Die vorliegende Dissertationarbeit ist dem Problem der analytischen Beschreibung des Confinement-Mechanismus in der QCD und in anderen Eichtheorien gewidment. Als Leitprinzip der Arbeit wurde das sogenannte Wilsonsche-Confinement-Kriterium gewählt, gemäss welchem diese Erscheinung durch eine effektive Stringtheorie beschrieben werden kann. Die entstehenden Strings des Eichfeldes verbinden farbige-Objekte (Quarks, Gluonen) miteinander und hindern ihr Auseinandergehen auf makroskopische Abstände. Es werden verschiedene Verfahren der Ableitung dieser Stringstheorien aus unterschiedlichen Eichtheorien, einschliesslich der QCD, vorgestellt. Kapitel 2 enthält die Untersuchung der nichtlokalen effektiven Stringwirkung, die im Rahmen des sogenannten stochastischen Vakuum-Modells der QCD entsteht, wobei die Wechselwirkung zwischen den Elementen der String-Weltfläche durch den phänomenologischen Background-Gluon-Propagator vermittelt wird. Durch Entwicklung dieser Wirkung nach Ableitungen wurden die ersten Terme niedrigster Ordnung bestimmt. Die ersten beiden Terme dieser Entwicklung sind die Nambu-Goto- und Rigidity-Terme mit Kopplungskonstanten, die sich durch das Gluon-Kondensat und die Korrelationlänge des QCD-Vakuums ausdrücken lassen. Die Vorzeichen dieser Konstanten zeigen, dass die durch dieses Verfahren erhaltenen Strings stabil sind. Danach wurde eine mögliche Lösung des ``Crumpling'' Problems auf der Basis eines zusätzlichen topologischen Stringtermes im Instantongas-Modell des QCD-Vakuums vorgestellt. Mittels Störungstheorie im nicht-störungstheoretischen QCD-Background berechneten wir zusätzliche-Korrekturen zur ursprünglichen nicht-störungstheoretischen Stringwirkung. Diese Korrekturen führen zu neuen Formen der nichtlokalen effektiven Stringwirkung, die den störungstheoretischen Gluon-Propagator im Backgroundfeld zwischen den Elementen der Weltfläche enthalten. Durch Ableitungsentwicklung dieser Wirkung bekommen wir eine Korrektur zur Kopplungskonstante des Rigidity-Terms; die Stringsspannung des Nambu-Goto-Terms jedoch bleibt unverändert. Am Ende dieses Kapitels wurde der Hamilton-Operator des QCD-Strings mit spinlosen Quarks hergeleitet, der der effektiven Stringwirkung mit Rigidity-Term entspricht. Dieser Hamilton-Operator liefert einen Korrekturterm zur Wechselwirkung im relativistischen Quarkmodell-Operator. Im Kapitel 3 untersuchten wir das Problem der Stringdarstellung von Abelsch-projezierten Eichtheorien. Als erstes wurde die Herleitung der Stringdarstellung der erzeugenden Funktion für das einfachste Modell dieser Art, d.h. die Abelsch-projezierte SU(2)-QCD gegeben, die einem dualen Abelschen Higgs-Modell mit äusseren elektrisch geladendenen Teilchen äquivalent ist. Der Vorteil dieses Stringszuganges im Vergleich zum Zugang des stochastischen Vakuum-Modells der QCD besteht in der Berücksichtigung der Integration über String-Weltflächen, die auf Grund der Integration über den Singulärteil der Higgsfeld-Phase entsteht. Zusätzlich zur Stringdarstellung der erzeugenden Funktion wurde im London-Limes die Stringdarstellung für die erzeugenden Funktionale der Feldstärke- und Monopolstromkorrelatoren hergeleitet. Dies gab uns die Möglichkeit, die entsprechenden bilokalen Kumulanten zu finden und zu zeigen, dass die bilokalen Kumulanten der Feldstärke für grosse Abstände das gleiche Verhalten wie die entsprechenden eichinvarianten Kumulanten der QCD zeigen. Das Letztere wurde durch das stochastische Vakuum-Modell vorhergesagt und durch Gitterexperimente berechnet. Dieses Ergebnis unterstützt einerseits die Methode der Abelschen Projektion und gibt anderseits dem stochastischen Vakuum-Modell der QCD einen neuen feldtheoretischen Status. Danach erweiterten wir unsere Analyse über den Rahmen des London-Limes hinaus untersuchten den Zusammenhang von quartischen Kumulanten und bilokalen Kumulanten. Anschliessend wurde die Stringdarstellung der SU(3)-Gluodynamik hergeleitet. Dabei wurde die Stringdarstellung für ein entsprechendes duales Modell formuliert, das drei Arten des magnetischen Higgs-Feldes enthält. Infolgedessen liefert das Modell drei Strings, von denen nur zwei wirklich unabhängig sind. Alle diese Strings wechselwirken untereinander durch Austausch zweier massiver dualer Eichbosonen. Ausserdem erhielten wir die bilokalen Kumulanten des effektiven dualen Modells der SU(3)-Gluodynamik. Die entsprechenden bilokalen Kumulanten zeigen für grosse Abstände ein Verhalten wie es durch das stochastische