In dieser Dissertation wird eine Methode zur Berechnung der genus-eins Korrekturen von offenen Strings zu Feldtheorie-Amplituden konstruiert. Hierzu werden Vektoren von Integralen definiert, die ein elliptisches Knizhnik-Zamolodchikov-Bernard (KZB) System auf dem punktierten Torus erfüllen, und die entsprechenden Matrizen aus dem KZB System berechnet. Der elliptische KZB Assoziator erzeugt eine Relation zwischen zwei regulierten Randwerten dieser Vektoren. Die Randwerte enthalten die genus-null und genus-eins Korrekturen. Das führt zu einer Rekursion im Genus und der Anzahl externer Zustände, die einzig algebraische Operationen der bekannten Matrizen aus dem KZB System umfasst. Geometrisch werden zwei externe Zustände der genus-null Weltfläche der offenen Strings zu einer genus-eins Weltfläche zusammengeklebt. Die Herleitung dieser genus-eins Rekursion und die Berechnung der relevanten Matrizen wird durch eine graphische Methode erleichtert, mit der die Kombinatorik strukturiert werden kann. Sie wurde durch eine erneute Untersuchung der auf Genus null bekannten Rekursion entwickelt, bei welcher der Drinfeld Assoziator Korrekturen offener Strings auf Genus null auf solche mit einem zusätzlichen externen Zustand abbildet. Diese genus-null Rekursion umfasst ebenfalls ausschliesslich Matrixoperationen und basiert auf einem Vektor von Integralen, der eine Knizhnik-Zamolodchikov (KZ) Gleichung erfüllt. Die in der Rekursion gebrauchten Matrizen aus der KZ Gleichung werden als Darstellungen einer Zopfgruppe identifiziert und rekursiv berechnet. Der elliptische KZB Assoziator ist die Erzeugendenreihe der elliptischen Multiplen Zeta-Werte. Die Konstruktion der genus-eins Rekursion benötigt verschiedene Eigenschaften dieser Werte und ihren definierenden Funktionen, den elliptischen Multiplen Polylogarithmen. So werden Relationen verschiedener Klassen von elliptischen Polylogarithmen und Funktionalrelationen erzeugt durch elliptische Funktionen hergeleitet. / In this thesis, a method to calculate the genus-one, open-string corrections to the field-theory amplitudes is constructed. For this purpose, vectors of integrals satisfying an elliptic Knizhnik-Zamolodchikov-Bernard (KZB) system on the punctured torus are defined and the matrices from the KZB system are calculated. The elliptic KZB associator is used to relate two regularised boundary values of these vectors. The boundary values are shown to contain the open-string corrections at genus zero and genus one. This yields a recursion in the genus and the number of external states, solely involving algebraic operations on the known matrices from the KZB system. Geometrically, two external states of the genus-zero, open-string worldsheet are glued together to form a genus-one, open-string worldsheet. The derivation of this genus-one recursion and the calculation of the relevant matrices is facilitated by a graphical method to structure the combinatorics involved. It is motivated by the reinvestigation of the recursion in the number of external states known at genus zero, where the Drinfeld associator maps the genus-zero, open-string corrections to the corrections with one more external state. This genus-zero recursion also involves matrix operations only and is based on a vector of integrals satisfying a Knizhnik-Zamolodchikov (KZ) equation. The matrices in the KZ equation and used in the recursion are shown to be braid matrices and a recursive method for their calculation is provided. The elliptic KZB associator is the generating series of elliptic multiple zeta values. The construction of the genus-one recursion requires various properties of these values and their defining functions, the elliptic multiple polylogarithms. Thus, the third part of this thesis consists of an analysis of elliptic multiple polylogarithms, which in particular leads to relations among different classes of elliptic polylogarithms and functional relations generated by elliptic functions.

