Thesis (Ph. D.)--University of Washington, 2008. / Vita. Includes bibliographical references (p. 113-122).

Thesis (Ph. D.)--University of Washington, 2006. / Vita. Includes bibliographical references (p. 81-85).

In this thesis we explore various aspects of Calabi-Yau (CY) manifolds and com- pactifications of the heterotic string over them. At first we focus on classifying symmetries and computing Hodge numbers of smooth CY quotients. Being non- simply connected, these quotients are an integral part of CY compactifications of the heterotic string, aimed at producing realistic string vacua. Discrete symmetries of such spaces that are generically present in the moduli space, are phenomenologically important since they may appear as symmetries of the associated low energy theory. We classify such symmetries for the class of smooth Complete Intersection CY (CICY) quotients, resulting in a large number of regular and R-symmetry examples. Our results strongly suggest that generic, non-freely acting symmetries for CY quotients arise relatively frequently. A large number of string derived Standard Models (SM) were recently obtained over this class of CY manifolds indicating that our results could be phenomenologically important. We also specialise to certain loci in the moduli space of a quintic quotient to produce highly symmetric CY quotients. Our computations thus far are the first steps towards constructing a sizeable class of highly symmetric smooth CY quotients. Knowledge of the topological properties of the internal space is vital in determining the suitability of the space for realistic string compactifications. Employing the tools of polynomial deformation and counting of invariant Kähler classes, we compute the Hodge numbers of a large number of smooth CICY quotients. These were later verified by independent cohomology computations. We go on to develop the machinery to understand the geometry of CY manifolds embedded as hypersurfaces in a product of del Pezzo surfaces. This led to an interesting account of the quotient space geometry, enabling the computation of Hodge numbers of such CY quotients. Until recently only a handful of CY compactifications were known that yielded low energy theories with desirable MSSM features. The recent construction of rank 5 line bundle sums over smooth CY quotients has led to several SU(5) GUTs with the exact MSSM spectrum. We derive semi-analytic results on the finiteness of the number of such line bundle models, and study the relationship between the volume of the CY and the number of line bundle models over them. We also imply a possible correlation between the observed number of generations and the value of the gauge coupling constants of the corresponding GUTs. String compactifications with underlying SO(10) GUTs are theoretically attractive especially since the discovery that neutrinos have non-zero mass. With this in mind, we construct tens of thousands of rank 4 stable line bundle sums over smooth CY quotients leading to SO(10) GUTs.

