Integrabilität hat sich als ein mächtiges Werkzeug zur Berechnung von Observablen in der AdS/CFT-Korrespondenz erwiesen. Zunächst für das planare Spektralproblem entdeckt, wurden auch Methoden zur Berechnung von Mehrpunktfunktionen entwickelt. In dieser Arbeit wird diese Korrespondenz für AdS5/CFT4 und AdS3/CFT2 betrachtet mit dem Ziel, den integrablen Formalismus zu erweitern. Teil I behandelt Integrabilität in der N=4 SYM-Theorie, wo der Hexagon-Formalismus die Berechnung von Dreipunktfunktionen ermöglicht. Dazu wird der Korrelator in zwei hexagonale Stücke zerlegt. Die lokalen Operatoren müssen im Spinkettenbild als Bethe-Zustand zerschnitten und ein verschränkter Zustand konstruiert werden. Der Hexagon-Formalismus wird hier auf Sektoren mit höherem Rang erweitert, wobei die operatorartige Struktur erhalten und nur minimale Informationen aus dem geschachtelten Bethe-Ansatz genutzt werden. Weiterhin erlaubt die Betrachtung von Doppelanregungen im Spinkettenbild die Realisierung aller Felder der N=4 SYM-Theorie. Der chirale Yang-Mills-Feldstärketensor wird aus vier Fermionen in führender Ordnung der Kopplung konstruiert, eine Methode zur Einsetzung des Lagrangeoperators im Hexagon-Formalismus wird vorgeschlagen und ein erster Test durchgeführt. Teil II behandelt den Hexagon-Formalismus für Superstrings auf AdS3xS3xT4 Hintergründen mit einer Mischung von Ramond-Ramond und Neveu-Schwarz-Neveu-Schwarz Flüssen. Der Formfaktor wird für Ein- und Zwei-Teilchen-Zustände konstruiert und lässt sich für viele Teilchen unter Nutzung der S Matrix verallgemeinern. Schließlich werden die thermodynamischen Bethe-Ansatz (TBA)-Gleichungen betrachtet, die von Frolov und Sfondrini für das Spektrum von Strings auf reinem Ramond-Ramond AdS3xS3xT4 Hintergrund konstruiert wurden. Bei schwacher Kopplung lassen sich die TBA-Gleichungen erheblich vereinfachen. Der Beitrag zu den anomalen Dimensionen in führender Ordnung ist auf masselose Anregungen zurückzuführen. / Integrability proved to be a powerful tool to calculate observables in the AdS/CFT correspondence. At first discovered in the planar spectral problem, methods have since been devised for calculating higher-point functions as well. In this thesis we will consider two instances of the correspondence, that is AdS5/CFT4 as well as AdS3/CFT2, aiming at extending the integrability framework. In Part I we focus on integrability in N=4 SYM theory, where the hexagon form factor provides a formalism to calculate three-point functions. For this, the correlator is cut into two hexagonal patches. Considering the local operators in the spin chain picture, the Bethe states also need to be cut, resulting in an entangled state. In this thesis, we extend the hexagon formalism to higher-rank sectors, while preserving its operator-like structure and importing a minimum of information from the nested Bethe ansatz. Moreover, considering double excitations in the spin chain picture allows us to accommodate for the full set of fields in N=4 SYM theory. We build the chiral Yang-Mills field strength tensor from four fermions at leading order in the coupling, put forward a Lagrangian insertion method in the hexagon formalism and perform a first test. In Part II we propose a hexagon formalism for superstrings in AdS3×S3×T4 backgrounds with an arbitrary mixture or Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz fluxes. We bootstrap the hexagon form factor for one- and two-particle states from symmetry and give a proposal for the evaluation of many particle states in terms of the theorie's S matrix. Finally, we consider the thermodynamic Bethe ansatz (TBA) equations constructed by Frolov and Sfondrini for the spectrum of strings on the pure-Ramond-Ramond AdS3×S3×T4 background. Here we study the small tension limit of the mirror TBA equations and find that the equations simplify considerably. We observe that the leading-order contribution to the anomalous dimensions is due to massless excitations.