In dieser Dissertation untersuchen wir die Niedrigenergieentwicklung von Streuamplituden geschlossener Strings auf Einschleifenniveau (d.h. auf Genus eins) in einem zehndimensionalen Minkowski-Hintergrund mit Hilfe einer speziellen Klasse von Funktionen, den sogenannten modularen Graphenformen. Diese erlauben eine systematische Berechnung der Niedrigenergieentwicklung und erfüllen viele nicht-triviale algebraische- und Differentialgleichungen. Wir studieren diese Relationen detailliert und leiten Basiszerlegungen für eine große Zahl modularer Graphenformen her. Eines der Ergebnisse dieser Dissertation ist ein Mathematica-Paket, welches diese Vereinfachungen automatisiert. Wir benutzen diese Techniken, um die führenden Niedrigenergieordnungen der Streuamplitude von vier Gluonen im heterotischen String auf Einschleifenniveau zu berechnen. Für Stringamplituden auf Baumniveau bildet die Einwertigkeitsabbildung multipler Zetawerte offene Stringamplituden auf geschlossene Stringamplituden ab. Wir zeigen, dass ein bestimmter Vorschlag für die Definition einer geeigneten einschleifen-Verallgemeinerung, der sogenannten elliptische Einwertigkeitsabbildung, nicht alle Terme im heterotischen String reproduzieren kann. Ferner studieren wir eine Erzeugendenfunktion, die vermutlich die Torusintegrale aller perturbativen Theorien geschlossener Strings enthält. Wir bestimmen eine Differentialgleichung, die von dieser Erzeugendenfunktion erfüllt wird und lösen sie mit Hilfe von pfadgeordneten Exponentialen, was auf iterierte Integrale von holomorphen Eisensteinreihen führt. Da eine ähnliche Konstruktion im offenen String zur Verfügung steht, eröffnet dies außerdem eine neue Perspektive auf die elliptische Einwertigkeitsabbildung. / In this thesis, we investigate the low-energy expansion of scattering amplitudes of closed strings at one-loop level (i.e. at genus one) in a ten-dimensional Minkowski background using a special class of functions called modular graph forms. These allow for a systematic evaluation of the low-energy expansion and satisfy many non-trivial algebraic and differential relations. We study these relations in detail, leading to basis decompositions for a large number of modular graph forms which greatly reduce the complexity of the expansions of the integrals appearing in the amplitude. One of the results of this thesis is a Mathematica package which automatizes these simplifications. We use these techniques to compute the leading low-energy orders of the scattering amplitude of four gluons in the heterotic string at one-loop level. For tree-level string amplitudes, the single-valued map of multiple zeta values maps open-string amplitudes to closed-string amplitudes. The definition of a suitable one-loop generalization, a so-called elliptic single-valued map, is an active area of research and we show that a certain conjectural definition for this map, which was successfully applied to maximally supersymmetric amplitudes, cannot reproduce all terms in the heterotic string which has half-maximal supersymmetry. In order to arrive at a more systematic treatment of modular graph forms and at a different perspective on the elliptic single-valued map, we then study a generating function which conjecturally contains the torus integrals of all perturbative closed-string theories. We determine a differential equation satisfied by this generating function and solve it in terms of path-ordered exponentials, leading to iterated integrals of holomorphic Eisenstein series. Since a similar construction is available for the open string, this opens a new perspective on the elliptic single-valued map.

Das Wechselspiel zwischen String- und Eichtheorien hat in den letzten Jahren zu vielen neuen Einsichten geführt. Das herausragendste Beispiel ist die sogenannte AdS/CFT Korrespondenz, eine Dualität zwischen Stringtheorien auf Anti-de Sitter-Räumen (AdS) und konformen Eichtheorien auf deren Rand. Die Untersuchung von Stringtheorie auf ebenfrontigen Gravitationswellen, die sich im sogenannten Penrose-Limes aus AdS-Raumzeiten gewinnen lassen, erlaubt es, diese Dualität über die niederenergetische Supergravitationsnäherung hinausgehend zu überprüfen. Verallgemeinerte ebenfrontige Gravitationswellen sind auch für sich gesehen interessant, da sie eine grosse Klasse von Raumzeiten bilden, die exakte klassische Lösungen der Stringtheorie sind. In dieser Arbeit werden Aspekte der Stringtheorie auf ebenfrontigen Gravitationswellen untersucht. Besonderes Interesse gilt dabei der Verbindung dieser Stringtheorien zu Eichtheorien. Wechselwirkungen von Strings in derjenigen Gravitationswellen-Raumzeit mit maximaler Supersymmetrie werden im Rahmen der Lichtkegel-Stringfeldtheorie behandelt. Viele Ergebnisse, die für den Fall der flachen Minkowski-Raumzeit bekannt sind, werden dabei vollständig auf die komplizierteren ebenfrontigen Gravitationswellen verallgemeinert. Die führenden nicht-planaren Korrekturen zu den anomalen Dimensionen von Operatoren in der Eichtheorie, die eine duale Beschreibung von Stringzuständen liefern, werden innerhalb der Lichtkegel-Stringfeldtheorie reproduziert. / The interplay between string and gauge theory has led to many new insights in recent years. The most prominent example is the AdS/CFT correspondence, a duality between string theory on Anti-de Sitter (AdS) spaces and conformal gauge theories defined on their boundary. The study of string theory on plane wave backgrounds, which are connected to AdS by the Penrose limit, opens up the possibility of testing this duality beyond the low-energy supergravity approximation. Generalized plane wave geometries are interesting in themselves, as they provide a large class of exact classical space-time backgrounds for string theory. In this thesis aspects of string theory on plane wave backgrounds are studied, with an emphasis on the connection to gauge theory. String interactions in the plane wave space-time with maximal supersymmetry are investigated in the framework of light-cone string field theory. In the process, many results that had been found for the case of flat Minkowski space-time are generalized to the more complex plane wave background. The leading non-planar corrections to the anomalous dimensions of gauge theory operators dual to string states are recovered within light-cone string field theory.