Boucle gravité quantique (LQG) est une tentative pour résoudre le problème de la gravité quantique. Boucle cosmologie quantique (LQC) est une tentative d'appliquer LQG à la cosmologie précoce. Le but de LQC est de se connecter LQG avec des observations. Il est très difficile d'observer les effets de la gravité quantique parce que la densité d'énergie énorme est très probablement nécessaire. Ceci est exactement pourquoi l'Univers est choisi comme une étape pour rechercher des phénomènes de gravité quantique.Le résultat central de LQC est que la grande singularité bang est remplacé par un gros rebond. Toutefois, ce ne sont pas quelque chose qui est possible d'observer aujourd'hui. Pour cette raison, nous avons étudié la façon dont les perturbations cosmiques sont affectées par LQC. Nous avons utilisé l'approche dite d'algèbre déformée, et nous avons calculé les spectres obtenus pour les deux perturbations scalaires et tenseurs. Les spectres que nous avons trouvé ne sont pas compatibles avec l'observation. Cependant cela ne peut abeille considérée comme très forte preuve contre LQG car il y a trop d'hypothèses sur le chemin. Plutôt cela est le résultat de cette interprétation spécifique de LQC.Nous avons également étudié la dynamique de fond (la partie homogène des équations) de LQC. Depuis lent-roll inflation est essentielle pour expliquer de nombreuses caractéristiques de l'univers, y compris le CMB, nous voulons savoir si lent-roll inflation est compatible avec LQC. Nous avons constaté que, en effet, il est. Si un champ d'inflation potentiel carré est ajouté à la théorie, le rebond va lever l'énergie potentielle suffisante pour fournir environ 145 e-plis de lent-roll inflation. Toutefois, lorsque anisotropies sont pris en compte, le montant de l'inflation diminue, et peut même disparaître complètement s'il y a trop de cisaillement au moment du rebond.Nous avons dérivé l'équation Friedman modifié pour anisotrope LQC. Cela nous a permis d'étudier anisotrope LQC pas seulement numériquement, mais aussi analytiquement, qui nous a donné une compréhension beaucoup plus complète de la situation que ce qui était connu auparavant.Enfin, nous avons étudié certains aspects géométriques de l'espace de Sitter, qui a donné lieu à deux considérations très différentes. Tout d'abord nous avons constaté que nous pouvons, pour une théorie générale de la cosmologie modifiée et sous certaines hypothèses assez conservatrices, tirer la dynamique d'un univers spatialement incurvée, étant donné la dynamique d'un un espace plat. Cela est pertinent dans les théories telles que LQC, où il est plus facile de trouver la solution plate que celle incurvée. Deuxièmement, nous proposons un mécanisme possible pour l'origine et la renaissance de l'Univers. / Loop quantum gravity (LQG) is an attempt to solve the problem of quantum gravity. Loop quantum cosmology (LQC) is an attempt to apply LQG to early cosmology. The purpose of LQC is to connect LQG with observations. It is very hard to observe any quantum gravity effects because enormous energy density is most likely required. This is exactly why the early Universe is chosen as a stage to search for quantum gravity phenomena.The central result of LQC is that the big bang singularity is replaced by a big bounce. However this is not something that is possible to observe today. For this reason, we have investigated how cosmic perturbations are affected by LQC. We have used the so called deformed algebra approach, and have calculated the resulting spectrums for both scalar and tensor perturbations.The spectrums that we have found are not compatible with observation. However this can not bee taken as very strong evidence against LQG since there are too many assumptions on the way. Rather this is a result for this specific interpretation of LQC.We have also studied the background dynamics (the homogenous part of the equations) of LQC. Since slow-roll inflation is essential in explaining many features of the universe, including the CMB, we want to know if slow-roll inflation is compatible with LQC. We have found that, indeed, it is. If a square potential inflation field is added to the theory, the bounce will lift the potential energy enough to provide around 145 e-folds of slow-roll inflation. However, when anisotropies are taken into account, the amount of inflation decreases, and can even disappear completely if there is too much shear at the time of the bounce.We have derived the modified Friedman equation for anisotropic LQC. This has allowed us to study anisotropic LQC not just numerically, but also analytically, which has given us a much more comprehensive understanding of the situation than what was known before.Finally, we have studied some geometric aspects of de Sitter space, which has resulted in two very different considerations. Firstly we found that we can, for a general theory of modified cosmology and under some quite conservative assumptions, derive the dynamics for a spatially curved universe, given the dynamics of a spatially flat one. This is relevant in theories such as LQC, where it is easier to find the flat solution than the curved one. Secondly, we propose a possible mechanism for the origin and rebirth of the Universe.

Title: Model Problems of the Theory of Gravitation Author: Marián Pilc Department: Institute of Theoretical Physics Faculty of Mathematics and Physics Supervisor: prof. RNDr. Jiří Bičák, DrSc., dr. h. c., Institute of Theoretical Physics Faculty of Mathematics and Physics Abstract: Equations of motion for general gravitational connection and orthonormal coframe from the Einstein-Hilbert type action of the Einstein-Cartan theory are derived. Ad- ditional gauge freedom is geometrically interpreted. Our formulation does not fix coframe to be tangential to spatial section hence Lorentz group is still present as part of gauge freedom. 3+1 decomposition introduces tangent Minkowski structures hence Hamilton-Dirac approach to dynamics works with Lorentz connection over spatial sec- tion. The second class constraints are analyzed and Dirac bracket is defined.Reduction of phase space is performed and canonical coordinates are introduced. The second part of this thesis is dedicated to quantum formulation of Einstein-Cartan theory. Point version of Einstein-Cartan phase space is introduced. Basic variables, crucial for quan- tization, are derived via groups acting on the phase space and their selfadjoint represen- tation is found. Representation of basic variables of Einstein-Cartan theory is derived via infinite...