In this thesis we review the gauge/gravity duality and how it can be used to compute the thermodynamic properties and low-energy excitations of holographic quantum liquids - strongly-interacting field theories with a non-zero density of matter. We then study in detail the charge density excitations of two such liquids, the D3/D7 theory and the RN-AdS₄ theory, by computing the poles of their charge density Green's functions, and their charge density spectral functions. Although it is not a Landau Fermi liquid, the charge density excitations of the D3/D7 theory display many of the same properties as one, including a collisionless/hydrodynamic crossover as the temperature is increased. In contrast to this, the charge density (and energy density) excitations of the RN-AdS₄ theory do not share these properties but behave in a way that cannot be explained by Landau's theory of interacting fermionic quasiparticles. This is consistent with other results which indicate that this is not a Landau Fermi liquid.

O objetivo desta dissertação é revisar os operadores laços de Wilson no contexto da correspondência AdS/CFT. Estes operadores, presentes em qualquer teoria de calibre, são importantes por nos fornecer um parâmetro de ordem para a transição de fase confinante/desconfinante. Além disso, eles são particularmente importantes no estudo da correspondência AdS/ CFT pois: i) Eles nos dão, em alguns casos, resultados exatos graças ao fato de poderem ser localizados em um modelo de matrizes, desta forma nos permitindo fazer testes altamente não triviais da correspondência; ii) Eles são os objetos da teoria de calibre que são duais as cordas propagando no interior do espaço, nos dando um rico dicionário entre quantidades no interior (AdS) e na borda do espaço (CFT). Depois de revisarmos os laços de Wilson em teorias de calibre e a correspondência Ads/CFT, introduziremos a definição dos laços de Wilson supersimétricos 1/2 BPS. Calcularemos eles para o caso de um acoplamento fraco e para qualquer outro valor da constante de acoplamento usando técnicas de modelos de matrizes. Finalmente, compararemos nossos resultados com computações de superfícies minimais no interior do espaço, encontrando uma concordância perfeita. / The aim of this thesis is to review Wilson loop operators in the contexto f the AdS/CFT correspondence. These operators, wich are present in any gauge theory, are important because they furnish an order parameter for confinement/deconfinement phase transitions. Besides this, they are particularly relevant in the study of the AdS/CFT correspondence because: i) they allow, in some cases, for exact results thanks to localization to matrix models and make it possible to perform highly non-trivial tests of the correspondence; ii) they are the gauge theory objects dual to strings propagating in the bulk of the space and give a rich dictionary between bulk (AdS) and boundary (CFT) quantities. After reviews of Wilson loops in gauge theories and of the Ads/CFT correspondence, we will introduce the definition of 1/2 BPS supersymmetric Wilson loops, we will compute them at weak coupling and then at any order in the coupling constant via matrix model techniques, and finally we will compare our results with minimal surface computations in the bulk, finding perfect agreement.

Elaboramos um estudo detalhado de alguns aspectos d(e uma versão d)a correspondência AdS/CFT, conjeturada por Maldacena e Witten, entre teorias quânticas de campo num fundo gravitacional dado por um espaço-tempo assintoticamente anti-de Sitter (AAdS), e teorias quânticas de campos conformalmente covariantes no infinito conforme (no sentido de Penrose) deste espaço-tempo, aspectos estes: (a) independentes d(o par d)e modelos específicos em Teoria Quântica de Campos, e (b) suscetíveis a uma reformulação em moldes matematicamente rigorosos. Adotamos como ponto de partida o teorema demonstrado por Rehren no contexto da Física Quântica Local (também conhecida como Teoria Quântica de Campos Algébrica) em espaços-tempos anti-de Sitter (AdS), denominado holografia algébrica ou dualidade de Rehren. O corpo do presente trabalho consiste em estender o resultado de Rehren para uma classe razoavelmente geral de espaços-tempos AAdS d-dimensionais (d>3), escrutinar