Vakuum-Modell vorhergesagt wurde. Zum Schluss dieses Kapitels geben wir eine nützliche Darstellung für erzeugende Funktionen von Abelsch-projezierten Theorien in Form von Integralen über Monopolströme an. Im Kapitel 4 wurde ein weiteres Modell untersucht, das eine analytische Beschreibung des Confinement-Mechanismus zulässt, nämlich die 3D kompakte QED. Für den Wilson-Loop der entsprechenden Theorie mit Monopoldichten wurde die Äquivalenz zur sogenannten Confining-Stringtheorie demonstriert. Ausserdem wurde das Verhalten der bilokalen Kumulante der Feldstärke im Limes schwacher Felder untersucht. Dieses Verhalten befindet sich ebenfalls in Übereinstimmung mit den Voraussagen des stochastischen Vakuum-Modells. Erwartungsgemäss sind die Stringdarstellungen der erzeugenden Funktionen der 3D kompakten QED im Limes schwacher Felder und der dualen Abelschen Higgs-Modelle sehr ähnlich. Wir zeigten ausserdem, dass diese Entsprechung nicht zufällig ist. Die 3D kompakte QED ergibt sich nämlich im Limes verschwindender Eichbosonmasse aus dem 3D Abelschen Higgs-Modell mit äusseren Monopolen. Zum Schluss wurde ein allgemeines Verfahren der Beschreibung der Anregungen der Stringweltfläche in Abelsch-projezierten Theorien (kompakte QED und QCD) ausgearbeitet. Es ist auf der Methode der nicht-linearen Sigma-Modelle gegründet und gibt eine Möglichkeit, die in diesen Fluktuationen quadratische Effektive Wirkung zu erhalten. In der Dissertation wurden analytische nicht-störungstheoretische Verfahren ausgearbeitet, die neue Informationen über den Confinement-Mechanismus in der QCD und anderen Eichtheorien liefern und zum besseren Verständnis der Vakuumstruktur dieser Theorien beitragen können. Sie sind insbesondere relevant für die Herleitung effektiver Stringtheorien aus Eichtheorien. / The main problem addressed in the present Dissertation was an attempt of an analytical description of confinement in QCD and other gauge theories. As a guiding principle for our investigations served the so-called Wilson's picture of confinement, according to which this phenomenon can be described in terms of some effective theory of strings, joining coloured objects to each other and preventing them from moving apart to macroscopic distances. In this Dissertation, we have proceeded with a derivation of such string theories corresponding to various gauge ones, including QCD, i.e. with the solution of the problem of string representation of gauge theories. We have started our analysis with the nonlocal string effective action, arising within the so-called Stochastic Vacuum Model of QCD, where the interaction between the string world-sheet elements is mediated by the phenomenological background gluon propagator. By performing the derivative expansion of this action, we have derived the first few terms of a string Lagrangian. The first two nontrivial of them turned out to be the Nambu-Goto and rigidity terms with the coupling constants expressed completely via the gluonic condensate and correlation length of the QCD vacuum. The signs of these constants ensure the stability of strings in the so-obtained effective string theory. After that, we have investigated the problem of crumpling for the string world-sheets by derivation of the topological string term in the instanton gas model of the gluodynamics vacuum. Next, by making use of perturbation theory in the nonperturbative QCD vacuum, we have calculated perturbative corrections to the obtained string effective action. Those lead to a new form of the nonlocal string effective action with the propagator between the elements of the world-sheet being the one of a perturbative gluon in the confining background. By the derivative expansion of this action, we got a correction to the rigidity term coupling constant, whereas the string tension of the Nambu-Goto term occurs to get no corrections due to perturbative gluonic exchanges. Finally, we have derived the Hamiltonian of QCD string with spinless quarks at the ends, associated with the obtained string effective action including the rigidity term. In the particular case of vanishing orbital momentum of the system, this Hamiltonian reduces to that of the so-called relativistic quark model, albeit with some modifications due to the rigidity term, which might have some influence on the dynamics of the QCD string with quarks. All these topics have been elaborated on in Section 2, and form the essence of the string representation of QCD within the Stochastic