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.

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In this dissertation we study several topics related to the geometry and physics of heterotic string compactification. After an introduction to some of the basic ideas of this field, we review the heterotic line bundle standard model construction and a complex structure mod- uli stabilization mechanism associated to certain hidden sector gauge bundles. Once this foundational material has been presented, we move on to the original research of this disser- tation. We present a scan over all known heterotic line bundle standard models to examine the frequency with which the particle spectrum is forced to change, or "jump," by the hidden sector moduli stabilization mechanism just mentioned. We find a significant percentage of forced spectrum jumping in those models where such a change of particle content is possible. This result suggests that one should consider moduli stabilization concurrently with model building, and that failing to do so could lead to misleading results. We also use state of the art techniques to study Yukawa couplings in these models. We find that a large portion of Yukawa couplings which naively would be expected to be non-zero actually vanish due to certain topological selection rules. There is no known symmetry which is responsible for this vanishing. In the final part of this dissertation, we study the Chern-Simons contribution to the superpotential of heterotic theories. This quantity is very important in determining the vacuum stability of these models. By explicitly building real bundle morphisms between vec- tor bundles over Calabi-Yau manifolds, we show that this contribution to the superpotential vanishes in many cases. However, by working with more complicated, and realistic geome- tries, we also present examples where the Chern-Simons contribution to the superpotential is non-zero, and indeed fractional. / Doctor of Philosophy / String theory is a candidate for a unified theory of all of the known interactions of nature. To be consistent, the theory needs to be formulated in 9 spatial dimensions, rather than the 3 of everyday experience. To connect string theory with reality, we need to reproduce the known physics of 3 dimensions from the 9 dimensional theory by hiding, or "compactifying," 6 directions on a compact internal space. The most common choice for such an internal space is called a Calabi-Yau manifold. In this dissertation, we study how the geometry of the Calabi-Yau manifold determines physical quantities seen in 3 dimensions such as the number of particle families, particle interactions and potential energy. The first project in this dissertation studies to what extent the process of making the Calabi-Yau manifold rigid, something which is required observationally, affects the particle spectrum seen in 3 dimensions. By scanning over a large model set, we conclude that computation of the particle spectrum and such "moduli stabilization" issues should be considered in concert, and not in isolation. We also showed that a large portion of the interactions that one would naively expect between the particles in such string models are actually absent. There is no known symmetry of the theory that accounts for this structure, which is linked to the topology of the extra spatial dimensions. In the final part of the dissertation, we show how to calculate previously unknown contributions to the potential energy of these string theory models. By linking to results from the mathematics literature, we show that these contributions vanish in many cases. However, we present examples where it is non-zero, a fact of crucial importance in understanding the vacua of heterotic string theories.

Estudamos as teorias não-relativísticas e não-comutativas de campos de spin zero e l/2 acoplado minimamente com o campo ele Chern-Simons em 2+ 1 dimensões. Na situação comutativa o modelo escalar foi usado para simular o efeito Aharonov-Bohrn na abordagem da teoria de campos. Na teoria escalar verificamos que, contrariamente ao resultado comutativo, a inclusão ele urna auto--interação quártica do campo escalar não ó necessária para garantir a renormalização ultravioleta do modelo. Entretanto, para obter um limite comutativo analítico a presença ele uma auto-interação quártica é exigida. Mostramos para o caso ele partículas ele spin 1/2 que a contribuição em um laço para a matriz ele espalhamento contendo o termo de Pauli é puramente não--planar. O termo de Pauli desempenha a mesma função ela auto-interação quártica como no caso escalar. Para valores pequenos do parâmetro da não--comutatividade determinamos as correções para o espalhamento Aharonov-Bohm e provamos que, até ordem de um laço, os modelos são livres de singularidades ultravioleta/infravermelha. / We study noncommutative nonrelativistic theories of spin 0 and 1/2 field coupled to thc Chern-Sirnons field in 2+1 dimensions. In the commutative situation the scalar model has been used to simulate the Aharonov-Bohm effect in the field theory context. We verified that, contrarily to the commutative result, the inclusion of a quartic self-interaction of the scalar field is not necessary to secure the ultraviolet renormalization of the model. However, to obtain a smooth commutative limit the presence of a quartic gauge invariant self-interaction is required. For the case of spin 1/2 particles we show that the one-loop contributions to the that scattering matrix the which contain the Pauli\'s term are purely nonplanar. Thc Pauli\'s term plays the same role of a quartic self-interaction in the scalar case. For small values of the noncommutative parameter we fix the corrections to the Aharonov-Bohm scattering and prove that up to one-loop the models are free from dangerous infrared/ultraviolet divergences.

