Cette thèse porte sur la géométrie du champ libre Gaussien. Le champ libre Gaussien est un objet central en théorie quantique des champs et représente entre autre les fluctuations naturelles d'un potentiel électrique ou d’un modèle de dimères. La thèse commence dans le discret avec la démonstration d'un principe de Donsker en dimension plus grande que 1. Ce résultat est établi grâce à une nouvelle façon de représenter le champ libre en exprimant son gradient comme la partie gradient d'un champ de bruits blancs. Ensuite, les processus d'exploration du champ libre - ou ensembles locaux - introduits par Schramm-Sheffield sont étudiés en détail. Ces ensembles locaux généralisent de façon naturelle le concept de temps d'arrêt. On formalise cette théorie d'une nouvelle manière en procédant par analogie au cas 1D. Pour mieux comprendre le comportement du champs libre près des points d'intersection des ensembles locaux, un étude fine des oscillations du champ libre 2D près du bord s'avère utile. Enfin, la partie principale de cette thèse étudie des processus d'explorations particuliers – les processus SLE qui sont couplés naturellement avec le champ libre. On peut donner par exemple un sens aux lignes de niveau en utilisant le processus SLE_4 (Schramm-Sheffield). Nous avons utilisé ce couplage pour mieux comprendre la relation dite de KPZ qui intervient dans la théorie de la gravité quantique de Liouville. A l ‘aide de résultats fins sur l’enroulement des SLEs, nous avons montré comment adapter la relation de KPZ à la famille ci-dessus de processus d’explorations du champ libre. On peut interpréter ces résultats aussi comme une description de la géométrie du champ libre près des ces lignes d’exploration. / In this thesis we study the geometry of the Gaussian free field (GFF). After a gentle general introduction, we describe what we call the Hodge decomposition of the white noise – a way to represent the white noise vector field as a sum of a gradient and a rotation of independent GFFs. This decomposition gives rise to the Donsker invariance principle for the GFF.Next, we revisit from a slightly different angle the theory of so-called local sets of the GFF, introduced by Schramm and Sheffield. These random sets allow one to study the geometry of the GFF in a Markovian way. We also go a step further in describing the behaviour of the field near the boundary of possibly several local sets. The first chapter ends with a study of boundary oscillations of the GFF.The GFF is only a generalized function, yet it comes out that one can still make sense of it as a „random landscape“. In particular, Schramm and Sheffield gave meaning to the level lines of the GFF in terms of a coupling with SLE_4 process. In chapter 2 we study this coupling and describe the existent proofs and a non-proof of measurability of the SLE_4 process in this coupling. The rest of this chapter contains one of the most technical parts of the thesis – we obtain fine estimates on the winding of the SLE curves, conditioned to pass closely by a fixed point.This technical work is put in use in chapter 3, where we study the so called KPZ relation. In this context, the KPZ formula relates fractal dimensions of sets under the Euclidean geometry and under the „quantum geometry“ given by the exponential of the GFF. So far the KPZ formula was derived for planar sets independent of the quantum geometry. Here, we determine the KPZ formulas for sets that are naturally coupled with the quantum geometry – for the flow and level lines of the GFF. The family of KPZ formulas obtained resemble but still differ from the KPZ formula for independent sets.

<div>We consider actions of the current Lie algebras $\gl_{n}[t]$ and $\gl_{k}[t]$ on the space $\mathfrak{P}_{kn}$ of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1}\lc z_{k})$ and $\bar{\alpha}=(\alpha_{1}\lc\alpha_{n})$, respectively.</div><div>We show that the images of the Bethe algebras $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}\subset U(\gl_{n}[t])$ and $\mathcal{B}_{\bar{z}}^{\langle k \rangle}\subset U(\gl_{k}[t])$ under these actions coincide.</div><div></div><div>To prove the statement, we use the Bethe ansatz description of eigenvectors of the Bethe algebras via spaces of quasi-exponentials. We establish an explicit correspondence between the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}$ and the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{z}}^{\langle k \rangle}$.</div><div></div><div>One particular aspect of the duality of the Bethe algebras is that the Gaudin Hamiltonians exchange with the Dynamical Hamiltonians. We study a similar relation between the trigonometric Gaudin and Dynamical Hamiltonians. In trigonometric Gaudin model, spaces of quasi-exponentials are replaced by spaces of quasi-polynomials. We establish an explicit correspondence between the spaces of quasi-polynomials describing eigenvectors of the trigonometric Gaudin Hamiltonians and the spaces of quasi-exponentials describing eigenvectors of the trigonometric Dynamical Hamiltonians.</div><div></div><div>We also establish the $(\gl_{k},\gl_{n})$-duality for the rational, trigonometric and difference versions of Knizhnik-Zamolodchikov and Dynamical equations.</div>

