O objetivo desse trabalho é fazer uma enumeração dos fibrados planos reais sobre algumas superfícies, como por exemplo, a esfera e o g-toro. Entre outras ferramentas, utilizamos a co-homologia das superfícies, com coeficientes locais, e também o método desenvolvido por Larmore para contar classes de homotopia de levantamento de funções. / The aim of this work is enumerate the plane bundles over some surfaces, for example the sphere and the g-torus. Among other tools we used cohomology of the surfaces with local coefficients and also the method developed by Larmore to count homotopy classes of lifting of functions.

Cette thèse est dédiée à l'étude des espaces de modules de fibrés sur une courbe algébrique et lisse sur le corps des nombres complexes.  Le texte est composé de deux parties : Dans la première partie, je m'intéresse à la géométrie liée aux classifications de fibrés quasi-paraboliques de rang 2 sur une courbe elliptique 2-pointée, à isomorphisme près. Les notions d'indécomposabilité, simplicité et stabilité de fibrés donnent lieu à des espaces de modules qui classifient ces objets.  La structure projective de ces espaces est décrite explicitement, et on prouve un théorème de type Torelli qui permet de retrouver la courbe elliptique 2-pointée.  Cet espace de modules est aussi mis en relation avec l'espace de modules de fibrés quasi-paraboliques sur une courbe rationnelle 5-pointée, qui apparaît naturellement comme revêtement double de l'espace de modules de fibrés quasi-paraboliques sur la courbe elliptique 2-pointée. Finalement, on démontre explicitement la modularité des automorphismes de cet espace de modules. Dans la deuxième partie, j'étudie l'espace de modules de fibrés semistables de rang 2 et déterminant trivial sur une courbe hyperelliptique. Plus précisément, je m'intéresse à l'application naturelle donnée par le fibré déterminant, générateur du groupe de Picard de cet espace de modules. Cette application  s'identifie à l'application theta, qui est de degré 2 dans notre cas. On définit une fibration de cet espace de modules vers un espace projective dont la fibre générique est birationnelle à l'espace de modules de courbes rationnelles 2g-épointées, et on décrit la restriction de theta aux fibres de cette fibration. On montre que cette restriction est, à une transformation birationnelle près, une projection osculatoire centrée en un point. En utilisant une description due à Kumar, on démontre que la restriction de l'application theta à cette fibration ramifie sur la variété de Kummer d'une certaine courbe hyperelliptique de genre g – 1. / This thesis is devoted to the study of moduli spaces of vector bundles over a smooth algebraic curve over field of complex numbers. The text consist of two main parts : In the first part, I investigate the geometry related to the classifications of rank 2 quasi-parabolic vector bundles over a 2-pointed elliptic curves, modulo isomorphism. The notions of indecomposability, simplicity and stability give rise to the corresponding moduli spaces classifying these objects. The projective structure of these spaces is explicitely described, and we prove a Torelli theorem that allow us to recover the 2-pointed elliptic curve. I also explore the relation with the moduli space of quasi-parabolic vector bundles over a 5-pointed rational curve, appearing naturally as a double cover of the moduli space of quasi-parabolic vector bundles over the 2-pointed elliptic curve. Finally, we show explicitely the modularity of the automorphisms of this moduli space. In the second part, I study the moduli space of semistable rank 2 vector bundles with trivial determinant over a hyperelliptic curve C. More precisely, I am interested in the natural map induced by the determinant line bundle, generator of the Picard group of this moduli space. This map is identified with the theta map, which is of degree 2 in our case. We define a fibration from this moduli space to a projective space whose generic fiber is birational to the moduli space of 2g-pointed rational curves, and we describe the restriction of the map theta to the fibers of this fibration. We show that this restriction is, up to a birational map, an osculating projection centered on a point. By using a description due to Kumar, we show that the restriction of the map theta to this fibration ramifies over the Kummer variety of a certain hyperelliptic curve of genus g - 1.