Die vorliegende Arbeit beinhaltet ein einführendes Kapitel über Orbifold-Konstruktionen in dem neben rudimentären Grundlagen bereits speziellere Themen wie Diskrete Torsion und asymmetrische Orbifold-Gruppen behandelt werden. Als Beispiele für Orbifolde werden Kompaktifizierungen auf Tori sowie das asymmetrische T^4/Z(3)^L x Z(3)^R Orbifold behandelt. Danach wird eine allgemein gehaltene Einführung in Orientifolde gegeben, einschließlich des offenen String Sektors samt Chan-Paton Freiheitsgraden. Die darauf folgenden Kapitel 4-7 behandeln von mir durchgeführte Forschungsarbeiten. Kapitel 4 beschäftigt sich mit der Quantisierung des offenen Strings mit linearen Randbedingungen, wie sie bei Strings in elektro-magnetischen Feldern auftreten. Weiterhin wird die Quantisierung der Null- und Impuls-Moden des offenen Strings in Torus-Kompaktifizierungen durchgeführt. Außerdem wird für den Fall allgemeiner konstanter Hintergrund Neveu-Schwarz U(1)-Hintergrundfelder der Kommutator der Stringkoordinaten berechnet. Dieser stützt bisherige Resultate zur Nicht-Kommutativität von offenen Stringtheorien in Neveu-Schwarz Hintergründen. Kapitel 5 gibt, zusammen mit einigen neuen Erkenntnissen, Resultate von [1] über asymmetrische Orientifolde, insbesondere deren D-Branen Inhalt wieder. Kapitel 6 faßt die Veröffentlichung [2] zusammen, in der untersucht wurde, inwieweit sich phänomenolgisch interessante Modelle in Orientifolden von Torus-Kompaktifizierungen finden lassen. Insbesondere tragen die D9-Branen magnetische Flüsse, womit chirale Fermionen im Spektrum auftreten. Die Rechnungen werden größtenteils im gleichwertigen, T-dualen Bild ausgeführt. In diesem ist die Anzahl der chiralen Fermionen durch die topologische Schnittzahl der D-Branen gegeben. Existieren auf Torus-Kompaktifizierungen entweder nur nicht-chirale oder nicht-supersymmetrische Modelle, so lassen sich auf gewissen Orbifolden beide Eigenschaften miteinander vereinbaren. Kapitel 7 behandelt das "sigma Omega"-Orientifold auf einem T^6/Z(4) Orbifold. Als besonders interessantes Beispiel wird ein supersymmetrisches U(4) x U(2)^3_L x U(2)^3_R Modell vorgestellt, daß durch Einschalten geeigneter Hintergrundfelder in der effektiven Niederenergie-Wirkung auf ein Modell gebrochen wird, daß dem MSSM (minimalem supersymmetrischen Standard Modell) sehr ähnlich ist. Dieses Kapitel basiert auf unserer Publikation [3]. Ferner ist der Arbeit ein Anhang beigefügt, der einige der verwendeten Formeln sowie Beweise zu zwei Sätzen enthält, die im Text verwendet wurden. / This thesis contains an introductory chapter on orbifolds. Besides rudimentary basics we discuss more advanced topics like discrete torsion and asymmetric orbifold groups. As examples we investigate torus compactifications and an asymmetric T^4/Z(3)^L x Z(3)^R orbifold. The following chapter explains the foundations of orientifolds, including open strings with Chan-Paton degrees of freedom. Chapters 4-7 present own research. In chapter 4 we quantize open strings with linear boundary conditions, as they show up in electro-magnetic fields. We quantize the zero- and momentum-modes for toroidal compactifications, too. As an application we calculate the commutator of the coordinate fields in the case of general constant Neveu-Schwarz U(1)-field strengths. Thereby we confirm previous results on non-commutativity of open string theories in Neveu-Schwarz backgrounds. Chapter 5 reviews the results of a former publication [1] on asymmetric orientifolds, supplemented by some recent insights in connection with the preceeding chapter. Chapter 6 is a summary of [2]. In this publication we investigated to what extend one can build phenomenologically interesting models from toroidal orientifolds. By turning on magnetic fluxes on D9-branes we induce chiral fermions. Most calculations are performed in an (equivalent) T-dual picture. Here the number of chiral fermions is given by the topological intersection number of D-branes. In orientifolds of toroidal compactifications one obtains either non-chiral or non-supersymmetric orientifold solutions. However both properties can be reconciled in orientifolds that are obtained from specific supersymmetric orbifold compactifications. In chapter 7 we present the "sigma Omega"-Orientifold on a T^6/Z(4) orbifold. As a very attractive example we investigate a supersymmetric U(4) x U(2)^3_L x U(2)^3_R model that is broken to an MSSM-like model by switching on suitable background fields in the low energy effective action. This chapter is based on our publication [3]. The thesis is supplemented by an appendix with formulas applied in the text, as well as proofs to two theorems that were used as well.