In a preamble, a quick summary of the line of thought from Noether's theorems to modern views on conserved charges in gauge theories is attempted. Most of the background material needed for the thesis is set out through a small survey of the literature. Emphasis is put on the concepts more than on the formalism, which is relegated to the appendices.<p><p>The treatment of exact conservation laws in Lagrangian gauge theories constitutes the main axis of the first part of the thesis. The formalism is developed as a self-consistent theory but is inspired by earlier works, mainly by cohomological results, covariant phase space methods and by the Hamiltonian formalism.<p>The thermodynamical properties of black holes, especially the first law, are studied in a general geometrical setting and are worked out for several black objects: black holes, strings and rings. Also, the geometrical and thermodynamical properties of a new family of black holes with closed timelike curves in three dimensions are described.<p><p><p>The second part of the thesis is the natural generalization of the first part to asymptotic analyses. We start with a general construction of covariant phase spaces admitting asymptotically conserved charges. The representation of the asymptotic symmetry algebra by a covariant Poisson bracket among the conserved charges is then defined and is shown to admit generically central extensions. The asymptotic structures of three three-dimensional spacetimes are then studied in detail and the consequences for quantum gravity in three dimensions are discussed. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

The Planck constant is one of the most important constants in nature, as it describes the world governed by quantum mechanics. However, it cannot be derived from other natural constants. We present a model from which it is possible to derive this constant without any free parameters. This is done utilizing the force between two oscillating electric dipoles described by an extension of Weber electrodynamics, based on a gravitational model by Assis. This leads not only to gravitational forces between the particles but also to a newly found Casimir-type attraction. We can use these forces to calculate the maximum point mass of this model which is equal to the Planck mass and derive the quantum of action. The result hints to a connection of quantum effects like the Casimir force and the Planck constant with gravitational ones and the origin of mass itself.

<p>Holographic duality as a rigorous approach to quantum gravity claims that a quantum gravitational system is exactly equal to a quantum theory without gravity in lower spacetime dimensions living on the boundary of the quantum gravitational system. The duality maps key questions about the emergence of spacetime to questions on the non-gravitational boundary system that are accessible to us theoretically and experimentally. Recently, various aspects of quantum information theory on the boundary theory have been found to be dual to the geometric aspects of the bulk theory. In this thesis, we study the exact and approximate quantum error corrections (QEC) in a general quantum system (von Neumann algebras) focused on QFT and gravity. Moreover, we study entanglement theory in the presence of conserved charges in QFT and the multiparameter multistate generalization of quantum relative entropy.</p>

In dieser Arbeit berechnen wir die gravitativen Ein-Schleifen-Korrekturen zu den Propagatoren und Wechselwirkungen der Felder des Standardmodells der Elementarteilchenphysik. Wir betrachten hierzu ein höherdimensionales brane-world-Modell: Wärend die Gravitonen, die Austauchteilchen der Gravitationswechselwirkung, in der gesamten D-dimensionalen Raumzeit propagieren können, sind die Materiefelder an eine d-dimensionale Untermanigfaltigkeit (brane) gebunden. Um die divergenten Anteile der Ein-Schleifen-Diagramme zu bestimmen, entwickeln wir ein neues Regularisierungschema welches einerseits die Wardidentitäten der Yang-Mills-Theorie respektiert anderseits sensitiv für potenzartige Divergenzen ist. Wir berechnen die gravitativen Beiträge zu den beta-Funktionen der Yang-Mills-Eichtheorie, der quartischen Selbst-Wechselwirkung skalarer Felder und der Yukawa-Wechselwirkung zwischen Skalaren und Fermionen. Im physikalisch besonders interessanten Fall einer vier-dimensionalen Materie-brane verschwinden die gravitativen Beiträge zum Laufen der Yang-Mills-Kopplungskonstante. Die führenden Beiträge zum Laufen der anderen beiden Kopplungskonstanten sind positiv. Diese Ergebnisse sind unabhängig von der Anzahl der Extradimensionen in denen die Gravitonen propagieren können. Des Weiteren bestimmen wir alle gravitationsinduzierten Ein-Schleifen-Konterterme mit höheren kovarianten Ableitungen für skalare Felder, Dirac-Fermionen und Eichbosonen. Ein Vergleich dieser Konterterme mit den höheren Ableitungsoperatoren des Lee-Wick-Standardmodells zeigt, dass die Gravitationskorrekturen nicht auf letzte beschränkt sind. Eine Beziehung zwischen Quantengravitation und dem Lee-Wick-Standardmodell besteht somit nicht. / In this thesis, we derive the gravitational one-loop corrections to the propagators and interactions of the Standard Model field. We consider a higher dimensional brane world scenario: Here, gravitons can propagate in the whole D dimensional space-time whereas the matter fields are confined to a d dimensional sub-manifold (brane). In order to determine the divergent part of the one-loop diagrams, we develop a new regularisation scheme which is both sensitive for polynomial divergences and respects the Ward identities of the Yang-Mills theory. We calculate the gravitational contributions to the beta functions of non-Abelian gauge theories, the quartic scalar self-interaction and the Yukawa coupling between scalars and fermions. In the physically interesting case of a four dimensional matter brane, the gravitational contributions to the running of the Yang-Mills coupling constant vanish. The leading contributions to the other two couplings are positive. These results do not depend on the number of extra dimensions. We further compute the gravitationally induced one-loop counterterms with higher covariant derivatives for scalars, Dirac fermions and gauge bosons. In is shown that these counterterms do not coincide with the higher derivative terms in the Lee-Wick standard model. A possible connection between quantum gravity and the latter cannot be inferred.