Vacuum Model. In Section 3, we have addressed the problem of string representation of Abelian-projected theories. In this way, we have started with the string representation for the partition function of the simplest model of this kind, namely the Abelian-projected SU(2)-QCD, which is argued to be the dual Abelian Higgs Model with external electrically charged particles. The advantage of this approach to the string representation of QCD w.r.t. the one based on the Stochastic Vacuum Model is a possibility to get an integration over the string world-sheets, resulting from the integration over the singular part of the phase of the Higgs field. After the string representation of the partition function in the London limit, we have proceeded with the string representation for the generating functionals of the field strength and monopole current correlators. Those enabled us to find the corresponding bilocal cumulants and demonstrate that the large-distance asymptotic behaviour of the bilocal field strength cumulant matches the one of the corresponding gauge-invariant cumulant in QCD, predicted by the Stochastic Vacuum Model and measured in the lattice experiments. This result supports the method of Abelian projection on the one hand and gives a new field-theoretical status to the Stochastic Vacuum Model on the other hand. After that, we have extended our analysis beyond the London limit, and studied the relation of the quartic cumulant, which appears there, with the bilocal one in the London limit. Next, by making use of the Abelian projection method, we have addressed the problem of string representation of the SU(3)-gluodynamics. Namely, we have casted the related dual model, containing three types of magnetic Higgs fields, into the string form. Consequently, the latter one turned out to contain three types of strings, among which only two ones were actually independent. As a result, we have found, that both the ensemble of strings as a whole and individual strings display confining properties in a sense that all types of strings (self)interact via the exchanges of the massive dual gauge bosons. We have also derived bilocal cumulants in the effective dual model of confinement, corresponding to the SU(3)-gluodynamics, and they turned out to be also in line with the ones predicted by the Stochastic Vacuum Model. In conclusion of this topic, we have obtained another useful representation for the partition functions of the Abelian-projected theories in the form of an integral over the monopole currents. In Section 4, we have studied another model, allowing for an analytical description of confinement, which is 3D compact QED. In this way, by virtue of the integral over the monopole densities, we have derived string representation for the Wilson loop in this theory and demonstrated the correspondence of this representation to another recently found one, the so-called confining string theory. After that, we have calculated the bilocal cumulant of the field strength tensors in the weak-field limit of the model under study. It also turned out to be in line with the general concepts of the Stochastic Vacuum Model and therefore matches the corresponding results known from the lattice measurements in QCD and found analytically for the effective Abelian-projected theories in the previous Section. Besides that, string representations of the partition functions of the weak-field limit of 3D compact QED and of the dual Abelian Higgs Model turned out to be also quite similar. We have illustrated later on that this correspondence is not accidental. Namely, we have shown that 3D compact QED is nothing else, but the limiting case of 3D Abelian Higgs Model with external monopoles, corresponding to the vanishing gauge boson mass. Finally, we have elaborated on a unified method of description of the string world-sheet excitations in the Abelian-projected theories, compact QED, and QCD, based on the techniques of nonlinear sigma-models, and obtained the effective action, quadratic in the world-sheet fluctuations. In conclusion, the proposed nonperturbative techniques provide us with some new information on the mechanisms of confinement in QCD and other gauge theories and shed some light on the vacuum structure of these theories. They also show the relevance of string theory to the description of this phenomenon and yield several prescriptions for the construction of the adequate string theories from the corresponding gauge ones.

