Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Document formatted into pages; contains x, 94 p.; also includes graphics Includes bibliographical references (p. 52-54). Available online via OhioLINK's ETD Center

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Les effets non-perturbatifs jouent un rôle extrêmement important dans la physique théorique contemporaine. Par exemple, ils sont connus d’être responsables de divers phénomènes physiques tels que le confinement des quarks et les dualités. Une arène particulièrement riche pour ces effets est fournie par des théories de jauge et des cordes. En particulier, pour les théories avec la supersymétrie N=2 à 4 dimensions, les années récentes ont marqué un progrès remarquable en compréhension leur dynamique non-perturbative. Peut-être un des résultats les plus intrigants est l'apparition de l’intégrabilité dans plusieurs problèmes de cette sorte. Ces résultats fournissent des relations non-triviales entre différents systèmes physiques et constructions mathématiques et donnent l’espoir que beaucoup de problèmes de longue date peuvent être en fait exactement solubles. La thèse est supposée d’explorer ces relations entre les effets non-perturbatifs et l’intégrabilité. Plus précisément, il est suggéré d'étudier le problème des compactifications des théories des cordes qui préservent la supersymétrie N=2. Leur action effective à basses énergies est complètement déterminée par une métrique sur un certain espace de modules qui est connu à recevoir des corrections instantoniques. Bien que beaucoup d'entre elles aient été déjà trouvées, la description non-perturbative complète est encore absente. D'autre part, sa importance va beaucoup au delà de l'action effective mentionnée ci-dessus puisqu'elle devrait contenir des informations sur la S-dualité, symétrie miroir non-perturbative, croisement de murs, trous noirs supersymétriques. Ainsi, le travail dans cette direction permettra non seulement obtenir des résultats intéressants, mais étudier également beaucoup de proches sujets dans la théorie des cordes, mathématiques et d'autres domaines de recherche. / Non-perturbative effects play an extremely important role in the moderntheoretical physics. For example, they are known to be responsible forvarious physical phenomena such as confinement and dualities.An especially rich arena for these effects is provided by gauge andstring theories. In particular, for theories with N=2 supersymmetry in4 dimensions, recent years marked a remarkable progress in understandingtheir non-perturbative dynamics. May be one of the most intriguing findingsis the appearance of integrability in several, sometimes differentlylooking problems. These results provide non-trivial relations betweendifferent physical systems and mathematical constructions and givea hope that many longstanding problems can be in fact exactly solvable.The thesis is supposed to explore these relations between non-perturbativeeffects and integrability. More precisely, it is suggested to study the problemof compactifications of string theories which preserve N=2 supersymmetry.Their low-energy effective action is completely determined by a metricon a certain moduli space which is known to receive instanton corrections.Although many of them have been already found, the complete non-perturbativedescription is still out of reach. On the other hand, its relevance goes muchbeyond the effective action mentioned above since it should encode informationabout S-duality, non-perturbative mirror symmetry, wall-crossing, supersymmetricblack holes. Thus, working in this direction would allow not only to obtaininteresting results, but also to study many related subjects in string theory,mathematics and other research areas.