como as propriedades desta extensão são enfraquecidas e/ou modificadas em relação ao espaço-tempo AdS, e como efeitos gravitacionais não-triviais se manifestam na teoria quântica no infinito conforme. Dentre os resultados obtidos, citamos: condições razoavelmente gerais sobre geodésicas nulas no interior (cuja plausibilidade justificamos por meio de resultados de rigidez geométrica) não só garantem que a nossa generalização é geometricamente consistente com causalidade, como também permite uma reconstrução ``holográfica\'\' da topologia do interior na ausência de horizontes e singularidades; a implementação das simetrias conformes na fronteira, que associamos explicitamente a uma família de isometrias assintóticas do interior construída de maneira intrínseca, ocorre num caráter puramente assintótico e é atingida dinamicamente por um processo de retorno ao equilíbrio, mediante condições de contorno adequadas no infinito; efeitos gravitacionais podem eventualmente causar obstruções à reconstrução da teoria quântica no interior, ou por torná-la trivial em regiões suficientemente pequenas ou devido à existência de múltiplos vácuos inequivalentes, que por sua vez levam à existência de excitações solitônicas localizadas ao redor de paredes de domínio no interior, similares a D-branas. As demonstrações fazem uso extensivo de geometria Lorentziana global. A linguagem empregada para as teorias quânticas relevantes para nossa generalização da dualidade de Rehren segue a formulação funtorial de Brunetti, Fredenhagen e Verch para a Física Quântica Local, estendida posteriormente por Sommer para incorporar condições de contorno. / We elaborate a detailed study of certain aspects of (a version of) the AdS/CFT correspondence, conjectured by Maldacena and Witten, between quantum field theories in a gravitational background given by an asymptotically anti-de Sitter (AAdS) spacetime, and conformally covariant quantum field theories in the latter\'s conformal infinity (in the sense of Penrose), aspects such that: (a) are independent from (the pair of) specific models in Quantum Field Theory, and (b) susceptible to a recast in a mathematically rigorous mould. We adopt as a starting point the theorem demonstrated by Rehren in the context of Local Quantum Physics (also known as Algebraic Quantum Field Theory) in anti-de Sitter (AdS) spacetimes, called algebraic holography or Rehren duality. The main body of the present work consists in extending Rehren\'s result to a reasonably general class of d-dimensional AAdS spacetimes (d>3), scrutinizing how the properties of such an extension are weakened and/or modified as compared to AdS spacetime, and probing how non-trivial gravitational effects manifest themselves in the conformal infinity\'s quantum theory. Among the obtained results, we quote: not only does the imposition of reasonably general conditions on bulk null geodesics (whose plausibility we justify through geometrical rigidity techniques) guarantee that our generalization is geometrically consistent with causality, but it also allows a ``holographic\'\' reconstruction of the bulk topology in the absence of horizons and singularities; the implementation of conformal symmetries in the boundary, which we explicitly associate to an intrinsically constructed family of bulk asymptotic isometries, have a purely asymptotic character and is dynamically attained through a process of return to equilibrium, given suitable boundary conditions at infinity; gravitational effects may cause obstructions to the reconstruction of the bulk quantum theory, either by making the latter trivial in sufficiently small regions or due to the existence of multiple inequivalent vacua, which on their turn lead to the existence of solitonic excitations localized around domain walls, similar to D-branes. The proofs make extensive use of global Lorentzian geometry. The language employed for the quantum theories relevant for our generalization of Rehren duality follows the functorial formulation of Local Quantum Physics due to Brunetti, Fredenhagen and Verch, extended afterwards by Sommer in order to incorporate boundary conditions. (An English translation of the full text can be found at arXiv:0712.0401)

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.