This work deals with the application of supermathematics to supersymmetrical problems arising in physics. Some recent developments are presented in detail. A reduction scheme for general supermanifolds to vector bundles is presented, which significantly simplifies their mathematical treatment in a physical context. Moreover, some applications of this new approach are worked out, such as the Fermi oscillator. / text

O objetivo desse trabalho é fazer uma enumeração dos fibrados planos reais sobre algumas superfícies, como por exemplo, a esfera e o g-toro. Entre outras ferramentas, utilizamos a co-homologia das superfícies, com coeficientes locais, e também o método desenvolvido por Larmore para contar classes de homotopia de levantamento de funções. / The aim of this work is enumerate the plane bundles over some surfaces, for example the sphere and the g-torus. Among other tools we used cohomology of the surfaces with local coefficients and also the method developed by Larmore to count homotopy classes of lifting of functions.

Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-22T22:29:57Z (GMT). No. of bitstreams: 1 Branco_LucasMagalhaesPereiraCastello_M.pdf: 1981391 bytes, checksum: 7ecd7674514f634b8bb527c0bcab1a06 (MD5) Previous issue date: 2013 / Resumo: Neste trabalho os aspectos básicos da teoria de calibre são abordados, incluindo as noções de conexão e curvatura em fibrados principais e vetoriais, considerações sobre o grupo de transformações de calibre e o espaço de moduli de soluções para a equação anti-auto-dual em dimensão quatro (o espaço de moduli de instantons). Posteriormente, mapas momento e redução são introduzidos. Primeiramente, no contexto clássico de geometria simplética e depois no contexto de geometria hyperkähler. Por fim, são apresentadas aplicações da teoria de mapas momento e redução em teoria de calibre. As equações ADHM são introduzidas e mostra se que estas podem ser dadas como o conjunto de zeros de um mapa momento hyperkähler. Além disso, considerações são feitas acerca da construção ADHM de instantons, que relaciona soluções dessas equações com as soluções da equação de anti-auto-dualidade. O espaço de moduli de conexões planas é também abordado. Neste caso, a curvatura é vista como um mapa momento e os cálculos podem ser generalizados para o espaço de moduli de conexões planas sobre variedades Kähler de dimensões mais altas e para o espaço de moduli de instantons sobre variedades hyperkähler de dimensão quatro / Abstract: In this work it is developed the basic concepts of gauge theory, including the notions of connections and curvature on principal bundles and vector bundles, considerations on the group of gauge transformations and the moduli space of anti-self-dual connections in dimension four (the instanton moduli space). After, moment maps and reduction are introduced. First in the classical context of symplectic geometry, then in hyperkähler geometry. At last, applications to the theory of moment maps and reduction in gauge theory are given. The ADHM equations are introduced and it is shown that solutions to these equations can be given by the zeros of a hyperkähler moment map. Furthermore, the ADHM construction, that relates the ADHM equations to instanton solutions, is discussed. The moduli space of flat connections over a Riemann surface is also treated. In this case, the curvature is seen as a moment map and the calculations can be generalized to flat connections over higher-dimensional Kähler manifolds and to the instanton moduli space over four dimensional hyperkähler manifolds / Mestrado / Matematica / Mestre em Matemática

In dieser Arbeit werden Aspekte der N = 1 Dualität zwischen dem Heterotischen String (der auf einer komplex dreidimensionalen Calabi-Yau Mannigfaltigkeit mit einem Vektorbündel kompaktifiziert wird) und der F-Theorie (die auf einer komplex vierdimensionalen Calabi-Yau Mannigfaltigkeit kompaktifiziert wird) diskutiert. Zu Beginn wird eine allgemeine Beschreibung der Stringdualitäten gegeben. Die Berech- nungen der notwendigen Calabi-Yau Mannigfaltigkeiten- und Vektorbündeldaten, welche Charakteristische Klassen und Bündelmoduli involvieren, werden im Detail durchgeführt. Die acht- bzw. sechsdimensionale Dualität zwischen dem Heterotis- chen String und der F-Theorie wird diskutiert. Im Anschluß erfolgt ein Vergleich der vierdimensionalen Spektren (dies involviert den Vergleich von N = 1 chiralen Multipletts) und ein Vergleich der Anomaliebedingungen (welche zu konsistenten vierdimensionalen Het/F-Theorie Kompaktifizierungen führen). Weiterhin werden vierdimensionale N = 1 Het/F-Theorie Beispiele konstruiert, insbesondere wird eine Klasse von elliptisch gefaserten Calabi-Yau's über del Pezzoflächen betrachtet. / We discuss aspects of N = 1 duality between the heterotic string compactified on a Calabi-Yau threefold with a vector bundle and F-theory on a Calabi-Yau fourfold. After a description of string duality intended for the non-specialist the framework and the constraints for heterotic/F-theory compactifications are presented. The computations of the necessary Calabi-Yau manifold and vector bundle data, involving characteristic classes and bundle moduli, are given in detail. The eight- and six- dimensional dualities are reviewed. The matching of the spectrum of chiral multiplets and of the number of heterotic five-branes respectively F-theory three-branes, needed for anomaly cancellation in four-dimensional vacua, is pointed out. Several examples of four-dimensional dual pairs are constructed where on both sides the geometry of the involved manifolds relies on del Pezzo surfaces.

