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Résumé anglais: This thesis consists of an introductory text followed by two separate parts which may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in the vicinity of spacelike singularities (the BKL-limit). In this limit, spatial points decouple and the dynamics exhibits ultralocal behaviour which may be mapped to an auxiliary problem given in terms of a (possibly chaotic) hyperbolic billiard. In all supergravities arising as low-energy limits of string theory or M-theory, the billiard dynamics takes place within the fundamental Weyl chambers of certain hyperbolic Kac-Moody algebras, suggesting that these algebras generate hidden infinite-dimensional symmetries of gravity. We investigate the modification of the billiard dynamics when the original gravitational theory is formulated on a compact spatial manifold of arbitrary topology, revealing fascinating mathematical structures known as galleries. We further use the conjectured hyperbolic symmetry E10 to generate and classify certain cosmological (S-brane) solutions in eleven-dimensional supergravity. Finally, we show in detail that eleven-dimensional supergravity and massive type IIA supergravity are dynamically unified within the framework of a geodesic sigma model for a particle moving on the infinite-dimensional coset space E10/K(E10). Part II of the thesis is devoted to a study of how (U-)dualities in string theory provide powerful constraints on perturbative and non-perturbative quantum corrections. These dualities are typically given by certain arithmetic groups G(Z) which are conjectured to be preserved in the effective action. The exact couplings are given by moduli-dependent functions which are manifestly invariant under G(Z), known as automorphic forms. We discuss in detail various methods of constructing automorphic forms, with particular emphasis on a special class of functions known as (non-holomorphic) Eisenstein series. We provide detailed examples for the physically relevant cases of SL(2,Z) and SL(3,Z), for which we construct their respective Eisenstein series and compute their (non-abelian) Fourier expansions. We also discuss the possibility that certain generalized Eisenstein series, which are covariant under the maximal compact subgroup K(G), could play a role in determining the exact effective action for toroidally compactified higher derivative corrections. Finally, we propose that in the case of rigid Calabi-Yau compactifications in type IIA string theory, the exact universal hypermultiplet moduli space exhibits a quantum duality group given by the emph{Picard modular group} SU(2,1;Z[i]). To verify this proposal we construct an SU(2,1;Z[i])-invariant Eisenstein series, and we present preliminary results for its Fourier expansion which reveals the expected contributions from D2-brane and NS5-brane instantons. / Résumé francais: Cette thèse est composée d'une introduction suivie de deux parties qui peuvent être lues indépendemment. Dans la première partie, nous analysons des structures hyperboliques apparaissant dans l'étude de la gravité au voisinage d'une singularité de type espace (la limite BKL). Dans cette limite, les points spatiaux se découplent et la dynamique suit un comportement ultralocal qui peut être reformulé en termes d'un billiard hyperbolique (qui peut être chaotique). Dans toutes les supergravités qui sont des limites de basse énergie de théories de cordes ou de la théorie M, la dynamique du billiard prend place à l'intérieur des chambres de Weyl fondamentales de certaines algèbres de Kac-Moody hyperboliques, ce qui suggère que ces algèbres correspondent à des symétries cachées de dimension infinie de la gravité. Nous examinons comment la dynamique du billard est modifiée quand la théorie de gravité originale est formulée sur une variété spatiale compacte de topologie arbitraire, révélant ainsi de fascinantes structures mathématiques appelées galleries. De plus, dans le cadre de la supergravité à onze dimensions, nous utilisons la symétrie hyperbolique conjecturée E10 pour engendrer et classifier certaines solutions cosmologiques (S-branes). Finalement, nous montrons en détail que la supergravité à onze dimensions et la supergravité de type IIA massive sont dynamiquement unifiées dans le contexte d'un modèle sigma géodesique pour une particule se déplaçant sur l'espace quotient de dimension infinie E10/K(E10). La deuxième partie de cette thèse est consacrée à étudier comment les dualités U en théorie des cordes fournissent des contraintes puissantes sur les corrections quantiques perturbatives et non perturbatives. Ces dualités sont typiquement données par des groupes arithmétiques G(Z) dont il est conjecturé qu'ils préservent l'action effective. Les couplages exacts sont donnés par des fonctions des moduli qui sont manifestement invariantes sous G(Z), et qu'on appelle des formes automorphiques. Nous discutons en détail différentes méthodes de construction de ces formes automorphiques, en insistant particulièrement sur une classe spéciale de fonctions appelées séries d'Eisenstein (non holomorphiques). Nous présentons comme exemples les cas de SL(2,Z) et SL(3,Z), qui sont physiquement pertinents. Nous construisons les séries d'Eisenstein correspondantes et leurs expansions de Fourier (non abéliennes). Nous discutons également la possibilité que certaines séries d'Eisenstein généralisées, qui sont covariantes sous le sous-groupe compact maximal, pourraient jouer un rôle dans la détermination des actions effectives exactes pour les théories incluant des corrections de dérivées supérieures compactifiées sur des tores.

