Doctor of Philosophy / Department of Mathematics / David R. Auckly / We study geometric variational problems for a class of models in quantum field theory known as Faddeev-Skyrme models. Mathematically one considers minimizing an energy functional on homotopy classes of maps from closed 3-manifolds into homogeneous spaces of compact Lie groups. The energy minimizers known as Hopfions describe stable configurations of subatomic particles such as protons and their strong interactions. The Hopfions exhibit distinct localized knot-like structure and received a lot of attention lately in both mathematical and physical literature. High non-linearity of the energy functional presents both analytical and algebraic difficulties for studying it. In particular we introduce novel Sobolev spaces suitable for our variational problem and develop the notion of homotopy type for maps in such spaces that generalizes homotopy for smooth and continuous maps. As the spaces in question are neither linear nor even convex we take advantage of the algebraic structure on homogeneous spaces to represent maps by gauge potentials that form a linear space and reformulate the problem in terms of these potentials. However this representation of maps introduces some gauge ambiguity into the picture and we work out 'gauge calculus' for the principal bundles involved to apply the gauge-fixing techniques that eliminate the ambiguity. These bundles arise as pullbacks of the structure bundles H[arrow pointing right with hook on tail]G[arrow pointing right]G/H of homogeneous spaces and we study their topology and geometry that are of independent interest. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces we obtain a weaker result on existence of minimizers only in each 2-homotopy class.

Quantum Chromo Dynamics (QCD) is a relativistic field theory of a non-abelian gauge field coupled to several flavors of fermions. Two dimensional (one space and one time) QCD serves as an interesting toy model that shares several features with the four dimensional physically relevant theory. The main aim of the research is to study two dimensional QCD using the lattice regularization. Two dimensional QCD without any fermion content is solved analytically using lattice regularization. Explicit expressions for the expectation values of Wilson loops and the correlation of two Polyakov loops oriented in two different directions are obtained. Physics of the QCD vacuum is explained using these results. The Hamiltonian formalism of lattice QCD with fermion content serves as an approach to study quark excitations out of the vacuum. The formalism is first developed and techniques to numerically evaluate the spectrum of physical particles, namely, meson and baryons are described. The Hybrid Monte Carlo technique was used to numerically extract the lowest meson and baryon masses as a function of the quark masses. It is shown that neither the lowest meson mass nor the lowest baryon mass goes to zero as the quark mass is taken to zero. This numerically establishes the presence of a mass gap in two dimensional QCD.

O comportamento infra-vermelho dos propagadores de glúons e de ghosts é de fundamental importância para o entendimento do limite de baixas energias da cromodinâmica quântica (QCD), especialmente no que diz respeito ao problema do confinamento de quarks e de glúons. O objetivo desta tese é implementar um novo método para o estudo do propagador de glúons no calibre covariante linear para a QCD na rede. Em particular, analisamos em detalhe a nova implementação proposta e estudamos os algoritmos para fixação numérica deste calibre. Note que a fixação numérica da condição de calibre de Feynman apresenta vários problemas não encontrados nos casos de Landau e de Coulomb, o que impossibilitou por longo tempo o seu estudo adequado. De fato, a definição considerada inicialmente, por Giusti et. al., é de difícil implementação numérica e introduz condições espúrias na fixação de calibre. Como consequência, os únicos estudos efetuados anteriormente referem-se aos propagadores de glúons e de quarks em redes relativamente pequenas, não permitindo uma análise cuidadosa do limite infra-vemelho da QCD neste calibre. A obtenção de novas soluções para a implementação do calibre de Feynman na rede é portanto de grande importância para viabilizar estudos numéricos mais sistemáticos dos propagadores e dos vértices neste calibre e, em geral, no calibre covariante linear. / The infrared behavior of gluon and ghost propagators is of fundamental importance for the understanding of the low-energy limit of quantum chromodynamics (QCD), especially with respect to the problem of the confinement of quarks and gluons. The goal of this thesis is to implement a new method to study the gluon propagator in the linear covariant gauge in lattice QCD. In particular, we analyze in detail the newly proposed implementation and study the algorithms for numerically fixing this gauge. Note that the numerical fixing of the Feynman gauge condition poses several problems that are not present in the Landau and Coulomb cases, which prevented it from being properly studied for a long time. In fact, the definition considered initially, by Giusti et. al., is of difficult numerical implementation and introduces spurious conditions into the gauge fixing. As a consequence, the only studies carried out previously involved gluon and quark propagators on relatively small lattices, hindering a careful analysis of the infrared limit of QCD in this gauge. Obtaining new solutions for the implementation of the Feynman gauge on the lattice is therefore of great importance to enable more systematic numerical studies of propagators and vertices in this gauge and, in general, in the linear covariant gauge.

