String theory is a unified framework for general relativity and quantum mechanics, thus being a theory of quantum gravity. In this thesis we discuss various aspects of quantum gravity for particular systems, having in common the existence of horizons. The main motivation is that one major challenge in theoretical physics today is in trying to understanding how time dependent backgrounds, with its resulting horizons and space-like singularities, should be described in a controlled way. One such system of particular importance is our own universe. We begin by discussing the information puzzle in de Sitter space and consequences thereof. A typical time-scale is encountered, which we interpreted as setting the thermalization time for the system. Then the question of closed time-like curves is discussed in the combined setting where we have a rotating black hole in a Gödel-like universe. This gives a unified picture of what previously was considered as independent systems. The last three projects concerns $c=1$ matrix models and their applications. First in relation to the RR-charged two dimensional type 0A black hole. We calculate the ground state energy on both sides of the duality and find a perfect agreement. Finally, we relate the 0A model at self-dual radius to the topological string on the conifold. We find that an intriguing factorization of the theory previously observed for the topological string is also present in the 0A matrix model.

This thesis consists of two papers on the spectral geometry of locally symmetric spaces of Riemannian and Lorentzian signature. Both works are concerned with the idea of relating analysis on such spaces to structures on their boundaries. The first paper is motivated by a conjecture of Patterson on the Selberg zeta function of Kleinian groups. We consider geometrically finite hyperbolic cylinders with non-compact Riemann surfaces of finite area as cross sections. For these cylinders, we present a detailed investigation of the Bunke-Olbrich extension operator under the assumption that the cross section of the cylinder has one cusp. We establish the meromorphic continuation of the extension of Eisenstein series and incomplete theta series through the limit set. Furthermore, we derive explicit formulas for the residues of the extension operator in terms of boundary values of automorphic eigenfunctions. The motivation for the second paper comes from conformal geometry in Lorentzian signature. We prove the existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymptotics of eigenfunctions on anti de Sitter spaces.

Un problema clásico en geometría lorentziana es la descripción de las inmersiones isométricas entre los espacios lorentzianos de curvatura constante. En este trabajo nos centramos en la clasificación de las inmersiones isométricas del plano lorentziano en el espacio anti-de Sitter tridimensional. Damos una fórmula de representación de estas inmersiones en términos de pares de curvas (con posibles singularidades) en el plano hiperbólico. Esto nos permite resolver los problemas propuestos por Dajczer y Nomizu en 1981. De entre todas las inmersiones isométricas del plano lorentziano en el espacio anti-de Sitter, algunas de ellas corresponden a toros lorentzianos (los ejemplos más sencillos son los toros de Hopf). Como aplicación de nuestra anterior descripción, probamos que todos estos toros pueden obtenerse a partir de dos curvas cerradas en el espacio hiperbólico. Finalmente, demostramos que los toros de Hopf son los únicos toros llanos lorentzianos inmersos en una amplia familia de sumersiones de Killing lorentzianas tridimensionales. / A classical problem in Lorentzian geometry is the description of the isometric immersions between Lorentzian spaces of constant curvature. We investigate the problem of classifying the isometric immersion from the Lorentz plane into the three-dimensional anti-de Sitter space, providing a representation formula of these isometric immersions in terms of pairs of curves (possibly with singularities) in the hyperbolic plane. We then give an answer to the open problems proposed by Dajczer and Nomizu in 1981. Among all isometric immersions of the Lorentz plane into the anti-de Sitter space, some of them are actually Lorentzian tori (the basic examples are the Hopf tori). As an application of our previous description, we prove that any such torus can be recovered from two closed curves in the hyperbolic plane. Finally, we prove that Lorentzian Hopf tori are the only immersed Lorentzian flat tori in a wide family of Lorentzian three-dimensional Killing submersions.

Submitted by isabela.moljf@hotmail.com (isabela.moljf@hotmail.com) on 2017-07-05T11:27:51Z No. of bitstreams: 1 rafaelleitefernandes.pdf: 295519 bytes, checksum: 990b7a9657e1e2be2738bb9b3dd7cf04 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-08-08T15:19:39Z (GMT) No. of bitstreams: 1 rafaelleitefernandes.pdf: 295519 bytes, checksum: 990b7a9657e1e2be2738bb9b3dd7cf04 (MD5) / Made available in DSpace on 2017-08-08T15:19:39Z (GMT). No. of bitstreams: 1 rafaelleitefernandes.pdf: 295519 bytes, checksum: 990b7a9657e1e2be2738bb9b3dd7cf04 (MD5) Previous issue date: 2013-02-27 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho vamos mostrar que o sistema chamado de “mundo brana” para representar o nosso Universo, introduzido por Randall-Sundrum (RS) no fim dos anos noventa, pode ser na verdade representado por branas curvas no espaço de de Sitter e Anti-de Sitter. Originalmente o sistema RS representaria um Universo com cinco dimensões onde o modelo padrão ficaria confinado em uma brana e os grávitons ficariam confinados na outra brana. Este sistema foi construído com uma métrica de Minkowski “torcida” com cinco dimensões onde os grávitons, responsáveis pelo campo gravitacional no modelo padrão, se deslocariam através da quinta dimensão, que por sua vez é infinita. E a ação padrão da relatividade geral, a ação de Einstein-Hilbert, foi a utilizada por RS. Neste trabalho de dissertação de mestrado vamos adotar uma ação modificada usada atualmente para explicar efeitos cosmológicos como a energia escura e a expansão do Universo por exemplo, ou seja, a ação com uma gravitação modificada chamada de f(R). Aqui vamos usar uma f(R) totalmente generalizada e suas consequências cosmológicas e viabilidade serão analisadas. Finalmente, vamos demonstrar que as partículas do modelo padrão estão confinadas nesta brana curva. Os resultados obtidos aqui generalizam totalmente outros resultados obtidos na literatura atual sobre mundo brana com branas grossas e são por isso, originais e serão submetidos à publicação. / In this work we will show that the so-called “brane world” framework introduced by Randall-Sundrum (RS), to represent our Universe, at the late nineties can be represented in fact by bent branes in de Sitter and Anti-de Sitter space with a generalized model for gravity. At the beginning, the RS scenario represent a Universe in five dimensions where the Standard Model is confined in one brane and the gravitons were confined in the another one. This model was build with a warped Minkowski metric with five dimensions where the gravitons, which are responsible by the gravitational field, are able to move through the fifth dimension, which is infinity. The standard general relativity action, namely, the Einstein-Hilbert action, was the one used by RS. In this Master dissertation we will adopt a modified action used at present to explain some cosmological effects like dark energy and Universe expansion, for example, i.e., we will use an action with a modified gravitation called f(R). However, here we will use a generalized f(R) and their cosmological consequences and viability will be analyzed. Finally, we will show that the particles of the Standard Model are confined at this bent brane. The results obtained here generalize altogether the others results obtained in the current literature concerning braneworlds with thick branes and are, consequently, new ones, which will be published elsewhere.