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Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T09:52:31Z (GMT). No. of bitstreams: 1 DeMartino_MarceloGoncalves_M.pdf: 1604556 bytes, checksum: be41ad6ca9fd66921624adce95bf0939 (MD5) Previous issue date: 2012 / Resumo: Este trabalho procurou apresentar os conhecimentos básicos necessários para trabalhar com a teoria de calibre em baixas dimensões e também mostrar algumas aplicações da mesma. Na parte básica da teoria, além de comentar aspectos da teoria de Hodge para variedades compactas, também se discute, com certo nível de detalhes, os conceitos de fibrados vetoriais e conexões, com ênfase dada para os cálculos locais com conexões e curvaturas. Duas aplicações mais concretas da teoria de calibre são apresentadas nesta dissertação. Primeiro, em dimensão quatro, discute-se a equação de Yang-Mills sobre 4-variedades e é apresentada uma solução para a equação anti-auto-dual, solução esta que é conhecida na literatura como ansatz de 't Hooft. Por fim, é apresentada a prova, baseado no artigo [DONALDSON, 1983], de um importante teorema devido a M. S. Narasimhan e C. S. Seshadri que relaciona os conceitos de estabilidade com o de existência de conexão unitária satisfazendo certa propriedade, em fibrados vetoriais complexos sobre superfícies de Riemann / Abstract: In this work it is developed the basic knowledge required to deal with gauge theory in low dimension and it is shown some applications of this theory. Regarding the basic knowledge, apart from discussing some aspects of Hodge theory over compact manifolds, it is also covered, with a certain deal of details, the concepts of vector bundles and connections, paying close attention to the local computations regarding connections and curvature. As for the applications of the theory, we start, in dimension four, by treating the Yang-Mills equation over 4-manifolds and it is showed a solution to the anti-self-dual Yang-Mills equation, solution that is known in the literature as the 't Hooft ansatz. At last, it is given a proof, following the paper [DONALDSON, 1983], of an important theorem due to M. S. Narasimhan and C. S. Seshadri that relates the algebro-geometric notion of stability to the differential-geometric notion of existence of unitary connection whose curvature satisfies a certain condition, on vector bundles over Riemann surfaces / Mestrado / Matematica / Mestre em Matemática

Orientador: Marcos Benevenuto Jardim / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T05:02:36Z (GMT). No. of bitstreams: 1 Prata_DanielaMoura_D.pdf: 839425 bytes, checksum: 24b3dac76766d8c843d040b951a4376a (MD5) Previous issue date: 2012 / Resumo: Neste trabalho relacionamos algumas classes de fibrados vetoriais...Observação: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: In this work we relate some classes of vector bundles on...Note: The complete abstract is available with the full electronic document / Doutorado / Matematica / Doutor em Matemática

The curvature of a contraction T in the Cowen-Douglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this thesis, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the Cowen-Douglas class. Secondly, we obtain an explicit formula for the curvature of the jet bundle of the Hermitian holomorphic bundle E f on a planar domain Ω. Here Ef is assumed to be a pull-back of the tautological bundle on gr(n, H ) by a nondegenerate holomorphic map f :Ω →Gr (n, H ). Clearly, finding relationships amongs the complex geometric invariants inherent in the short exact sequence 0 → Jk(Ef ) → Jk+1(Ef ) →J k+1(Ef )/ Jk(Ef ) → 0 is an important problem, whereJk(Ef ) represents the k-th order jet bundle. It is known that the Chern classes of these bundles must satisfy c(Jk+1(Ef )) = c(Jk(Ef )) c(Jk+1(Ef )/ Jk(Ef )). We obtain a refinement of this formula: trace Idnxn ( KJk(Ef )) - trace Idnxn ( KJk-1(Ef ))= KJk(Ef )/ Jk-1(Ef )(z).