odd symmetry. For Seifert manifolds, we show how to use this odd symmetry to localize the path integral of Chern-Simons theory. The formulation of three-dimensional Chern-Simons theory using a contact structure admits natural generalizations to higher dimensions. We introduce and study these theories. The focus is on the five-dimensional theory, which can be understood as a topologically twisted version of N=1 supersymmetric Yang-Mills theory. When formulated on contact manifolds that are circle fibrations over a symplectic manifold, it localizes to contact instantons. For the theory on the five-sphere, we show that the perturbative part of the partition function is given by a matrix model. In the second part of the thesis, we study supersymmetric sigma models in the Hamiltonian formalism, both in a classical and in a quantum mechanical setup. We argue that the so called Chiral de Rham complex, which is a sheaf of vertex algebras, is a natural framework to understand quantum aspects of supersymmetric sigma models in the Hamiltonian formalism. We show how a class of currents which generate symmetry algebras for the classical sigma model can be defined within the Chiral de Rham complex framework, and for a six-dimensional Calabi-Yau manifold we calculate the equal-time commutators between the currents and show that they generate the Odake algebra.

Orientador: Prof. Dr. Alysson Fábio Ferrari / Tese (doutorado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2016. / Nesta tese nosso objetivo foi estudar a Quebra Dinâmica de Simetria (QDS) em uma teoria de Chern-Simons supersimétrica em (2 + 1) dimensões acoplada a N supercampos de matéria, no formalismos dos supercampos. Neste sentido, desenvolvemos um mecanismo para calcular o superpotencial efetivo Keff (¿cl, ¿) , onde ¿cl é um supercampo de fundo e ¿ o parâmetro de calibre que é introduzido no processo de quantização da teoria. Potenciais efetivos dependentes do parâmetro de calibre já foram estudados no contexto da teoria quântica de campos, e podem levar a consequências não triviais para o estudo da QDS em teorias de calibre. Portanto, nós não assumimos de princípio a invariância de calibre dos resultados em nosso modelo, como usualmente é feito na literatura. Nós desenvolvemos o formalismo das identidades de Nielsen na linguagem dos supercampos, que é o formalismo apropriado para o estudo da QDS quando o potencial efetivo depende do calibre. Nós também discutimos como podemos calcular o potencial efetivo a partir da Equação de Grupo de Renormalização (EGR), a partir do conhecimento das funções de grupo de renormalização, i.e., funções ¿ e dimensão anômala ". Desenvolvemos um cálculo detalhado destas funções a dois laços da teoria de perturbação, encontrando que estas não dependem do parâmetro ¿, e portanto, usando a EGR, calculamos o superpotencial Keff, mostrando que ele é também independente de ¿. Então nós discutimos o aprimoramento no cálculo de Keff somando os termos "logaritmos líderes", e comparamos este aprimoramento com aquele obtido na versão não supersimétrica do modelo. Finalmente, fizemos o estudo da QDS encontrando que ela é operacional para todos os valores razoáveis dos parâmetros livres, enquanto que o aprimoramento obtido pela EGR em geral só produz uma pequena correção quantitativa nos resultados, ao invés da dramática mudança qualitativa encontrada em modelos não supersimétricos. / In this thesis we study the Dynamical Symmetry Breaking (DSB) mechanism in a supersymmetric Chern-Simons theory in (2 + 1) dimensions coupled to N matter superfields in the superfield formalism. For this purpose, we developed a mechanism to calculate the effective superpotencial Keff (¿cl, ¿), where ¿cl is a background superfield, and ¿ a gaugefixing parameter that is introduced in the quantization process. The possible dependence of the effective potential on the gauge parameter have been studied in the context of quantum field theory, and it can have nontrivial consequences to the study of DSB in gauge theories. Therefore, we did not assume from the start the gauge independence of the effective potential in our model, as it is customary in the literature. We developed the formalism of the Nielsen identities in the superfield language, which is the appropriate formalism to study DSB when the effective potential is gauge dependent. We also discuss how to calculate the effective superpotential via the Renormalization Group Equation (RGE) from the knowledge of the renormalization group functions of the theory, i.e., ¿ functions and anomalous dimensions ". We perform a detailed calculation of these functions at two loops, finding that these do not depend on ¿, and therefore, by using the RGE, we calculate the effective superpotencial Keff, showing that it is also independen of ¿. Then we discuss the improvement of the calculation of Keff by summing up leading logarithms, and we compare this improvement with the one obtained in the non supersymmetric version of the model. Finally, we study the DSB finding that it is operational for all reasonable values of the free parameters, while the improvement obtained from the RGE in general only produces a small quantitative correction in the results, instead of the more dramatic qualitative change found in non supersymmetric models.