Diese Arbeit befasst sich mit Quantenfeldtheorien auf nicht-kommutativen Räumen. Solche Modelle treten im Zusammenhang mit der Stringtheorie und mit der Quantengravitation auf. Ihre nicht-störungstheoretische Behandlung ist üblicherweise schwierig. Hier untersuchen wir jedoch drei nicht-kommutative Quantenfeldtheorien nicht-perturbativ, indem wir die Wirkungsfunktionale in eine äquivalente Matrixformulierung übersetzen. In der Matrixdarstellung kann die jeweilige Theorie dann numerisch behandelt werden. Als erstes betrachten wir ein regularisiertes skalares Modell auf der nicht-kommutativen Ebene und untersuchen den Kontinuumslimes bei festgehaltener Nicht-Kommutativität. Dies wird auch als Doppelskalierungslimes bezeichnet. Insbesondere untersuchen wir das Verhalten der gestreiften Phase. Wir finden keinerlei Hinweise auf die Existenz dieser Phase im Doppelskalierungslimes. Im Anschluss daran betrachten wir eine vier-dimensionale U(1) Eichtheorie. Hierbei sind zwei der räumlichen Richtungen nicht-kommutativ. Wir untersuchen sowohl die Phasenstruktur als auch den Doppelskalierungslimes. Es stellt sich heraus, dass neben den Phasen starker und schwacher Kopplung eine weitere Phase existiert, die gebrochene Phase. Dann bestätigen wir die Existenz eines endlichen Doppelskalierungslimes, und damit die Renormierbarkeit der Theorie. Weiterhin untersuchen wir die Dispersionsrelation des Photons. In der Phase mit schwacher Kopplung stimmen unsere Ergebnisse mit störungstheoretischen Berechnungen überein, die eine Infrarot-Instabilität vorhersagen. Andererseits finden wir in der gebrochenen Phase die Dispersionsrelation, die einem masselosen Teilchen entspricht. Als dritte Theorie betrachten wir ein einfaches, in seiner Kontinuumsform supersymmetrisches Modell, welches auf der "Fuzzy Sphere" formuliert wird. Hier wechselwirken neutrale skalare Bosonen mit Majorana-Fermionen. Wir untersuchen die Phasenstruktur dieses Modells, wobei wir drei unterschiedliche Phasen finden. / This work deals with three quantum field theories on spaces with noncommuting position operators. Noncommutative models occur in the study of string theories and quantum gravity. They usually elude treatment beyond the perturbative level. Due to the technique of dimensional reduction, however, we are able to investigate these theories nonperturbatively. This entails translating the action functionals into a matrix language, which is suitable for numerical simulations. First we explore a scalar model on a noncommutative plane. We investigate the continuum limit at fixed noncommutativity, which is known as the double scaling limit. Here we focus especially on the fate of the striped phase, a phase peculiar to the noncommutative version of the regularized scalar model. We find no evidence for its existence in the double scaling limit. Next we examine the U(1) gauge theory on a four-dimensional spacetime, where two spatial directions are noncommutative. We examine the phase structure and find a new phase with a spontaneously broken translation symmetry. In addition we demonstrate the existence of a finite double scaling limit which confirms the renormalizability of the theory. Furthermore we investigate the dispersion relation of the photon. In the weak coupling phase our results are consistent with an infrared instability predicted by perturbation theory. If the translational symmetry is broken, however, we find a dispersion relation corresponding to a massless particle. Finally, we investigate a supersymmetric theory on the fuzzy sphere, which features scalar neutral bosons and Majorana fermions. The supersymmetry is exact in the limit of infinitely large matrices. We investigate the phase structure of the model and find three distinct phases. Summarizing, we study noncommutative field theories beyond perturbation theory. Moreover, we simulate a supersymmetric theory on the fuzzy sphere, which might provide an alternative to attempted lattice formulations.