Gauge-string duality has provided a powerful framework for the study of strongly coupled gauge theories and non-perturbative string models. This thesis analyzes the holographic description of non-local gauge theory operators and some aspects of the Bagger-Lambert theory. The latter, as a proposal for a multiple M2-branes effective theory, is conjectured to be the holographic dual of a compactification of M-theory. We show that all half-BPS Wilson loop operators in N=4 SYM - which are labeled by Young tableaus - have a gravitational dual description in terms of D5-branes or alternatively in terms of D3-branes in AdS5xS5. We prove that the insertion of a half-BPS Wilson loop operator in the N=4 SYM path integral is achieved by integrating out the degrees of freedom on a configuration of bulk D5-branes or alternatively on a configuration of bulk D3-branes. We construct a new class of supersymmetric surface operators in N=4 SYM and find the corresponding dual supergravity solutions. Consistency requires constructing N=4 SYM in the D7 supergravity background and not in flat space. This enlarges the class of holographic gauge theories dual to string theory backgrounds to gauge theories in non-trivial supergravity backgrounds. We write down a maximally supersymmetric one parameter deformation of the field theory action of Bagger and Lambert and we show that this theory on RxT2 is invariant under the superalgebra of the maximally supersymmetric Type IIB plane wave. It is argued that this theory holographically describes the Type IIB plane wave in the discrete light-cone quantization (DLCQ). Finally, we show by explicit computation that the Bagger-Lambert Lagrangian realizes the M2-brane superalgebra, including also two p-form central charges that encode the M-theory intersections involving M2-branes.

Gauge-string duality has provided a powerful framework for the study of strongly coupled gauge theories and non-perturbative string models. This thesis analyzes the holographic description of non-local gauge theory operators and some aspects of the Bagger-Lambert theory. The latter, as a proposal for a multiple M2-branes effective theory, is conjectured to be the holographic dual of a compactification of M-theory. We show that all half-BPS Wilson loop operators in N=4 SYM - which are labeled by Young tableaus - have a gravitational dual description in terms of D5-branes or alternatively in terms of D3-branes in AdS5xS5. We prove that the insertion of a half-BPS Wilson loop operator in the N=4 SYM path integral is achieved by integrating out the degrees of freedom on a configuration of bulk D5-branes or alternatively on a configuration of bulk D3-branes. We construct a new class of supersymmetric surface operators in N=4 SYM and find the corresponding dual supergravity solutions. Consistency requires constructing N=4 SYM in the D7 supergravity background and not in flat space. This enlarges the class of holographic gauge theories dual to string theory backgrounds to gauge theories in non-trivial supergravity backgrounds. We write down a maximally supersymmetric one parameter deformation of the field theory action of Bagger and Lambert and we show that this theory on RxT2 is invariant under the superalgebra of the maximally supersymmetric Type IIB plane wave. It is argued that this theory holographically describes the Type IIB plane wave in the discrete light-cone quantization (DLCQ). Finally, we show by explicit computation that the Bagger-Lambert Lagrangian realizes the M2-brane superalgebra, including also two p-form central charges that encode the M-theory intersections involving M2-branes.

We report on the first systematic study of correlation functions in N=4 super Yang-Mills theory using integrability techniques. In particular, we show how to compute three- and four- point functions of single-trace gauge-invariant operators at tree level in the SU(2) sector of the theory. Using the AdS/CFT correspondence, the correlation functions that we compute can be thought of as the joining or splitting of strings moving in AdS5 × S5. We show that when one (two) of the operators in the three-(four-)point function are taken to be small BPS operators, our weak coupling results match perfectly with the strong coupling results in the Frolov-Tseytlin limit. We conclude by presenting some results that will be needed to extend the methods presented in this thesis beyond the SU(2) sector of N=4 super Yang-Mills.

Cryptons are metastable bound states of fractionally-charged particles that arise generically in the hidden sectors of models derived from heterotic string. We study their properties and decay modes in a specific flipped SU(5) model with long-lived four-particle spin-zero bound states called tetrons. The expected masses and lifetimes of the neutral tetrons make them good candidates for cold dark matter (CDM), and a potential source of the ultra-high energy cosmic rays (UHECRs) which have been observed, whereas the charged tetrons would have decayed in the early Universe. We calculate the spectra of ultra-high-energy cosmic rays (UHECRs) in an ex¬plicit top-down model based on the decays of metastable neutral ‘crypton’ states. For all the decay operators, the total UHECR spectra are compatible with the available data. Also, the fractions of photons are compatible with all the published upper limits, but may be detectable in future experiments. We also construct several intersecting D-brane models on a variety of orientifold backgrounds. In particular, we construct flipped SU(5), Pati-Salam, and MSSM-like models. The phenomenological properties of these models are studied. For one model in particular, we find that we may explain the quark masses and mixings, the tau lepton mass, and generate small neutrino masses via the see-saw mechanism.