Static, spherically-symmetric particle-like solutions to the coupled Einstein-Dirac and Einstein-Dirac-Maxwell equations have been studied by Finster, Smoller and Yau (FSY). In their work, FSY left the fermion mass as a parameter set to ±1. This thesis generalises these equations to include the Higgs field, letting the fermion mass become a function through coupling, μ. We discuss the dynamics associated with the Higgs field and find that there exist qualitatively similar solutions to those found by FSY, with well behaved, non-divergent metric components and electrostatic potential, close to the origin, going over to the point-particle solutions for large r; the Schwarzschild or Reissner-Nordström metric, and the Coulomb potential. We then go on to discuss an alternative gravity theory, conformal gravity, (CG), and look for solutions of the CG equations of motion coupled to the Dirac, Higgs and Maxwell equations. We obtain asymptotically nonvanishing, yet fully normalisable Dirac spinor components, resembling those of FSY, and, in the case where charge is included, non-divergent electrostatic potential close to the origin, matching onto the Coulomb potential for large r.

This thesis studies the geometry of objects from 2-dimensional statistical physics in the continuum. Chapter 1 is an introduction to Schramm-Loewner evolutions (SLE). SLEs are the canonical family of non-self-intersecting, conformally invariant random curves with a domain-Markov property. The family is indexed by a parameter, usually denoted by κ, which controls the regularity of the curve. We give the definition of the SLEκ process, and summarise the proofs of some of its properties. We give particular attention to the Rohde-Schramm theorem which, in broad terms, tells us that an SLEκ is a curve. In Chapter 2 we introduce the Gaussian free field (GFF), a conformally invariant random surface with a domain-Markov property. We explain how to couple the GFF and an SLEκ process, in particular how a GFF can be unzipped along a reverse SLEκ to produce another GFF. We also look at how the GFF is used to define Liouville quantum gravity (LQG) surfaces, and how thick points of the GFF relate to the quantum gravity measure. Chapter 3 introduces a diffusion on LQG surfaces, the Liouville Brownian motion (LBM). The main goal of the chapter is to complete an estimate given by N. Berestycki, which gives an upper bound for the Hausdor dimension of times that a γ-LBM spends in α-thick points for γ, α ∈ [0, 2). We prove the corresponding, tight, lower bound. In Chapter 4 we give a new proof of the Rohde-Schramm theorem (which tells us that an SLEκ is a curve), which is valid for all values of κ except κ = 8. Our proof uses the coupling of the reverse SLEκ with the free boundary GFF to bound the derivative of the inverse of the Loewner flow close to the origin. Our knowledge of the structure of the GFF lets us find bounds which are tight enough to ensure continuity of the SLEκ trace.