<p>We discuss various aspects of D-branes in string theory and holography in string theory and loop quantum gravity. </p><p>One way to study D-branes is from a microscopic perspective, using conformal field theory techniques. For example, we investigate the question of how D-branes can be introduced into orbifolded theories. Another way to study D-branes is from a space-time perspective. An example is provided by unstable D-branes, where we compute an effective action describing the decay of a bosonic D-brane. </p><p>The holographic principle is a proposed duality which suggests that a theory in any region has a dual description on the boundary. We explore two examples: (1) The area law for the entropy of a black hole in the framework of loop quantum gravity, related to particular regularizations of the area operator. (2) The AdS/CFT correspondence proposal, where we investigate a string pulsating on AdS using spin chains.</p>

We discuss various aspects of D-branes in string theory and holography in string theory and loop quantum gravity. One way to study D-branes is from a microscopic perspective, using conformal field theory techniques. For example, we investigate the question of how D-branes can be introduced into orbifolded theories. Another way to study D-branes is from a space-time perspective. An example is provided by unstable D-branes, where we compute an effective action describing the decay of a bosonic D-brane. The holographic principle is a proposed duality which suggests that a theory in any region has a dual description on the boundary. We explore two examples: (1) The area law for the entropy of a black hole in the framework of loop quantum gravity, related to particular regularizations of the area operator. (2) The AdS/CFT correspondence proposal, where we investigate a string pulsating on AdS using spin chains.

Les travaux de ma thèse s'articulent en trois parties distinctes.Dans la première partie j'étudie les mesures de Malliavin-Shavguldize sur les difféomorphismes du cercle et de l'intervalle. Il s'agit de mesures de type « Haar » pour ces groupes de dimension infinie : elles furent introduites il a une vingtaine d'années pour permettre une étude de leur théorie des représentations. Un premier chapitre est dédié à recueillir les résultats présents dans la littérature et et les représenter dans une forme plus étendue, avec un regard particulier sur les propriétés de quasi-invariance de ces mesures. Ensuite j'étudie de problèmes de nature plus dynamique : quelle est la dynamique qu'on doit s'attendre d'un difféomorphisme choisi uniformément par rapport à une mesure de Malliavin-Shavguldize ? Je démontre en particulier qu'il y a une forte présence des difféomorphismes de type Morse-Smale.La partie suivante vient de mon premier travail publié, obtenu en collaboration avec Andrés Navas. Inspirés d'un théorème récent de Avila et Kocsard sur l'unicité des distributions invariantes par un difféomorphisme lisse minimal du cercle, nous analysons le même problème en régularité faible, avec des argument plus géométriques.La dernière partie est constituée des résultats récemment obtenus avec Mikhail Khristoforov et Victor Kleptsyn. Nous abordons les problèmes reliés à la gravité quantique de Liouville en étudiant des espaces auto-similaires qui sont la limite de graphes finis. Nous démontrons qu'il est possible de trouver des distances aléatoires non-triviales sur ces espaces qui sont compatibles avec la structure auto-similaire. / This thesis is divided into three different parts.In the first part, we study the Malliavin-Shavgulidze measure on circle and interval diffeomorphisms. They are Haar-like measures for these infinite-dimensional groups: they were introduced about twenty years ago to help to study their represantation theory. The first chapter collects the results that were obtained in the past years and in some cases we present them under a renewed point of view, with particular attention on quasi-invariance properties for this measures. Then we study some questions of dynamical nature: which is the typical dynamics that we must expect described by a diffeomorphism chosen randomly according to some Malliavin-Shavguldize measure? In particular, we prove that there is a strong presence of Morse-Smale diffeomorphisms.The third chapter comes from the published joint work with Andrés Navas. Inspired by a recent theorem by Avila and Kocsard about the uniqueness of the invariant distribution for a minimal smooth circle diffeomorphism, we analyse the same problem in low regularity, with more geometric arguments.The last part corresponds to the recent results obtained with Mikhail Khristoforov and Victor Kleptsyn. We consider problems in relation with Liouville quantum gravity, by studying self-similar metric spaces which are the limit of finite graphs. We prove that it is possible to find nontrivial random distances on these spaces which are compatible with the self-similar structure.