We consider linear perturbations of the background type IIB SUGRA solutions and find the equations of motion for the moduli. In particular, we allow for spacetime fluctuations of the positions of D3-branes in the compact dimensions. We postulate an ansatz for the 5-form flux due to the motion of the D3-branes, and a corresponding first-order part of the metric. The movement of the D3-branes is then shown to affect the warp factor at linear order. Using the equations of motion for the D3-branes, the universal volume modulus, and the universal axion, we construct a second-order, effective action. Finally, based on the form of the effective action, we examine a Kahler potential for the moduli space.

This thesis is concerned with so called chiral gauge theories, also known as self dual gauge theories. In particular, the aim of this thesis to investigate the role that chiral gauge theories play in duality symmetries in lower dimensions through dimensional reduction. Chapter one serves as an introduction to the notions of duality in field and string theory. The problems of formulating well defined actions for self-dual gauge theories are introduced as well as a brief presentation of the different approaches used to over come these problems. Chapter two introduces dimensional reduction and demonstrates how duality symmetries arise from the dimensional reduction of self-dual theories in a variety of dimensions and on different compact spaces. Examples are presented where the couplings of the resulting theories are calculated explicitly in terms of the geometrical data of the compact space. The duality generators acting on these couplings are also calculated explicitly and related to the geometry/topology of the compact space. Chapter three deals with the idea of duality manifest actions and their relation to the self-dual theories in higher dimensions. Non-linear Born-Infeld type actions are introduced and again dimensional reduction is shown to play a role in the duality of the Born-Infeld action. This leads to a duality manifest version of the Born-Infeld action. Chapter four describes perhaps the main application of this thesis. The effective action of the M-theory five brane wrapped on a torus is identified with the effective action of the IIB D-3 brane dimensionally reduced on a circle (after some appropriate world volume dualizations). The IIB S-duality then arises as a result of the modular symmetry of the torus. The final chapter contains a brief summary and a hint of further directions for research that were outside the scope of this thesis.

We report on the first systematic study of correlation functions in N=4 super Yang-Mills theory using integrability techniques. In particular, we show how to compute three- and four- point functions of single-trace gauge-invariant operators at tree level in the SU(2) sector of the theory. Using the AdS/CFT correspondence, the correlation functions that we compute can be thought of as the joining or splitting of strings moving in AdS5 × S5. We show that when one (two) of the operators in the three-(four-)point function are taken to be small BPS operators, our weak coupling results match perfectly with the strong coupling results in the Frolov-Tseytlin limit. We conclude by presenting some results that will be needed to extend the methods presented in this thesis beyond the SU(2) sector of N=4 super Yang-Mills.

This dissertation covers two different but related topics: the construction of consistent models in type IIB and heterotic string theories, and the higher derivative couplings for D-brane action, which will enable us to relate some models of type IIB to the heterotic side through duality chain. In the first part, we describe an alternative to the KKLT scenario, in which one can achieve de-Sitter space after fixing all moduli. We fix complex structure moduli and the axio-dilaton by deriving the stability conditions for the critical points of the no-scale scalar potential that governs the dynamics of the complex structure moduli and the axio-dilaton in compactifications of type IIB string theory on Calabi-Yau three-folds. In the second part, we show the existence of a class of flux backgrounds in heterotic string theory. The background metric we will consider is a T2 fibration over a K3 base times four-dimensional Minkowski space. Unbroken space-time supersymmetry determines all background fields except one scalar function which is related to the dilaton. The heterotic Bianchi identity gives the same differential equation for the dilaton, and we will discuss in detail the solvability of this equation for backgrounds preserving an N=2 supersymmetry. In the third part, we obtain the higher derivative D-brane action by using both linearized T-duality and string disc amplitude computation. We evaluate disc amplitude of one R-R field C^(p-3) and two NS-NS fields in the presence of a single Dp-brane in type II string theory. We obtain the action for the higher derivative brane interactions among one R-R field C^(p-3) and two NS-NS B-fields after carefully comparing the supergravity amplitudes with the corresponding string amplitude up to alpha^r2 order. We also show that these higher derivative brane couplings are invariant under both R-R and NS-NS B-field gauge transformations, and compatible with linear T-duality.