Die im infraroten Impulsbereich der Quantenchromodynamik (QCD) berechneten Gluon- und Ghost-Propagatoren spielen eine große Rolle für das sogenannte Confinement der Quarks und Gluonen. Sie sind Gegenstand intensiver Foschungen dank nicht-perturbativer Methoden basierend auf Dyson-Schwinger- (DS) und funktionalen Renormierungsgruppen-Gleichungen (FRG). Darüber hinaus sollte es deren Verhalten bei endlichen Temperaturen erlauben, den chiralen und Deconfinement-Phasenübergang bzw. das Crossover in der QCD besser aufzuklären. Unser Zugang beruht auf der gitter-diskretisierten QCD (LQCD), die es als ab-initio-Methode gestattet, verschiedenste störungstheoretisch nicht zugängliche QCD-Observablen der hadronischen Welt zu berechnen. Wir untersuchen das Temperaturverhalten der Gluon- und Ghost-Propagatoren in der Landau-Eichung für die reine Gluodynamik und die volle QCD. Für den Gluon-Propagator berechnen wir deren longitudinale (DL) sowie transversale (DT) Komponenten. Ziel ist es, Datensätze in Form von Fit-Formeln zu liefern, welche als Input für die DS- (oder FRG-) Gleichungen verwendet werden können. Wir beschäftigen uns mit der vollen (Nf=2) LQCD unter Verwendung der sogenannten twisted mass Fermiondiskretisierung. Von der tmfT-Kollaboration wurden uns dafür Eichfeldkonfigurationen für Temperaturen im Crossover-Bereich sowie jeweils für drei fixierte Pion-Massenwerte im Intervall [300, 500] MeV bereitgestellt. Schließlich berechnen wir innerhalb der reinen SU(3) Eichtheorie (bei T=0) den Landau Gluon-Propagator unter Verwendung verschiedener Eichfixierungskriterien. Unser Ziel ist es, den Einfluss von Eich-Kopien mit minimalen (nicht-trivialen) Eigenwerten des Faddeev-Popov-Operators zu verstehen. Eine solche Studie soll klären, wie Gribov-Kopien das Verhalten der Gluon- und Ghost-Propagatoren im infraroten Bereich prinzipiell beeinflussen. / Gluon and ghost propagators in quantum chromodynamics (QCD) computed in the infrared momentum region play an important role to understand quark and gluon confinement. They are the subject of intensive research thanks to non-perturbative methods based on Dyson-Schwinger (DS) and functional renormalization group (FRG) equations. Moreover, their temperature behavior might also help to explore the chiral and deconfinement phase transition or crossover within QCD at non-zero temperature. Our prime tool is the lattice discretized QCD (LQCD) providing a unique ab-initio non-perturbative approach to deal with the computation of various observables of the hadronic world. We investigate the temperature dependence of Landau gauge gluon and ghost propagators in pure gluodynamics and in full QCD. Regarding the gluon propagator, we compute its longitudinal DL as well its transversal DT components. The aim is to provide a data set in terms of fitting formulae which can be used as input for DS (or FRG) equations. We deal with full (Nf=2) LQCD with the twisted mass fermion discretization. We employ gauge field configurations provided by the tmfT collaboration for temperatures in the crossover region and for three fixed pion mass values in the range [300,500] MeV. Finally, within SU(3) pure gauge theory (at T=0) we compute the Landau gauge gluon propagator according to different gauge fixing criteria. Our goal is to understand the influence of gauge copies with minimal (non-trivial) eigenvalues of the Faddeev-Popov operator.