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.

<p>The fractional quantum Hall effect (FQHE), discovered in 1982 in a two-dimensional electron system, has generated a wealth of successful theory and new concepts in condensed matter physics, but is still not fully understood. The possibility of having nonabelian quasiparticle statistics has recently attracted attention on purely theoretical grounds but also because of its potential applications in topologically protected quantum computing.</p><p>This thesis focuses on the quasiparticles using three different approaches. The first is an effective Chern-Simons theory description, where the noncommutativity imposed on the classical space variables captures the incompressibility. We propose a construction of the quasielectron and illustrate how many-body quantum effects are emulated by a classical noncommutative theory.</p><p>The second approach involves a study of quantum Hall states on a torus where one of the periods is taken to be almost zero. Characteristic quantum Hall properties survive in this limit in which they become very simple to understand. We illustrate this by giving a simple counting argument for degeneracy 2<i>n</i>^-1, pertinent to nonabelian statistics, in the presence of 2<i>n</i> quasiholes in the Moore-Read state and generalise this result to 2<i>n</i>-<i>k</i> quasiholes and <i>k </i>quasielectrons.</p><p>In the third approach, we study the topological nature of the degeneracy 2<i>n</i>^-1 by using a recently proposed analogy between the Moore-Read state and the two-dimensional spin-polarized p-wave BCS state. We study a version of this problem where one can use techniques developed in the context of high-<i>T</i>c superconductors to turn the vortex background into an effective gauge field in a Dirac equation. Topological arguments in the form of index theory gives the degeneracy 2<i>n</i>^-1 for 2<i>n</i> vortices.</p>

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.

The fractional quantum Hall effect (FQHE), discovered in 1982 in a two-dimensional electron system, has generated a wealth of successful theory and new concepts in condensed matter physics, but is still not fully understood. The possibility of having nonabelian quasiparticle statistics has recently attracted attention on purely theoretical grounds but also because of its potential applications in topologically protected quantum computing. This thesis focuses on the quasiparticles using three different approaches. The first is an effective Chern-Simons theory description, where the noncommutativity imposed on the classical space variables captures the incompressibility. We propose a construction of the quasielectron and illustrate how many-body quantum effects are emulated by a classical noncommutative theory. The second approach involves a study of quantum Hall states on a torus where one of the periods is taken to be almost zero. Characteristic quantum Hall properties survive in this limit in which they become very simple to understand. We illustrate this by giving a simple counting argument for degeneracy 2n-1, pertinent to nonabelian statistics, in the presence of 2n quasiholes in the Moore-Read state and generalise this result to 2n-k quasiholes and k quasielectrons. In the third approach, we study the topological nature of the degeneracy 2n-1 by using a recently proposed analogy between the Moore-Read state and the two-dimensional spin-polarized p-wave BCS state. We study a version of this problem where one can use techniques developed in the context of high-Tc superconductors to turn the vortex background into an effective gauge field in a Dirac equation. Topological arguments in the form of index theory gives the degeneracy 2n-1 for 2n vortices.