Στην παρούσα διδακτορική διατριβή μελετάμε εφαρμογές οι οποίες σχετίζονται με την μη-Αβελιανή Τ-δυϊκότητα και την αντιστοιχία AdS/CFT. Στο πρώτο μέρος, το οποίο αντιστοιχεί στο πρώτο κεφάλαιο της διατριβής, παρουσιάζουμε συνοπτικά τα απαραίτητα μαθηματικά εργαλεία που απαιτούνται για την καλύτερη κατανόηση των κεφαλαίων που ακολουθούν. Στο δεύτερο μέρος, το οποίο αποτελείται από τα κεφάλαια 2,3 και 4, ασχολούμαστε με την έννοια της μη-Αβελιανής Τ-δυϊκότητας. Ποιο συγκεκριμένα, στο δεύτερο κεφάλαιο παρουσιάζουμε τους κανόνες Buscher της Αβελιανής Τ-δυϊκότητα καθώς και την γενίκευση τους στην μη-Αβελιανή περίπτωση. Επίσης στο κεφάλαιο αυτό δείχνουμε τον τρόπο με τον οποίο μπορούμε να εφαρμόσουμε τους κανόνες της μη-Αβελιανής Τ-δυϊκότητας σε υπόβαθρα υπερβαρύτητας τύπου II τα οποία περιλαμβάνουν πεδία Ramond-Ramond. Η διαδικασία αυτή μπορεί να θεωρηθεί σαν μια τεχνική κατασκευής νέων λύσεων υπερβαρύτητας. Στο τρίτο κεφάλαιο θεωρούμε μια γενική κατηγορία υποβάθρων υπερβαρύτητας με ισομετρία SO(4) στα οποία εφαρμόζουμε τον μετασχηματισμό της μη-Αβελιανής Τ-δυϊκότητας ως προς την υποομάδα SU(2) της ομάδας ισομετρίας. Πραγματοποιώντας διαστατική ελάττωση στην αρχική και την δυϊκή θεωρία καταλήγουμε στην ίδια επταδιάστατη θεωρία. Ως αποτέλεσμα, οποιαδήποτε λύση αυτής της επταδιάστατης θεωρίας μπορεί να ανυψωθεί ταυτόχρονα στο αρχικό και στο δυϊκό υπόβαθρο. Η παρατήρηση αυτή μας παρέχει μια αντιστρεπτή απεικόνιση μεταξύ δυο λύσεων υπερβαρύτητας τύπου II οι οποίες συνδέονται με έναν μετασχηματισμό μη-Αβελιανής Τ-δυϊκότητας. Επίσης, για την συγκεκριμένη περίπτωση υποβάθρων αποδεικνύουμε ότι το δυϊκό υπόβαθρο διατηρεί τη μισή υπερσυμμετρία σε σχέση με το αρχικό. Στο τέταρτο κεφάλαιο μελετάμε τη δράση της μη-Αβελιανής Τ-δυϊκότητας σε μια σειρά από υπόβαθρα με υπερσυμμετρία N=1 των οποίων οι δυϊκές θεωρίες πεδίου είναι γνωστές. Σκοπός του κεφαλαίου αυτού είναι η μελέτη της μη-Αβελιανής Τ-δυϊκότητας στα πλαίσια της αντιστοιχίας AdS/CFT. Αυτό το επιτυγχάνουμε μελετώντας διάφορες ποσότητες των θεωριών πεδίου που αντιστοιχούν στο αρχικό και στο δυϊκό υπόβαθρο. Τέλος, στο πέμπτο κεφάλαιο κάνουμε χρήση τεχνικών ολογραφίας προκειμένου να μελετήσουμε το φαινόμενο της εισαγωγής φερμιονικών προσμίξεων σε τρισδιάστατες θεωρίες ύλης τύπου Chern-Simons, οι οποίες περιλαμβάνουν μεγάλο αριθμό γεύσεων. / In this thesis, we study applications which are related to the non-Abelian T-duality and the AdS/CFT correspondence. In the first part, which corresponds to the first chapter of the present thesis, we briefly present the basic mathematical tools required for the better understanding of the material included in the next chapters. In the second part, which consists of the chapters 2,3 and 4, we deal with the concept of non-Abelian T-duality. More concretely, in the second chapter we present the Buscher rules of the Abelian T-duality and we generalize them to the non-Abelian case. We also show how to implement the rules of non-Abelian T-duality in backgrounds of type II supergravity with non-vanishing Ramond-Ramond fields. This proccess can be seen as a generating technique of new supergravity backgrounds. In the third chapter we consider a general class of supergravity backgrounds with SO(4) isometry and we perform the non-Abelian T-duality transforation with respect to the SU(2) subgroup of SO(4). After a consistent truncation to seven dimensions of both the original and the dual background we reproduce the same seven dimensional theory. As a result, any solution of this seven dimensional theory can be uplifted simultaneously to both the original and the dual background. This provides an invertible map between two solutions of type II supergravity which are related with a non-Abelian T-duality transformation. Also, in the case of supergravity backgrounds considered here, we show that the dual background preserves the half supersymmetry with respect to the original one. In chapter 4, we study the action of non-Abelian T-duality on a series of N=1 supersymmetric backgrounds whose field theory duals are well understood. The aim of this chapter is to study the transformation of non-Abelian T-duality within the framework of the AdS/CFT correspondence. This is done by considering several observables of the field theories that correspond to both the original and the dual backgrounds. Finally, in chapter 5 we use holographic techniques in order to study the effect of the addition of fermionic impurities to the three dimensional Chern-Simons matter theories with a large number of flavors.