<p>Entanglement entropy (EE) is a quantum information theoretic measure that quantifies the correlations between a region and its surroundings. We study this quantity in the following two setups : </p> <ul> <li>We look at the dynamics of a free minimally coupled, massless scalar field in a deSitter expansion, where the expansion stops after some time (i.e. we quench the expansion) and transitions to flat spacetime. We study the evolution of entanglement entropy (EE) and the Rényi entropy of a spatial region during the expansion and, more interestingly, after the expansion stops, calculating its time evolution numerically. The EE increases during the expansion but the growth is much more rapid after the expansion ends, finally saturating at late times, with saturation values obeying a volume law. The final state of the subregion is a partially thermalized state, reminiscent of a Gibbs ensemble. We comment on application of our results to the question of when and how cosmological perturbations decohere.</li> <li>We study the EE in a theory that is holographically dual to a BTZ black hole geometry in the presence of a scalar field, using the Ryu-Takayangi (RT) formula. Gaberdiel and Gopakumar had conjectured that the theory of N free fermions in 1+1 dimensions, for large N, is dual to a higher spin gravity theory with two scalar fields in 2+1 dimensions. So, we choose our boundary theory to be the theory of N free Dirac fermions with a uniformly winding mass, m e^iqx, in two spacetime dimensions (which describes for instance a superconducting current in an N-channel wire). However, to O(m²), thermodynamic quantities can be computed using Einstein gravity. We aim to check if the same holds true for entanglement entropy (EE). Doing calculations on both sides of the duality, we find that general relativity does indeed correctly account for EE of single intervals to O(m²).</li> </ul>

<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>

The thesis treats aspects of string/M-theory in anti-de Sitter spacetimes and their supersymmetric completions. By applying the AdS/CFT correspondence, as well as models of spin chains and singletons, we try to attain a better understanding of the underlying symmetries and the unbroken phases of string/M-theory. Tensionless string/M-theory in anti-de Sitter spacetime is argued to imply a higher spin gauge symmetry enhancement and to be described by gauged sigma models of multi-singletons as well as by closed singleton strings. Vasiliev's weakly projected equations of symmetric massless higher spin gauge fields in the vector oscillator formulation is shown to follow from a deformation of the singleton model. Various four dimensional minimal as well as non-minimal supersymmetric higher spin gauge theories in the spinor formulation are examined. The minimal higher spin gauge theory based on the symmetry algebra hs(1|4) is elaborated on in an N=1 superspace, illustrating the remarkable fact that the choice of base manifold is not fixed in unfolded dynamics. The importance of the representations saturating the unitarity bounds in anti-de Sitter spacetime is stressed throughout the thesis, with particular emphasis on the singleton and the massless representations. Singletons, and hence massless states, are shown to appear as bound states on the string or p-brane and are localized at cusps. Furthermore, we examine semiclassical string solutions in Type IIB String Theory in AdS(5) x S(5) and their boundary duals in N=4 Super Yang-Mills Theory in d=4 which are constituted out of thermodynamic composite operators. By using integrable spin chain techniques and Bäcklund transformations in the field theory and in the string theory, respectively, the one-loop anomalous dimensions as well as the tower of conserved charges of the composite operators are shown to be in agreement with the energies and the tower of conserved charges associated with the dual string states.