This thesis studies N=(2,2) gauged linear sigma models (GLSMs) and three-dimensional N=2 Chern-Simons-matter theories and their mathematical applications. After a brief review of GLSMs, we systematically study nonabelian GLSMs for symplectic and orthogonal Grassmannians, following up a proposal in the math community. As consistency checks, we have compared global symmetries, Witten indices, and Calabi-Yau conditions to geometric expectations. We also compute their nonabelian mirrors following the recently developed nonabelian mirror symmetry. In addition, for symplectic Grassmannians, we use the effective twisted superpotential on the Coulomb branch of the GLSM to calculate the ordinary and equivariant quantum cohomology of the space, matching results in the math literature. Then we discuss 3d gauge theories with Chern-Simons terms. We propose a complementary method to derive the quantum K-theory relations of projective spaces and Grassmannians from the corresponding 3d gauge theory with a suitable choice of the Chern-Simons levels. In the derivation, we compare to standard presentations in terms of Schubert cycles, and also propose a new description in terms of shifted Wilson lines, which can be generalized to symplectic Grassmannians. Using this method, we are able to obtain quantum K-theory relations, which match known math results, as well as make predictions. / Doctor of Philosophy / In this thesis, we study two specific models of supersymmetric gauge theories, namely two-dimensional N=(2,2) gauged linear sigma models (GLSMs) and three-dimensional N=2 Chern-Simons-matter theories. These models have played an important role in quantum field theory and string theory for decades, and generated many fruitful results, improving our understanding of Nature by drawing on many branches in mathematics, such as complex differential geometry, intersection theory, quantum cohomology/quantum K-theory, enumerative geometry, and many others. Specifically, this thesis is devoted to studying their applications in quantum cohomology and quantum K-theory. In the first part of this thesis, we systematically study two-dimensional GLSMs for symplectic and orthogonal Grassmannians, generalizing the study for ordinary Grassmannians. By analyzing the Coulomb vacua structure of the GLSMs for symplectic Grassmannians, we are able to obtain the ordinary and equivariant quantum cohomology for these spaces. A similar methodology applies to 3d Chern-Simons-matter theories and quantum K-theory, for which we propose a new description in terms of shifted Wilson lines.

O caso finito de teorias topológicas de 1-gauge, quando nenhuma simetria global está presente, é bastante bem compreendido e classificado. Nos últimos anos, as tentativas de generalizar as teorias de 1-gauge através das chamadas teorias de 2-gauge abriram a porta para novos modelos interessantes e novas fases topológicas, as quais não são descritas pelos esquemas de classificação anteriores. Nesta tese, vamos além da construção de 2-gauge, e consideramos uma classe de modelos que vivem em maiores dimensões. Esses modelos estão inseridos em uma estrutura de complexos de cadeia de grupos abelianos, forçando a generalizar o conceito usual de configurações de gauge. A vantagem de tal abordagem é que, a ordem topológica fica manifestamente explcita. Isto é feito em ter- mos de uma cohomologia com coeficientes em um complexo de cadeia finita. Além disso, mostramos que a degenerescência do estado fundamental suporta um conjunto conve- niente de números quânticos que indexam os estados e que, além, foram completamente caracterizados. Consequentemente, nós também mostramos que muitos dos exemplos abelianos de teorias de 1 -gauge 2-gauge são recuperados como casos especiais desta construção. / The finite case of 1-gauge topological theories, when no global symmetries are present, is fairly well understood and classified. In recent years, attempts to generalize the latter situation through the so called 2-gauge theories have opened the door to interesting new models and new topological phases, not described by the previous schemes of classifica- tion. In this paper we go even beyond the 2-gauge construction by considering a class of models that live in arbitrary higher dimensions. These models are embedded in a structure of chain complexes of abelian groups, forcing to generalize the usual notion of gauge configurations. The advantage of such an approach is that, the topological order is explicitly manifest when the ground state space of these models is described. This is done in terms of a cohomology with coefficients in a finite chain complex. Furthermore, we show that the ground state degeneracy underpins a convenient set of quantum num- bers that label the states and that have been completely characterized. We also show that abelian examples of 1-gauge 2-gauge theories are recovered as special cases of this construction.

O caso finito de teorias topológicas de 1-gauge, quando nenhuma simetria global está presente, é bastante bem compreendido e classificado. Nos últimos anos, as tentativas de generalizar as teorias de 1-gauge através das chamadas teorias de 2-gauge abriram a porta para novos modelos interessantes e novas fases topológicas, as quais não são descritas pelos esquemas de classificação anteriores. Nesta tese, vamos além da construção de 2-gauge, e consideramos uma classe de modelos que vivem em maiores dimensões. Esses modelos estão inseridos em uma estrutura de complexos de cadeia de grupos abelianos, forçando a generalizar o conceito usual de configurações de gauge. A vantagem de tal abordagem é que, a ordem topológica fica manifestamente explcita. Isto é feito em ter- mos de uma cohomologia com coeficientes em um complexo de cadeia finita. Além disso, mostramos que a degenerescência do estado fundamental suporta um conjunto conve- niente de números quânticos que indexam os estados e que, além, foram completamente caracterizados. Consequentemente, nós também mostramos que muitos dos exemplos abelianos de teorias de 1 -gauge 2-gauge são recuperados como casos especiais desta construção. / The finite case of 1-gauge topological theories, when no global symmetries are present, is fairly well understood and classified. In recent years, attempts to generalize the latter situation through the so called 2-gauge theories have opened the door to interesting new models and new topological phases, not described by the previous schemes of classifica- tion. In this paper we go even beyond the 2-gauge construction by considering a class of models that live in arbitrary higher dimensions. These models are embedded in a structure of chain complexes of abelian groups, forcing to generalize the usual notion of gauge configurations. The advantage of such an approach is that, the topological order is explicitly manifest when the ground state space of these models is described. This is done in terms of a cohomology with coefficients in a finite chain complex. Furthermore, we show that the ground state degeneracy underpins a convenient set of quantum num- bers that label the states and that have been completely characterized. We also show that abelian examples of 1-gauge 2-gauge theories are recovered as special cases of this construction.