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-05-15T19:23:47Z No. of bitstreams: 1 tiberiodepaulanetto.pdf: 1926871 bytes, checksum: 17bceffda5c85de37a0d50a14f4f3f04 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-05-22T14:37:09Z (GMT) No. of bitstreams: 1 tiberiodepaulanetto.pdf: 1926871 bytes, checksum: 17bceffda5c85de37a0d50a14f4f3f04 (MD5) / Made available in DSpace on 2018-05-22T14:37:09Z (GMT). No. of bitstreams: 1 tiberiodepaulanetto.pdf: 1926871 bytes, checksum: 17bceffda5c85de37a0d50a14f4f3f04 (MD5) Previous issue date: 2017-02-22 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nesta tese, exploram-se diferentes aspectos e aplicações das teorias gravitacionais com correções quânticas. O texto é dividido em três partes principais. Na primeira parte, são consideradas as soluções linearizadas em diferentes teorias de gravitação com derivadas superiores. O potencial Newtoniano é calculado nos modelos locais, super-renormalizáveis no nível quântico, e mostra-se que a singularidade Newtoniana é cancelada devido a contribuição dos modos massivos extras. Logo depois, o colapso gravitacional de uma pequena massa é estudado na gravitação não-local livre de fantasmas, sendo o principal resultado a ausência da singularidade na solução do campo gravitacional e a possibilidade da não formação do miniburaco negro como resultado do colapso. Na segunda parte, algumas questões sobre a inflação induzida pela anomalia conforme são estudadas. É discutida a possibilidade da transição entre os períodos de inflação estável para instável. É mostrado que esta transição é automática se as correções quânticas nesse período forem desprezadas. Em seguida, considera-se o efeito de termos que violam as simetrias de CPT e Lorentz na inflação induzida pela anomalia conforme. É demonstrado que os novos termos responsáveis por violar essas simetrias não afetam a dinâmica do fator de escala da métrica. Por fim, na terceira parte as correções quânticas para o modelo dos Galileons e para as teorias dos campos massivos tensoriais antissimétricos são obtidas. É mostrado que o propagador da teoria dos Galileons recebe correções quânticas com derivadas superiores e que o teorema de não-renormalização do modelo dos Galileons permanece, de uma maneira generalizada, válido na região das baixas energias. Depois, por meio de cálculos explícitos das correções quânticas semiclássicas não-locais é confirmada a equivalência quântica entre os modelos dos campos tensoriais antissimétricos massivos com a teoria de Proca e com o modelo do campo escalar massivo mínimo. / In this thesis, different aspects and applications of gravitational theories with quan-tum corrections are explored. The text is divided into three main parts. In the first part, the linearized solutions in different gravity theories with higher derivatives are considered. The Newtonian potential is calculated in the local models, super-renormalizable at the quantum level, and it is shown that the Newtonian singularity is cancelled due to the contributions of the extra massive modes. Then the gravitational collapse of a small mass is studied in non-local ghost-free gravity, being the main result the absence of singularity in the gravitational field solution and the possibility of non-mini black hole formation as the collapse result. In the second part, some issues about anomaly-induced inflation are studied. It is discussed the possibility of the transition between stable to unstable periods of inflation. It is shown that this transition is automatic if the quantum corrections in this period are neglected. In the following, we consider the effect of CPT and Lorentz-violating terms in the conformal anomaly-induced inflation. It is shown the new terms responsible to violate these symmetries do not affect the dynamics of the metric scale factor. Finally, in the third part, the quantum corrections for the Galileon model and for the theory of the massive antisymmetric tensor fields are obtained. It is shown that the propaga-tor of Galilean theory receives quantum corrections with higher derivatives and that the non-renormalization theorem for Galileon models remains, in a generalized way, valid in the low-energy region. Then, by means of explicit calculations of non-local semiclassical quantum corrections, the quantum equivalence between the massive antisymmetric tensor field models with the Proca theory and minimal massive scalar field model is confirmed.