Une théorie de gravitation quantique propose de décrire l'interaction gravitationnelle à toutes les échelles de distance et d'énergie. Cependant, comprendre l'émergence de notre espace-temps classique reste un problème toujours ouvert. Cette thèse s'y attaque en gravité quantique à boucles à partir d'outils de l'information quantique.Ceci est fait en plusieurs étapes. La gravité quantique à boucles étant toujours une théorie en cours de développement, un point de vue pragmatique est adopté en étudiant une classe d'état physique du champ gravitationnel, motivée à la fois par des intuitions simples et les résultats de la physique à N corps. Une analyse de la reconstruction de la géométrie à partir des corrélations peut être faite et des leçons peuvent être tirées sur la forme de la dynamique fondamentale. Dans un second temps, la physique des sous-systèmes est analysée en commençant d'abord par évaluer l'entropie d'intrication entre l'intérieur et l'extérieur de la région, permettant ainsi de retrouver la loi holographique de l'entropie des trous noirs et donnant une forme possible des états holographiques de la théorie. Plusieurs dynamiques de la frontière, vu comme un système isolé ou ouvert, sont ensuite analysées, éclairant de nouveau la forme de la dynamique fondamentale. Enfin, la dernière étape de ces recherches étudie la dynamique de la frontière en interaction avec un environnement formé des degrés de liberté (de matière ou gravitationnels) formant le reste de l'Univers et la décohérence sur la frontière qu'il induit. Ceci permet de discuter la transition quantique/classique et de mettre en lumière, dans un modèle donné, les états pointeurs de la géométrie. / A quantum theory of gravitation aims at describing the gravitational interaction at every scales of energy and distance. However, understanding the emergence of our classical spacetime is still an open issue in many proposals. This thesis analyzes this problem in loop quantum gravity with tools borrowed from quantum information theory.This is done in several steps. Since loop quantum gravity is still under construction, a pragmatic point of view is advocated and an ansazt for physical states of the gravitational field is studied at first, motivated from condensed matter physics and simple intuitions. We analyze the proposal of reconstructing geometry from correlations. Lessons on the quantum dynamics and the Hamiltonian constraint are extracted. The second aspect of this work focuses on the physics of sub-systems and especially the physics of their boundary. We begin by calculating the entanglement entropy between the interior and the exterior of the region, recovering the holographic law known from classical black hole physics. Then different boundary dynamics are studied, both in the isolated and open cases, which shed lights again on the fundamental dynamics. Finally, the last aspect of this research studies the dynamics of the boundary interacting with an environment whose degrees of freedom (gravitational or matter) forming the rest of the Universe and especially the decoherence it induces. This allows to discuss the quantum to classical transition and understand, in a given model, the pointer states of geometry.

Probabilistic Spacetime is a simple generalization of the classical model of spa- cetime in General Relativity, such that it allows to consider multiple metric field realizations endowed with probabilities. The motivation for such a generalization is a possible application in the context of some quantum gravity approaches, na- mely those using the path integral. It is argued that this model might be used to restrict the precision of the geometry on small scales without postulating discrete structure; or it may be used as an effective description of a probabilistic geometry resulting from a full-fledged quantum gravity computation.

Ce travail de thèse se concentre sur l'approche non-perturbative canonique à la formulation d'une théorie quantique de la gravitation dans le cadre de la Gravitation quantique à boucles (LQG), répondant à deux problèmes majeurs. Dans la première partie, nous étudions la possible quantification, dans le cadre de la LQG, de la gravité en trois dimensions avec constante cosmologique et nous essayons de prendre contact avec autres approches de quantification déjà existantes dans la littérature. Dans la deuxième partie, nous nous concentrons sur une application très importante de la LQG: la définition et le comptage des états microscopiques d'un ensemble en mécanique statistique qui fournit une description de l'entropie des trous noirs. Notre analyse s'appuie fortement sur et s'étend à un traitement manifestement SU(2) invariant les travaux fondateurs de Ashtekar et al. / This thesis work concentrates on the non-perturbative canonical approach to the formulation of a quantum theory of gravity in the framework of Loop Quantum Gravity (LQG), addressing two major problems. In the first part, we investigate the possible quantization, in the context of LQG, of three dimensional gravity in the case of non-vanishing cosmological constant and try to make contact with alternative quantization approaches already existing in the literature. In the second part, we concentrate on a very important application of LQG: the definition and the counting of microstates of a statistical mechanical ensemble which provides a description and accounts for the black hole entropy. Our analysis strongly relies on and extends to a manifestly SU(2) invariant treatment the seminal work of Ashtekar et al.