O principal objetivo do presente trabalho é expor o procedimento de quantização de teorias de Yang-Mills, através do método de Faddeev-Popov, no formalismo de 1a Ordem, e investigar num primeiro momento sua equivalência (clássica e quântica) ao formalismo usual (2a Ordem) e algumas de suas aplicações, principalmente no cálculo de correções quânticas. Para isso, ideias gerais a respeito do processo de quantização via formalismo de Faddeev-Popov foram expostas, e posteriormente utilizadas no processo de quantização de teorias de Yang-Mills no formalismo de 1a Ordem. Apresenta-se também as ideias gerais relativas ao método de regularização dimensional utilizado no cálculo de correções quânticas à nível de 1-loop para a teoria de Yang-Mills no formalismo de 1a ordem, utilizando-se, para isso, computação simbólica. Foi demonstrado que via formalismo de 1a Ordem, a estrutura ultravioleta encontrada no propagador do bóson de gauge é consistente com a renormalizabilidade da teoria. Embora tenhamos diferenças quanto a estrutura das interações neste novo formalismo, a estrutura das divergências ultravioletas continua a mesma do formalismo usual. / The main objective of the present work is to expose the quantization procedure of Yang- Mills theories in first order formalism, by Faddeev Popov\'s method. We want to investigate the classical and quantum equivalence between first and second order formalism, and look and analyze the differences in practical calculations of quantum corrections. Therefore, the general ideas about quantizantion by Faddeev-Popov\'s method was exposed, and used later in first order theory. It is also presented in this work, the main ideas concerning to dimensional regularization used in quantum corrections calculations at one-loop order for Yang-Mills theories, using for that, symbolic computation. It has been shown that upon using the first order formalism, the ultraviolet structre found in gauge boson propagator is also consistent to the theory\'s renormalizability. Although we have differences concerning to interactions structures in this new formalism, the ultraviolet structures from usual formalism is also found in it.

In dieser Arbeit behandeln wir exakte, nicht-perturbative Ergebnisse, die mithilfe der superkonformen Index-Technik, in supersymmetrischen Eichtheorien mit vier Superladungen (d. h. N=1 Supersymmetrie in vier Dimensionen und N=2 in drei Dimensionen) gewonnen wurden. Wir benutzen die superkonforme Index-Technik um mehrere Dualitäts Vermutungen in supersymmetrischen Eichtheorien zu testen. Wir führen Tests der dreidimensionalen Spiegelsymmetrie und Seiberg ähnlicher Dualitäten durch. Das Ziel dieser Promotionsarbeit ist es moderne Fortschritte in nicht-perturbativen supersymmetrischen Eichtheorien und ihre Beziehung zu mathematischer Physik darzustellen. Im Speziellen diskutieren wir einige interessante Identitäten der Integrale, denen einfache und hypergeometrische Funktionen genügen und ihren Bezug zu supersymmetrischen Dualitäten in drei und vier Dimensionen. Methoden der exakten Berechnungen in supersymmertischen Eichtheorien sind auch auf integrierbare statistische Modelle anwendbar. Dies wird im letzten Kapitel der vorliegenden Arbeit behandelt. / In this thesis we discuss exact, non-perturbative results achieved using superconformal index technique in supersymmetric gauge theories with four supercharges (which is N = 1 supersymmetry in four dimensions and N = 2 supersymmetry in three). We use the superconformal index technique to test several duality conjectures for supersymmetric gauge theories. We perform tests of three-dimensional mirror symmetry and Seiberg-like dualities. The purpose of this thesis is to present recent progress in non-perturbative supersymmetric gauge theories in relation to mathematical physics. In particular, we discuss some interesting integral identities satisfied by basic and elliptic hypergeometric functions and their relation to supersymmetric dualities in three and four dimensions. Methods of exact computations in supersymmetric theories are also applicable to integrable statistical models, which we discuss in the last chapter of the thesis.