Le mécanisme d'instabilité gravitationnelle d'un champ de Yang-Mills est étudié via l'intégration numérique de la formulation hamiltonienne du système Einstein-Yang-Mills décrivant le couplage d'un champ de jauge à la gravitation. Une évolution en deux temps est mise en évidence: une dilution des fluctuations, conséquence de l'invariance conforme du champ, apparaît en premier lieu ; elle est suivie d'un régime d'oscillations croissantes lorsque l'on s'éloigne suffisamment de la solution homogène. Une comparaison instructive avec le mécanisme d'instabilité gravitationnelle du champ scalaire est également envisagée. Enfin, nous avons étudié l'influence de champs scalaires de quintessence sur la formation d'amas de matière noire grâce à la modification d'un code à N particules. Ceux-ci inhibent la formation des amas, en privilégiant des structures moins nombreuses et de faible masse, tout en produisant des différences assez significatives que pour permettre de discerner non seulement un modèle avec quintessence d'un autre plus conventionnel (avec constante cosmologique) mais également les divers modèles de quintessence entre eux. / The gravitational instability of Yang-Mills fields is studied by means of a numerical integration of the hamiltonian formulation of Einstein-Yang-Mills equations, which describe the coupling between gravitation and a gauge field. A two-step evolution appears to rule this mechanism: the fluctuations first dilute, as a sequel of the conformal invariance of the gauge theory; then, the fluctuations undergo oscillations of increasing amplitude as the solution moves away from the homogeneous one. An interesting comparison with the gravitational instability of a scalar field has also been made. Finally, we have established that the quintessence scalar fields inhibit the formation of dark matter halos. By analysing the results of a modified N-body code, we show that those fields produce less structures and lighter halos, and lead to significative differences that allow to distinguish either a quintessence scenario from a more conventional one with a cosmological constant either different quintessence models.

We calculate the string tensions, mass spectrum, and deconfining temperatures of <i>SO(N</i>) gauge theories in 2+1 dimensions. After a review of lattice field theory, we describe how we simulate the corresponding lattice gauge theories, construct operators to project on to specific states, and extrapolate values to the continuum limit. We discuss how to avoid possible complications such as finite size corrections and the bulk transition. <i>SO(N</i>) gauge theories have become recently topical since they do not have a fermion sign problem, are orbifold equivalent to <i>SU(N</i>) gauge theories, and share a common large-<i>N</i> limit in their common sector of states with <i>SU(N</i>) gauge theories. This motivates us to compare the physical properties of <i>SO(N</i>) and <i>SU(N</i>) gauge theories between 'group equivalences', which includes Lie algebra equivalences such as <i>SO</i>(6) and <i>SU</i>(4), and particularly a large-<i>N</i> equivalence. We discuss the large-<i>N</i> orbifold equivalence between <i>SO(N</i>) and <i>SU(N</i>) gauge theories, which relates the large-<i>N</i> gauge theories perturbatively. Using large-<i>N</i> extrapolations at fixed 't Hooft coupling, we test to see if <i>SO(N</i>) gauge theories and <i>SU(N</i>) gauge theories share non-perturbative properties at the large-<i>N</i> limit. If these group equivalences lead to similar physics in the gauge theories, then we could imagine doing finite chemical potential calculations that are currently intractable in <i>SU(N</i>) gauge theories by calculating equivalent quantities in the corresponding <i>SO(N</i>) gauge theories. We show that the <i>SO(N</i>) and <i>SU(N</i>) values match between group equivalences and at the large-<i>N</i> limit.