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.

The idea of holography [1, 2] finds a concrete realization in form of the AdS/CFT correspondence [3, 4]. This duality relates a field theory with conformal symmetries to quantum gravity living in one higher dimension. In this thesis we study aspects of black hole quasinormal modes, higher spin theories and entanglement entropy in the context of this duality. In almost all cases we have been able to subject the duality to some precision tests. Quasinormal modes encode the spectrum of black holes and the time-scale of pertur- bations therein [5]. From the dual CFT viewpoint they are the poles of retarded Green's function (or peaks in the spectral function) [6]. Quasinormal modes were previously studied for scalar, gauge field and fermion fluctuations [7]. We solve for these quasinormal modes of higher spin (s _ 2) fields in the background of the BTZ black hole [8, 9]. We obtain an exact solution for a field of arbitrary spin s (integer or half-integer) in the BTZ background. This implies that the BTZ is perhaps the only known black hole background where such an analysis can be done analytically for all bosonic and fermionic fields. The quasinormal modes are shown to match precisely with the poles of the corresponding Green's function in the CFT living on the boundary. Furthermore, we show that one-loop determinants of higher spin fields can also be written as a product form [10] in terms of these quasinormal modes and this agrees with the same obtained by integrating the heat-kernel [11]. We then turn our attention to dualities relating higher-spin gravity to CFTs with W algebra symmetries. Since higher spin gravity does go beyond diffeomorphism invariance, one needs re_ned notions of the usual concepts in differential geometry. For example, in general relativity black holes are defined by the presence of the horizon. However, higher spin gravity has an enlarged group of symmetries of which the diffeomorphisms form a subgroup. The appropriate way of thinking of solutions in higher spin gravity is via characterizations which are gauge invariant [12, 13]. We study classical solutions embedded in N = 2 higher spin supergravity. We obtain a general gauge-invariant condition { in terms of the odd roots of the superalgebra and the eigenvalues of the holonomy matrix of the background { for the existence of a Killing spinor such that these solutions are supersymmetric [14]. We also study black holes in higher spin supergravity and show that the partition function of these black holes match exactly with that obtained from a CFT with the same asymptotic symmetry algebra [15]. This involved studying the asymptotic symmetries of the black hole and thereby developing the holographic dictionary for the bulk charges and chemical potentials with the corresponding quantities of the CFT. We finally investigate entanglement entropy in the AdS3/CFT2 context. Entanglement entropy is an useful non-local probe in QFT and many-body physics [16]. We analytically evaluate the entanglement entropy of the free boson CFT on a circle at finite temperature (i.e. on a torus) [17]. This is one of the simplest and well-studied CFTs. The entanglement entropy is calculated via the replica trick using correlation functions of bosonic twist operators on the torus [18]. We have then set up a systematic high temperature expansion of the Renyi entropies and determined their finite size corrections. These _nite size corrections both for the free boson CFT and the free fermion CFT were then compared with the one-loop corrections obtained from bulk three dimensional handlebody spacetimes which have higher genus Riemann surfaces (replica geometry) as its boundary [19]. One-loop corrections in these geometries are entirely determined by the spectrum of the excitations present in the bulk. It is shown that the leading _nite size corrections obtained by evaluating the one-loop determinants on these handlebody geometries exactly match with those from the free fermion/boson CFTs. This provides a test for holographic methods to calculate one-loop corrections to entanglement entropy. We also study conformal field theories in 1+1 dimensions with W-algebra symmetries at _nite temperature and deformed by a chemical potential (_) for a higher spin current. Using OPEs and uniformization techniques, we show that the order _2 correction to the Renyi and entanglement entropies (EE) of a single interval in the deformed theory is universal [20]. This universal feature is also supported by explicit computations for the free fermion and free boson CFTs { for which the EE was calculated by using the replica trick in conformal perturbation theory by evaluating correlators of twist fields with higher spin operators [21]. Furthermore, this serves as a verification of the holographic EE proposal constructed from Wilson lines in higher spin gravity [22, 23]. We also examine relative entropy [24] in the context of higher-spin holography [25]. Relative entropy is a measure of distinguishability between two quantum states. We confirm the expected short-distance behaviour of relative entropy from holography. This is done by showing that the difference in the modular Hamiltonian between a high-temperature state and the vacuum matches with the difference in the entanglement entropy in the short-subsystem regime.