Neste trabalho estudamos as teorias de gauge puras (sem campo de matéria) na rede em três dimensões. Em especial, estudamos a subclasse das teorias topológicas. A maneira como denimos e tratamos as teorias de gauge e diferente, mas equivalente, à forma usual apresentada em [2, 3]. Definimos estas teorias via o formalismo de Kuperberg, que é um formalismo puramente matemático de um invariante topológico de variedades tridimensionais. Este formalismo, embora bastante abstrato, pode ser adaptado para descrever as classes de modelos das teorias de gauge na rede, e traz várias vantagens, pois possibilita que tratemos de teorias topológicas e não topológicas, além da fácil identicação dos limites topológicos da função de partição. Estudamos também a classe das teorias chamadas quase topológicas, que podem ser pensadas como deformações de teorias topológicas. Em particular, consideramos teorias de gauge com grupo de gauge Z2, que é o grupo de gauge mais simples possível com dinâmica não trivial. Dentro das teorias de gauge, identicamos as classes de modelos que são quase topológicos, além de outras classes nas quais a função de partição pode ser trivialmente calculada. A função de partição foi calculada explicitamente no caso quase topológico em duas situações: sobre a esfera tridimensional S3 e sobre o toroS1x S1x S1x, que representa uma rede com condições periódicas de contorno. Dois modelos físicos de teorias de gauge, ainda com grupo de gauge Z2, foram estudados: o modelo com ação de Wilson SW = Pfaces [Tr(g) - 1] e o modelo com ação spin-gauge SSG = Pfaces Tr(g). No limite de baixa temperatura ambos os modelos mostram-se ser topológicos, enquanto que no limite de alta temperatura mostraram-se ser trivialmente calculáveis. / In this work we studied the class of models of pure lattice gauge theories (without matter elds) in three dimensions. Especially, we studied the subclass of topological theories. Lattice gauge theories were dened in an unusual way, unlike the description shown in [2, 3]. We dened lattice gauge theories via the Kuperberg\'s formalism [4], which is a mathematical model for a topological invariant of 3-manifolds. Such formalism, although completely abstract, can describe the class of models of lattice gauge theories because it can describe both topological and non topological theories, besides it provides an easy identication of the partition function topological limits. We also studied the class of theories called quasi topological, which can be thought as deformations of topological theories. As an example, we consider Z2 as gauge group, because it is the simplest group that does not imply trivial dynamics. Inside this class of models we identify the subclasses of quasi topological theories and also other classes in which the partition function can be trivially computed. The partition function was explicitly computed in two situations: on the 3-sphere S3 and on the 3-manifold S1 x S1 x S1 that represents periodic boundary conditions. Two physical models were studied: the model with Wilson\'s action SW(conf)1 and the model with spin-gauge action SSG(conf)2. In the low temperature limit both models shown to be topological and in the high temperature limit they could be trivially computed.