Desenvolvemos aqui um estudo das formulações alternativas-equivalentes da Relatividade Geral, baseada no formalismo de conexões de Ashtekar. Iniciamos discutindo a estrutura matemática necessária de fibrados e conexões, e a teoria de sistemas Hamiltonianos vinculados. Em seguida, damos uma breve introdução ao formalismo métrico de Einstein e então passamos ao formalismo geometrodinâmico canônico (formalismo ADM). Introduzimos as transformações no espaço de fase que geram as formulações alternativas, de forma generalizada tal que possamos obter ambas as variáveis complexas de Ashtekar ou as variáveis reais de Barbero, ou mesmo qualquer forma intermediária por meio do parâmetro de Immirzzi. / We develop here a study of the alternative-equivalent formulations of General Relativity, based on Ashtekars connexion formalism. We begin discussing the mathematical structure needed of fibre bundles and connexions, and the theory of constrained Hamiltonian systems. Next, we give a brief introduction for Einsteins metric formalism and then we pass to the canonical geometrodynamic formalism (ADM formalism). We introduce the transformations of the phase space which generate the alternative formulations, in a generalized form such that we can obtain both Ashtekars complex variables or Barberos real variables, or even any intermediary form by using the Immirzzi parameter.

Nós trabalhamos em três problemas relacionados com as teorias de gauge a temperatura finita. O primeiro discute a invariância de gauge da massa física do elétron num espaço de dimensão arbitrária a temperatura zero. Obtivemos a massa física a partir do polo do propagador fermiônico e demonstramos que a maneira usual de definir este propagador funciona para gauges covariantes mas não para gauges não covariantes. Em seguida propusemos um novo propagador e verificamos de duas formas diferentes que a massa física obtida a partir deste funciona para um gauge definido com parâmetros de controle tais que ele possa ser generalizado para as duas classes estudadas. O segundo problema é sobre a interação de n fótons num espaço de (1+1) dimensões no limite de altas temperaturas. Usando o formalismo de tempo imaginário e o modelo de Schwinger, mostramos que todos os termos das amplitudes causais retardadas com um ou mais loops têm contribuição nula. Interpretamos fisicamente este resultado e fizemos um paralelo de como ele se relaciona com a invariância CPT da teoria. A última parte é relacionada à gravitação quântica em (3+1) dimensões. Discutimos a possibilidade de obtermos as funções de n grávitons 1PI nos limites estático e de comprimento de onda longo em função de polinômios que podem ser escritos e relacionados de uma maneira simples. Para tanto, usamos as identidades de Ward e a invariância de Weyl de forma a relacionar as funções de n e (n+1) grávitons. Em seguida, utilizamos o formalismo da equação de transporte de Boltzmann para compreender melhor os resultados. / We have worked in three problems related to finite temperature gauge theory. The first one discusses the gauge invariance of the electron physical mass in an arbitrary dimension space at zero temperature. We have obtained the physical mass from the pole of the fermion propagator and we have demonstrated that the usual form to define this propagator works well for covariant gauges, but not for non covariant gauges. Then, we have proposed anew fermion propagator and we verified in two different ways that the physical mass obtained from this new one works for a gauge defined with control parameters so that it could be generalized for both classes studied. The second problem is on the n photon interaction in a space with (1+1) dimensions at hard thermal loops. Using the imaginary time formalism and the Schwinger\'s model, we have shown that all terms of the retarded causal amplitudes with one or more loops have null contribution. We have got a physical interpretation of this result and we have done a parallel of how it relates with the CPT invariance of this theory. The last one is related with quantum gravitation in (3+1) dimensions. We have discussed the possibility to obtain the 1PI n graviton functions in static and long-wavelength limits from polynomials which could be written and related in a simple manner. To this end, we used the Ward identities and the Weyl invariance to relate the n and (n+1) graviton functions. Then, we used the Boltzmann transport equation formalism to get a better understanding of the results.