La théorie des spins élevés est le domaine de la physique théorique au centre de cette thèse. Le contexte général de la naissance de cette théorie est présentée dans l’introduction. La première partie est axée sur les ingrédients (méthode de Noether, fonctions génératrices et formalisme ambiant) permettant la construction de vertex cubiques entre un champ scalaire de matière et un champ de jauge de spin élevé dans un espace-temps à courbure constante à partir des courants conservés en espace-temps plat. Dans un second temps, nous préparons les éléments pour un futur test de la correspondance holographique à l’ordre cubique voire quartique en la constante de couplage. Plus précisément, nous révisons en détail le calcul de certains propagateurs, ce qui nous mène à calculer les fonctions à trois points impliquant deux scalaires. La dernière partie, bien que concernant toujours l’holographie des spins élevés, traite de la physique non-relativiste. Les symétries et les courants d’un gaz parfait/unitaire de Fermi y sont étudiés. Le lien entre physiques relativiste et non-relativiste est obtenue grâce à la réduction dimensionnelle de Bargmann. / The higher spin theory is the field of theoretical physics at the center of this thesis. The general context of the birth of this theory is presenting in the introduction. The first part focuses on the ingredients (Noether method, generating functions and ambient formalism) for the construction of cubic vertices between a scalar matter field and a higher spin gauge field in a constant curvature space-time from conserved currents in flat space-time. In a second step, we prepare the around for a future test of the holographic correspondence in the cubic or quartic order in the coupling constant. More specifically, we review in detail the computation of some propagators, which leads us to calculate three-point functions involving two scalars. The last part, although always on the higher spin holography, deals with non-relativistic physics. Symmetries and currents of an ideal or unitary Fermi gas are studied. The link between relativistic and non-relativistic physics is obtained by Bargmann dimensional reduction.

Elaboramos um estudo detalhado de alguns aspectos d(e uma versão d)a correspondência AdS/CFT, conjeturada por Maldacena e Witten, entre teorias quânticas de campo num fundo gravitacional dado por um espaço-tempo assintoticamente anti-de Sitter (AAdS), e teorias quânticas de campos conformalmente covariantes no infinito conforme (no sentido de Penrose) deste espaço-tempo, aspectos estes: (a) independentes d(o par d)e modelos específicos em Teoria Quântica de Campos, e (b) suscetíveis a uma reformulação em moldes matematicamente rigorosos. Adotamos como ponto de partida o teorema demonstrado por Rehren no contexto da Física Quântica Local (também conhecida como Teoria Quântica de Campos Algébrica) em espaços-tempos anti-de Sitter (AdS), denominado holografia algébrica ou dualidade de Rehren. O corpo do presente trabalho consiste em estender o resultado de Rehren para uma classe razoavelmente geral de espaços-tempos AAdS d-dimensionais (d>3), escrutinar como as propriedades desta extensão são enfraquecidas e/ou modificadas em relação ao espaço-tempo AdS, e como efeitos gravitacionais não-triviais se manifestam na teoria quântica no infinito conforme. Dentre os resultados obtidos, citamos: condições razoavelmente gerais sobre geodésicas nulas no interior (cuja plausibilidade justificamos por meio de resultados de rigidez geométrica) não só garantem que a nossa generalização é geometricamente consistente com causalidade, como também permite uma reconstrução ``holográfica\'\' da topologia do interior na ausência de horizontes e singularidades; a implementação das simetrias conformes na fronteira, que associamos explicitamente a uma família de isometrias assintóticas do interior construída de maneira intrínseca, ocorre num caráter puramente assintótico e é atingida dinamicamente por um processo de retorno ao equilíbrio, mediante condições de contorno adequadas no infinito; efeitos gravitacionais podem eventualmente causar obstruções à reconstrução da teoria quântica no interior, ou por torná-la trivial em regiões suficientemente pequenas ou devido à existência de múltiplos vácuos inequivalentes, que por sua vez levam à existência de excitações solitônicas localizadas ao redor de paredes de domínio no interior, similares a D-branas. As demonstrações fazem uso extensivo de geometria Lorentziana global. A linguagem empregada para as teorias quânticas relevantes para nossa generalização da dualidade de Rehren segue a formulação funtorial de Brunetti, Fredenhagen e Verch para a Física Quântica Local, estendida posteriormente por Sommer para incorporar condições de contorno. / We elaborate a detailed study of certain aspects