<p>In this thesis, two applications of dimensional deconstruction are studied. The first application is a model for neutrino oscillations in the presence of a large decon- structed extra dimension. In the second application, Kaluza{Klein dark matter from a latticized universal extra dimension is studied. The goal of these projects have been twofold. First, to see whether it is possible to reproduce the relevant features of the higher-dimensional continuum theory, and second, to examine the effect of the latticization in experiments. In addition, an introduction to the the- ory of dimensional deconstruction as well as to the theory of continuous extra dimensions is given. Furthermore, the various higher-dimensional models, such as Arkani-Hamed{Dvali{Dimopolous (ADD) models and models with universal extra dimensions, that have been intensively studied in recent years, are discussed.</p>

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.

Les calculs de précision dans les théories de jauge jouent un rôle très important pour l’étude de la physique du Modèle Standard et au-delà dans les super-collisionneurs de particules comme le LHC, TeVatron et ILC. Par conséquent, il est extrêmement important de fournir des outils du calcul d’amplitudes à une boucle stables, rapides, efficaces et hautement automatisés. Cette thèse a pour but de développer la bibliothèque d’intégrales Golem95. Cette bibliothèque est un programme écrit en Fortran95, qui contient tous les ingrédients nécessaires pour calculer une intégrale scalaire ou tensorielle à une boucle avec jusqu’à six pattes externes. Golem95 utilise une méthode traditionnelle de réduction (réduction à la Golem) qui réduit les facteurs de forme en des intégrales de base redondantes qui peuvent être scalaires (sans paramètres de Feynman au numérateur) ou tensorielles (avec des paramètres de Feynman au numérateur); ce formalisme permet d’éviter les problèmes de l’instabilité numérique engendrés par des singularités factices dues à l’annulation des déterminants de Gram. En plus, cette bibliothèque peut être interfacée avec des programmes du calcul automatique basés sur les méthodes d’unitarité comme GoSam par exemple. Les versions antérieures de Golem95 ont été conçues pour le calcul des amplitudes sans masses internes. Le but de ce travail de thèse est de généraliser cette bibliothèque pour les configurations les plus générales (les masses complexes sont incluses), et de fournir un calcul numériquement stable dans les régions problématique en donnant une représentation intégrale unidimensionnelle stable pour chaque intégrale de base de Golem95. / Higher order corrections in gauge theories play a crucial role in studying physics within the standard model and beyond at TeV colliders, like LHC, TeVatron and ILC. Therefore, it is of extreme importance to provide tools for next-to-leading order amplitude computation which are fast, stable, efficient and highly automatized. This thesis aims at developing the library of integrals Golem95. This library is a program written in Fortran95, it contains all the necessary ingredients to calculate any one-loop scalar or tensorial integral with up to six external legs. Golem95 uses the traditional reduction method (Golem reduction) to reduce the form factors into redundant basic integrals, which can be scalar (without Feynman parameters in the numerator) or tensorial (with Feynman parameter in the numerator); this formalism allows us to avoid the problems of numerical instabilities generated by the spurious singularities induced by the vanishing of the Gram determinants. In addition, this library can be interfaced with automatic programs of NLO calculation based on the unitarity inspired reduction methods as GoSam for example. Earlierversions of Golem95 were designed for the calculation of amplitudes without internal masses. The purpose of this thesis is to extend this library for more general configurations (complex masses are supported); and to provide numerically stable calculation in the problematic regions (det(G) → 0), by providing a stable one-dimensional integral representation for each Golem95 basic integral.