Este trabalho busca explorar a definição dada por Connes em [Con01] do caráter de Chern para a geometria não-comutativa. Construímos os funtores K 0 e K 1 com os principais resultados para demonstrarmos a Sequência Exata de Seis Termos e a Sequência de Mayer-Vietoris. Calculamos os grupos de K-teoria de algumas álgebras de operadores pseudo-diferenciais clássicos de ordem zero. Posteriormente usamos as sequências exatas para calcular explicitamente o caráter de Chern-Connes nos C -sistemas dinâmicos. / This work intends to explore the definition given by Connes in [Con01] of the Chern charac- ter for noncommutative geometry. We construct the functors K 0 and K 1 with the main results to demonstrate the Exact Sequence of Six Terms and the Sequence of Mayer Vietoris. We compute the K-groups of some algebras of classical zero-order pseudo-differential operators. We then use the exact sequences to explicitly calculate the Chern-Connes Character of C -dynamic systems.

Depuis 1980, il est un problème ouvert de donner des formules cohomologiques pour l'indice basique d'un opérateur différentiel basique transversalement elliptique sur un fibré vectoriel au dessus d'une variété feuilletée. Dans les années 1990, El Kacimi-Alaoui a proposé d'utiliser la théorie de Molino pour étudier cette indice. Molino a montré qu'à tout feuilletage Riemannien transversalement orienté, nous pouvons associer une variété, appelée variété basique, qui est munie d'une action du groupe orthogonal, El Kacimi-Alaoui a montré comment associer à l'opérateur basique transversalement elliptique un opérateur sur un fibré vectoriel, appelé fibré utile, au dessus de la variété basique.L'idée est d'obtenir la formule cohomologique espérée à partir des résultats sur l'opérateur sur le fibré utile. Cette thèse est une première étape dans cette direction. Lorsque le feuilletage Riemannien est de Killing, Goertsches et Töben ont remarqué qu'il existe un isomorphisme cohomologique naturel entre la cohomologie basique équivariante du feuilletage de Killing et la cohomologie équivariante de la variété basique.Le résultat principal de cette thèse est de donner une réalisation géométrique de l'isomorphisme cohomologique ci-dessus à travers les caractères de Chern sous certaine Hypothèse. / From 1980s, it is an open problem of proposing cohomologic formula for the basic index of a transversally elliptic basic differential operator on a vector bundle over a foliated manifold. In 1990s, El Kacimi-Alaoui has proprosed to use the Molino theory for study this index. Molino has proved that to every transversally oriented Riemannien foliation, we can associate a manifold, called basique manifold, which is équiped with an action of orthogonal group, El Kacimi-Alaoui has shown how to associate a transversally elliptic basic differential operator an operator on a vector bundle, called useful bundle, over the basique manifold.The idea is to obtain the desired cohomologic formula from résultats about the operator on the useful bundle. This thesis is a first step in this direction. While the Riemannien foliation is Killing, Goertsches et Töben have remarked that there exists a naturel cohomologic isomorphism between the equivariant basique cohomology of the Killing foliation and the equivariant cohomology of the basique manifold.The principal result of this thesis is the geometric realisation of the cohomologic isomorphism by Chern characters under some hypothèses.