of (a version of) the AdS/CFT correspondence, conjectured by Maldacena and Witten, between quantum field theories in a gravitational background given by an asymptotically anti-de Sitter (AAdS) spacetime, and conformally covariant quantum field theories in the latter\'s conformal infinity (in the sense of Penrose), aspects such that: (a) are independent from (the pair of) specific models in Quantum Field Theory, and (b) susceptible to a recast in a mathematically rigorous mould. We adopt as a starting point the theorem demonstrated by Rehren in the context of Local Quantum Physics (also known as Algebraic Quantum Field Theory) in anti-de Sitter (AdS) spacetimes, called algebraic holography or Rehren duality. The main body of the present work consists in extending Rehren\'s result to a reasonably general class of d-dimensional AAdS spacetimes (d>3), scrutinizing how the properties of such an extension are weakened and/or modified as compared to AdS spacetime, and probing how non-trivial gravitational effects manifest themselves in the conformal infinity\'s quantum theory. Among the obtained results, we quote: not only does the imposition of reasonably general conditions on bulk null geodesics (whose plausibility we justify through geometrical rigidity techniques) guarantee that our generalization is geometrically consistent with causality, but it also allows a ``holographic\'\' reconstruction of the bulk topology in the absence of horizons and singularities; the implementation of conformal symmetries in the boundary, which we explicitly associate to an intrinsically constructed family of bulk asymptotic isometries, have a purely asymptotic character and is dynamically attained through a process of return to equilibrium, given suitable boundary conditions at infinity; gravitational effects may cause obstructions to the reconstruction of the bulk quantum theory, either by making the latter trivial in sufficiently small regions or due to the existence of multiple inequivalent vacua, which on their turn lead to the existence of solitonic excitations localized around domain walls, similar to D-branes. The proofs make extensive use of global Lorentzian geometry. The language employed for the quantum theories relevant for our generalization of Rehren duality follows the functorial formulation of Local Quantum Physics due to Brunetti, Fredenhagen and Verch, extended afterwards by Sommer in order to incorporate boundary conditions. (An English translation of the full text can be found at arXiv:0712.0401)

O objetivo desta dissertação é revisar os operadores laços de Wilson no contexto da correspondência AdS/CFT. Estes operadores, presentes em qualquer teoria de calibre, são importantes por nos fornecer um parâmetro de ordem para a transição de fase confinante/desconfinante. Além disso, eles são particularmente importantes no estudo da correspondência AdS/ CFT pois: i) Eles nos dão, em alguns casos, resultados exatos graças ao fato de poderem ser localizados em um modelo de matrizes, desta forma nos permitindo fazer testes altamente não triviais da correspondência; ii) Eles são os objetos da teoria de calibre que são duais as cordas propagando no interior do espaço, nos dando um rico dicionário entre quantidades no interior (AdS) e na borda do espaço (CFT). Depois de revisarmos os laços de Wilson em teorias de calibre e a correspondência Ads/CFT, introduziremos a definição dos laços de Wilson supersimétricos 1/2 BPS. Calcularemos eles para o caso de um acoplamento fraco e para qualquer outro valor da constante de acoplamento usando técnicas de modelos de matrizes. Finalmente, compararemos nossos resultados com computações de superfícies minimais no interior do espaço, encontrando uma concordância perfeita. / The aim of this thesis is to review Wilson loop operators in the contexto f the AdS/CFT correspondence. These operators, wich are present in any gauge theory, are important because they furnish an order parameter for confinement/deconfinement phase transitions. Besides this, they are particularly relevant in the study of the AdS/CFT correspondence because: i) they allow, in some cases, for exact results thanks to localization to matrix models and make it possible to perform highly non-trivial tests of the correspondence; ii) they are the gauge theory objects dual to strings propagating in the bulk of the space and give a rich dictionary between bulk (AdS) and boundary (CFT) quantities. After reviews of Wilson loops in gauge theories and of the Ads/CFT correspondence, we will introduce the definition of 1/2 BPS supersymmetric Wilson loops, we will compute them at weak coupling and then at any order in the coupling constant via matrix model techniques, and finally we will compare our results with minimal surface computations in the bulk, finding perfect agreement.