O comportamento infra-vermelho dos propagadores de glúons e de ghosts é de fundamental importância para o entendimento do limite de baixas energias da cromodinâmica quântica (QCD), especialmente no que diz respeito ao problema do confinamento de quarks e de glúons. O objetivo desta tese é implementar um novo método para o estudo do propagador de glúons no calibre covariante linear para a QCD na rede. Em particular, analisamos em detalhe a nova implementação proposta e estudamos os algoritmos para fixação numérica deste calibre. Note que a fixação numérica da condição de calibre de Feynman apresenta vários problemas não encontrados nos casos de Landau e de Coulomb, o que impossibilitou por longo tempo o seu estudo adequado. De fato, a definição considerada inicialmente, por Giusti et. al., é de difícil implementação numérica e introduz condições espúrias na fixação de calibre. Como consequência, os únicos estudos efetuados anteriormente referem-se aos propagadores de glúons e de quarks em redes relativamente pequenas, não permitindo uma análise cuidadosa do limite infra-vemelho da QCD neste calibre. A obtenção de novas soluções para a implementação do calibre de Feynman na rede é portanto de grande importância para viabilizar estudos numéricos mais sistemáticos dos propagadores e dos vértices neste calibre e, em geral, no calibre covariante linear. / The infrared behavior of gluon and ghost propagators is of fundamental importance for the understanding of the low-energy limit of quantum chromodynamics (QCD), especially with respect to the problem of the confinement of quarks and gluons. The goal of this thesis is to implement a new method to study the gluon propagator in the linear covariant gauge in lattice QCD. In particular, we analyze in detail the newly proposed implementation and study the algorithms for numerically fixing this gauge. Note that the numerical fixing of the Feynman gauge condition poses several problems that are not present in the Landau and Coulomb cases, which prevented it from being properly studied for a long time. In fact, the definition considered initially, by Giusti et. al., is of difficult numerical implementation and introduces spurious conditions into the gauge fixing. As a consequence, the only studies carried out previously involved gluon and quark propagators on relatively small lattices, hindering a careful analysis of the infrared limit of QCD in this gauge. Obtaining new solutions for the implementation of the Feynman gauge on the lattice is therefore of great importance to enable more systematic numerical studies of propagators and vertices in this gauge and, in general, in the linear covariant gauge.

On étudie les théories de Yang-Mills. Pour ce faire, nous appliquons une nouvelle procédure de fixation de jauge qui vise à prendre en compte la présence des copies de Gribov. Ces copies correspondent à des solutions supplémentaires de la condition de jauge et ne sont pas prises en compte dans la procédure standard de Faddeev-Popov. Cette nouvelle procédure de fixation de jauge a d'abord été implémenté dans la jauge de Landau, où le régime de basse énergie a pu être étudié simplement par la théorie de perturbation et les propagateurs des gluons et des ghosts ont été trouvé en bon accord avec les résultats du réseau. Dans une première partie, nous appliquons cette procédure à une classe de jauges covariantes et non-linéaires (les jauges de Curci-Ferrari-Delbourgo-Jarvis). Nous montrons que ces jauges sont renormalisables en dimension quatre et donnons explicitement les expressions des constantes de renormalisation à une boucle. Nous calculons en théorie de perturbation les propagateurs de la théorie à l'ordre d'une boucle et implémentons le groupe de renormalisation. La seconde partie concerne l'étude du cas à température finie et de la transition de phase confinement-déconfinement. Nous travaillons dans une extention massive de la jauge de Landau-DeWitt. Nous calculons les propagateurs à une boucle et montrons qu'ils présentent de clairs signaux de la transition de phase à la différence de la jauge de Landau. / We investigate Yang-Mills theories. In particular, we follow a recently proposed new gauge-fixing procedure that aims at dealing with the presence of the so-called Gribov copies. These copies correspond to additional solutions to the gauge equation that are disregarded in the standard Faddeev-Popov procedure. This novel gauge-fixing approach was first implemented in the Landau gauge, where the low momentum regime was investigable by means of simple perturbation theory and the one-loop gluon and ghost propagators were found in good agreement with lattice results. In a first part, we extend this proposal to a class of nonlinear covariant (the Curci-Ferrari-Delbourgo-Jarvis) gauges . We prove that these gauges are renormalizable in four dimensions. We provide explicit expression of the renormalization constants at one-loop order. Then we compute the various propagators of the theory at one-loop order with and without renormalization group improvement. The second part of the thesis concerns the finite temperature case and in particular the study of the confinement-deconfinement phase transition. We work in the Landau-DeWitt gauge (a background extention of the Landau gauge) which allows for an explicit presence of an order parameter of the phase transition. This gauge is implemented following the previous gauge-fixing procedure. In particular it has been shown that the phase transition can be studied in perturbation theory. Here, we compute at one-loop order the gluon and ghost propagators (for SU(2) gauge group) and show that they display strong signals of the phase transition. This is to be put in regards with the results obtained for the Landau gauge propagators.