Neste trabalho estudamos as teorias de gauge puras (sem campo de matéria) na rede em três dimensões. Em especial, estudamos a subclasse das teorias topológicas. A maneira como denimos e tratamos as teorias de gauge e diferente, mas equivalente, à forma usual apresentada em [2, 3]. Definimos estas teorias via o formalismo de Kuperberg, que é um formalismo puramente matemático de um invariante topológico de variedades tridimensionais. Este formalismo, embora bastante abstrato, pode ser adaptado para descrever as classes de modelos das teorias de gauge na rede, e traz várias vantagens, pois possibilita que tratemos de teorias topológicas e não topológicas, além da fácil identicação dos limites topológicos da função de partição. Estudamos também a classe das teorias chamadas quase topológicas, que podem ser pensadas como deformações de teorias topológicas. Em particular, consideramos teorias de gauge com grupo de gauge Z2, que é o grupo de gauge mais simples possível com dinâmica não trivial. Dentro das teorias de gauge, identicamos as classes de modelos que são quase topológicos, além de outras classes nas quais a função de partição pode ser trivialmente calculada. A função de partição foi calculada explicitamente no caso quase topológico em duas situações: sobre a esfera tridimensional S3 e sobre o toroS1x S1x S1x, que representa uma rede com condições periódicas de contorno. Dois modelos físicos de teorias de gauge, ainda com grupo de gauge Z2, foram estudados: o modelo com ação de Wilson SW = Pfaces [Tr(g) - 1] e o modelo com ação spin-gauge SSG = Pfaces Tr(g). No limite de baixa temperatura ambos os modelos mostram-se ser topológicos, enquanto que no limite de alta temperatura mostraram-se ser trivialmente calculáveis. / In this work we studied the class of models of pure lattice gauge theories (without matter elds) in three dimensions. Especially, we studied the subclass of topological theories. Lattice gauge theories were dened in an unusual way, unlike the description shown in [2, 3]. We dened lattice gauge theories via the Kuperberg\'s formalism [4], which is a mathematical model for a topological invariant of 3-manifolds. Such formalism, although completely abstract, can describe the class of models of lattice gauge theories because it can describe both topological and non topological theories, besides it provides an easy identication of the partition function topological limits. We also studied the class of theories called quasi topological, which can be thought as deformations of topological theories. As an example, we consider Z2 as gauge group, because it is the simplest group that does not imply trivial dynamics. Inside this class of models we identify the subclasses of quasi topological theories and also other classes in which the partition function can be trivially computed. The partition function was explicitly computed in two situations: on the 3-sphere S3 and on the 3-manifold S1 x S1 x S1 that represents periodic boundary conditions. Two physical models were studied: the model with Wilson\'s action SW(conf)1 and the model with spin-gauge action SSG(conf)2. In the low temperature limit both models shown to be topological and in the high temperature limit they could be trivially computed.

Efeitos térmicos em teorias de calibre em (2+1) dimensões são estudados em espaços onde as coordenadas podem ou não comutar. No caso comutativo, a dependência com a temperatura do tensor de polarização é calculada a um laço em teorias envolvendo tanto bósons quanto férmions. Como aplicação, são calculados os processos de blindagem em tais modelos, chegando ao interessante resultado de que cargas magnéticas não sofrem tais efeitos na QED3. Uma prova válida em qualquer ordem de perturbação é desenvolvida, confirmando este comportamento. Em teorias não comutativas, são estudadas as correções a um laço ao coeficiente de Chern-Simons, sendo encontrado que não existe o fenômeno da mistura UV/IR na teoria Chern-Simons-Higgs. O comportamento assintótico de tal coeficiente é analisado no regime de altas temperaturas. Vários outros aspectos envolvendo os efeitos térmicos em teorias de Chern-Simons são explorados. / Thermal effects in (2+1)-dimensional gauge theories are studied in both commutative as well as noncommutative manifolds. In the first situation, the finite temperature polarization tensor is computed a tone loop for fermionic and bosonic couplings. As an application, the screening masses are evaluated and it is found the surprising result that magnetic charges are not screened in QED3. It is demonstrated that this result holds to any order in pertubationtheory. In the noncommutative case, the one loop correction to the Chern-Simons coefficient is studied, and it is found that there is no UV/IR mixing in the Chern-Simons-Higgs model. The asymptotic behavior of such coefficient is analised in the high temperature regime. Several other interesting aspects involving thermal effects of Chern-Simons theories are also discussed.