In dieser Dissertation werden Aspekte von superkonformen Quantenfeldtheorien untersucht, die für die sogenannte AdS/CFT Korrespondenz relevant sind. Die AdS/CFT Korrespondenz beschreibt eine Dualität zwischen Stringtheorien im Anti-de Sitter Raum und superkonformen Quantenfeldtheorien im Minkowskiraum. In diesem Kontext wurde die sog. Wilsonschleifen / Amplituden Dualität entdeckt, die die Übereinstimmung von n-Gluon MHV Amplituden und n-seitigen polygonalen Wilsonschleifen in der N=4 supersymmetrischen Yang-Mills (SYM) Theorie beschreibt. Im ersten Teil dieser Dissertation wird die Wilsonschleifenseite einer solchen möglichen Dualität in der N=6 superkonformen Chern-Simons (ABJM) Theorie untersucht. Das Hauptergebnis dieser Untersuchungen ist, dass der Erwartungswert der n-seitigen polygonalen Wilsonschleifen auf Einschleifenebene verschwindet, während er auf Zweischleifenebene in seiner funktionalen Form identisch zu der analogen Wilsonschleife in N=4 SYM auf Einschleifenniveau ist. Außerdem wird eine anomale konforme Wardidentität für Wilsonschleifen in Chern-Simons Theorie berechnet. Zudem werden die damit im Zusammenhang stehenden Entwicklungen für Amplituden und Korrelatoren in der ABJM Theorie diskutiert. Im zweiten Teil dieser Dissertation werden Dreipunktfunktionen von zwei geschützten Operatoren und einem Twist-Zwei Operator mit beleibigem Spin j in der N=4 SYM Theorie berechnet. Dafür werden die Indizes des Spin j Operators auf den Lichtkegel projiziert und der Korrelator wird in einem Grenzfall untersucht in dem der Impuls der bei dem Spin j Operator einfließt verschwindet. Dieser Grenzfall vereinfacht die perturbative Berechnung erheblich, da alle Dreipunktdiagramme effektiv auf Zweipunktdiagramme reduziert werden und die Abhängigkeit der Mischungsmatrix auf Einschleifenebene herausfällt. Das Ergebnis stimmt mit der Analyse der Operatorproduktentwicklung von Vierpunktfunktionen geschützter Operatoren von Dolan und Osborn aus dem Jahre 2004 überein. / In this thesis aspects of superconformal field theories that are of interest in the so-called AdS/CFT correspondence are investivated. The AdS/CFT correspondence states a duality between string theories living on Anti-de Sitter space and superconformal quantum field theories in Minkowski space. In the context of the AdS/CFT correspondence the so-called Wilson loop / amplitude duality was discovered, stating the equality of the finite parts of n-gluon MHV amplitudes and n-sided lightlike polygonal Wilson loops in N=4 supersymmetric Yang-Mills (SYM) theory. It is the subject of the first part of this thesis to investigate the Wilson loop side of a possible similar duality in N=6 superconformal Chern-Simons matter (ABJM) theory. The main result is, that the expectation value of n-sided lightlike polygonal Wilson loops vanishes at one-loop order and at two-loop order is identical in its functional form to the Wilson loop in N=4 SYM theory at one-loop order. Furthermore, an anomalous conformal Ward identity for Wilson loops in Chern-Simons theory is derived. Related developments and symmetries of amplitudes and correlators in ABJM theory are discussed as well. In the second part of this thesis we calculate three-point functions of two protected operators and one twist-two operator with arbitrary even spin j in N =4 SYM theory. In order to carry out the calculations, the indices of the spin j operator are projected to the light-cone and the correlator is evaluated in a soft-limit where the momentum coming in at the spin j operator becomes zero. This limit largely simplifies the perturbative calculation, since all three-point diagrams effectively reduce to two-point diagrams and the dependence on the one-loop mixing matrix drops out completely. The result is in agreement with the analysis of the operator product expansion of four-point functions of half-BPS operators by Dolan and Osborn in 2004.