La géométrie complexe est un outil puissant pour étudier les systèmes intégrables classiques, la physique statistique sur réseau aléatoire, les problèmes de matrices aléatoires, la théorie topologique des cordes, …Tous ces problèmes ont en commun la présence de relations, appelées équations de boucle ou contraintes de Virasoro. Dans le cas le plus simple, leur solution complète a été trouvée récemment, et se formule naturellement en termes de géométrie différentielle sur une surface de Riemann : la "courbe spectrale", qui dépend du problème. Cette thèse est une contribution au développement de ces techniques et de leurs applications.Pour commencer, nous abordons les questions de développement asymptotique à tous les ordres lorsque N tend vers l’infini, des intégrales N-dimensionnelles venant de la théorie des matrices aléatoires de taille N par N, ou plus généralement des gaz de Coulomb. Nous expliquons comment établir, dans les modèles de matrice beta et dans un régime à une coupure, le développement asymptotique à tous les ordres en puissances de N. Nous appliquons ces résultats à l'étude des grandes déviations du maximum des valeurs propres dans les modèles beta, et en déduisons de façon heuristique des informations sur l'asymptotique à tous les ordres de la loi de Tracy-Widom beta, pour tout beta positif. Ensuite, nous examinons le lien entre intégrabilité et équations de boucle. En corolaire, nous pouvons démontrer l'heuristique précédente concernant l'asymptotique de la loi de Tracy-Widom pour les matrices hermitiennes.Nous terminons avec la résolution de problèmes combinatoires en toute topologie. En théorie topologique des cordes, une conjecture de Bouchard, Klemm, Mariño et Pasquetti affirme que des séries génératrices bien choisies d'invariants de Gromov-Witten dans les espaces de Calabi-Yau toriques, sont solution d'équations de boucle. Nous l'avons démontré dans le cas le plus simple, où ces invariants coïncident avec les nombres de Hurwitz simples. Nous expliquons les progrès récents vers la conjecture générale, en relation avec nos travaux. En physique statistique sur réseau aléatoire, nous avons résolu le modèle O(n) trivalent sur réseau aléatoire introduit par Kostov, et expliquons la démarche à suivre pour résoudre des modèles plus généraux.Tous ces travaux soulignent l'importance de certaines "intégrales de matrices généralisées" pour les applications futures. Nous indiquons quelques éléments appelant à une théorie générale, encore basée sur des "équations de boucles", pour les calculer / Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory, … All these topics share certain relations, called "loop equations" or "Virasoro constraints". In the simplest case, the complete solution of those equations was found recently : it can be expressed in the framework of differential geometry over a certain Riemann surface which depends on the problem : the "spectral curve". This thesis is a contribution to the development of these techniques, and to their applications.First, we consider all order large N asymptotics in some N-dimensional integrals coming from random matrix theory, or more generally from "log gases" problems. We shall explain how to use loop equations to establish those asymptotics in beta matrix models within a one cut regime. This can be applied in the study of large fluctuations of the maximum eigenvalue in beta matrix models, and lead us to heuristic predictions about the asymptotics of Tracy-Widom beta law to all order, and for all positive beta. Second, we study the interplay between integrability and loop equations. As a corollary, we are able to prove the previous prediction about the asymptotics to all order of Tracy-Widom law for hermitian matrices.We move on with the solution of some combinatorial problems in all topologies. In topological string theory, a conjecture from Bouchard, Klemm, Mariño and Pasquetti states that certain generating series of Gromov-Witten invariants in toric Calabi-Yau threefolds, are solutions of loop equations. We have proved this conjecture in the simplest case, where those invariants coincide with the "simple Hurwitz numbers". We also explain recent progress towards the general conjecture, in relation with our work. In statistical physics on the random lattice, we have solved the trivalent O(n) model introduced by Kostov, and we explain the method to solve more general statistical models.Throughout the thesis, the computation of some "generalized matrices integrals" appears to be increasingly important for future applications, and this appeals for a general theory of loop equations.