Estudamos as teorias não-relativísticas e não-comutativas de campos de spin zero e l/2 acoplado minimamente com o campo ele Chern-Simons em 2+ 1 dimensões. Na situação comutativa o modelo escalar foi usado para simular o efeito Aharonov-Bohrn na abordagem da teoria de campos. Na teoria escalar verificamos que, contrariamente ao resultado comutativo, a inclusão ele urna auto--interação quártica do campo escalar não ó necessária para garantir a renormalização ultravioleta do modelo. Entretanto, para obter um limite comutativo analítico a presença ele uma auto-interação quártica é exigida. Mostramos para o caso ele partículas ele spin 1/2 que a contribuição em um laço para a matriz ele espalhamento contendo o termo de Pauli é puramente não--planar. O termo de Pauli desempenha a mesma função ela auto-interação quártica como no caso escalar. Para valores pequenos do parâmetro da não--comutatividade determinamos as correções para o espalhamento Aharonov-Bohm e provamos que, até ordem de um laço, os modelos são livres de singularidades ultravioleta/infravermelha. / We study noncommutative nonrelativistic theories of spin 0 and 1/2 field coupled to thc Chern-Sirnons field in 2+1 dimensions. In the commutative situation the scalar model has been used to simulate the Aharonov-Bohm effect in the field theory context. We verified that, contrarily to the commutative result, the inclusion of a quartic self-interaction of the scalar field is not necessary to secure the ultraviolet renormalization of the model. However, to obtain a smooth commutative limit the presence of a quartic gauge invariant self-interaction is required. For the case of spin 1/2 particles we show that the one-loop contributions to the that scattering matrix the which contain the Pauli\'s term are purely nonplanar. Thc Pauli\'s term plays the same role of a quartic self-interaction in the scalar case. For small values of the noncommutative parameter we fix the corrections to the Aharonov-Bohm scattering and prove that up to one-loop the models are free from dangerous infrared/ultraviolet divergences.

Cette thèse est faite de deux parties. La première partie est un article rédigé conjointementavec Martin Puchol et Jialin Zhu. La deuxième partie est une série de résultats obtenus par moi-même liés au théorème de Riemann-Roch-Grothendieck pour les fibrés vectoriels plats. Dans la première partie, nous donnons une preuve analytique d'un résultat décrivant le comportement de la torsion analytique en théorie de de Rham lorsque la variété considérée est séparée en deux par une hypersurface. Plus précisément, nous donnons une formule liant la torsion analytique de la variété entière aux torsions analytiques associées aux variétés à bord avec des conditions limites relative ou absolue le long de l'hypersurface. Dans la deuxième partie de cette thèse, nous raffinons les résultats de Bismut-Lott pour les images directes des fibrés vectoriels plats au cas où le fibré vectoriel plat en question est lui-même la cohomologie holomorphe d'un fibré vectoriel le long d'une fibration plate à fibres complexes. Dans ce contexte, nous donnons une formule de Riemann-Roch-Grothendieck dans laquelle la classe de Todd du fibré tangent relatif apparaît explicitement. En remplaçant les classes de cohomologie par des formes explicites qui les représentent en théorie de Chern-Weil, nous généralisons ainsi des constructions de Bismut-Lott. / This thesis consists of two parts. The first part is an article written jointly with Martin Puchol and Jialin Zhu, the second part is a series of results obtained by myself in connection with the Riemann-Roch-Grothendieck theorem for flat vector bundles. In the first part, we give an analytic approach to the behavior of classical Ray-Singer analytic torsion in de Rham theory when a manifold is separated along a hypersurface. More precisely, we give a formula relating the analytic torsion of the full manifold, and the analytic torsion associated with relative or absolute boundary conditions along the hypersurface. In the second part of this thesis, we refine the results of Bismut-Lott on direct images of flat vector bundles to the case where the considered flat vector bundle is itself the fiberwise holomorphic cohomology of a vector bundle along a flat fibration by complex manifolds. In this context, we give a formula of Riemann-Roch-Grothendieck in which the Todd class of the relative holomorphic tangent bundle appears explicitly. By replacing cohomology classes by explicit differential forms in Chern-Weil theory, we extend the constructions of Bismut-Lott in this context.

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