Maxim Kontsevich a défini un invariant $Z$ des sphères d'homologie rationnelle orientées de dimension $3$ en 1992, en poursuivant l'étude initiée par Edward Witten du développement perturbatif de la théorie de Chern-Simons.L'invariant $Z$ de Kontsevich est gradué. Il s'écrit $Z=(Z_n)_{nin NN }$, où $Z_n$ prend ses valeurs dans un espace $CA_n$ engendré par des diagrammes trivalents à $2n$ sommets appelésdiagrammes de Feynman-Jacobi de degré $n$.L'invariant $Z$ apparait d'abord comme un invariant $Z(M,tau)$ des sphères d'homologie rationnelle $M$ de dimension $3$ munies d'une parallélisation $tau$.Il est l'exponentielle d'un invariant $z(M,tau)=(z_n(M,tau))_{nin NN }$dont la partie de degré $n$ compte algébriquement les plongements des diagrammes de Feynman-Jacobi connexes à $2n$ sommets assujettis à vérifier certaines conditions.On peut associer un invariant homotopique entier $p_1(tau)$ aux parallélisations $tau$ des variétés orientées de dimension $3$, et il existe un élément $beta=(beta_n)_{nin NN}$ de $CA_n$ appelé anomalie tel que$$z_n(M,tau)-p_1(tau)beta_n$$ soit indépendant de $tau$ et noté $z_n(M)$.$$Z(M)=expleft((z_n(M))_{nin NN}right).$$On sait depuis l'introduction de cette constante par Greg Kuperberg et Dylan Thurston en 1999 que $beta_n=0$ si $n$ est pair et que $beta_1 neq 0$.Cette thèse porte sur le calcul de la première valeur inconnue $beta_3$. Elle en présente des expressions très simplifiées et implémentables sur ordinateur. / The Kontsevich invariant $Z$ of rational homology $3-$ sphere was constructed by Maxim Kontsevich in 1992 using configuration space integrals.This invariant is graduated. It can be written as $Z=(Z_n)_{nin NN}$, where $Z_n$ values in the space $mathcal{A}_n$ of jacobi diagram with order $n$. A Jacobi diagram with order $n$ is a trivalent graph with $2n$ vertices. At a first point, we can see $Z$ as an invariant $Z(M,tau)$ of rational homology $3-$spheres equipped with a trivialisation $tau$ so that $Z$ is the exponential of an invariant $z(M,tau)=(z_n(M,tau))_{ninNN}$. In fact, we can say that $z_n(M,tau)$ counts the number of embeddings of connected jacobi diagrams with order $n$ with some additionnal conditions. We can associate an homotopic integer invariant $p_1(tau)$ to each trivialisation $tau$ of oriented $3-$manifolds and it exists $beta=(beta_n)_{ninNN}$, where $beta_ninmathcal{A}_n$ that is called anomaly so that $$z_n(M,tau) - p_1(tay)$$ is independant of $tau$. We name it $z_n(M)$ and $$Z(M)=exp((z_n(M)_{nin NN})).$$Greg Kuperberg and Dylan Thurston introduced this constant in 1999. We already know that $beta_n=0$ if $n$ is even and $beta_1neq 0$. This thesis is about the computation of $beta_3$. It describes simplified expressions of $beta_3$, and this expressions can be compute with a computer.

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.

The goal of the original research presented in this thesis is to study the strong coupling regime of Quantum Field Theories (QFTs) with different methods, making concrete predictions about the phase structure and the dynamics of these theories, and on their observables. The focus is on (gauge) field theories in three spacetime dimensions, which are an interesting laboratory to understand the properties of strong coupling in setups that are usually simpler than in the more familiar case of gauge theories in four dimensions. Importantly, topological effects play a relevant role in three dimensions, thanks to the presence of the so-called Chern-Simons term.The thesis contains a short introduction to QFTs in 3d, principles and applications of infrared dualities, large N techniques, and holography. Indeed, the web of infrared dualities, the large N expansion, and the holographic correspondence between QFT and gravity are the main tools which we use to investigate the strongly coupled regimes of 3d QFTs.Then, the original material is presented. In a first line of research, we focus on the study of the phase diagram of a 3d gauge theory making use of conjectured infrared dualities, extending such dualities to the case where more than one mass parameter can be dialed. In a second line of research, we study a class of 3d gauge theories by engineering their gravity dual in a string theory setup. We prove the existence of multiple phase transitions between phases characterized by both massless particles and topological sectors. In a third line of research, we use holography as a tool to explore the interplay between the physics of 4d QCD and 3d gauge theories. In particular, we analyze the properties of 3d domain walls, which appear as soliton-like solutions of 4d QCD in specific parametric regimes. Finally, we propose a boundary construction of 3d large N vector models, which appear as critical points of theories obtained by coupling degrees of freedom localized on a 3d boundary to a 4d bulk theory. This construction allows to prove new dualities and uncovers a new computational tool for 3d vector models. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished