This dissertation is devoted to investigation of quantum chaos in the Sachdev-Ye-Kitaev (SYK) and supersymmetric SYK models. First, a unified minimal scheme is developed to classify quantum chaos in the SYK and supersymmetric SYK models and also work out the structure of the energy levels in one periodic table. The SYK with even q-body or supersymmetric SYK with odd q-body interaction, with N even or odd number of sites, are put on an equal footing in the minimal Hilbert space; N (mod 8), q (mod 4) double Bott periodicity, and a reflection relation are identified. Then, exact diagonalizations are performed to study both the bulk energy level statistics and hard-edge behaviors. Excellent agreements between the exact diagonalization results and the symmetry classifications are demonstrated. This compact and systematic method can be transformed to map out more complicated periodic tables of SYK models with more degrees of freedom, tensor models, or symmetry protected topological phases.

No description available.

We present a method of quantum renormalization group, which makes it possible to construct a bulk theory for a general conformal field theory in the context of anti-de Sitter/conformal field theory correspondence. We demonstrate that within this method it is possible to construct scalar field theory in anti-de Sitter space. We also demonstrate that from a conformal field theory possessing global symmetry, it is possible to construct non-abelian gauge theory in anti-de Sitter space. / Thesis / Master of Science (MSc)

El excelente comportamiento de las columnas tubulares rellenas de hormigón (en adelante columnas CFT) y el aumento del empleo de hormigón de alta resistencia (en adelante HAR), unido a que la normativa actual en algunos países y en especial en Europa, a través del Eurocódigo 4 (en adelante EC4), no recoge la posibilidad de diseño columnas CFT rellenas con este material, pone de manifiesto la necesidad de actualizar dicha normativa para incluir también la posibilidad de diseñar columnas CFT con HAR. Esta Tesis Doctoral nace, por tanto, con el objetivo de actualizar la formulación existente en el EC4 de manera que se puedan diseñar columnas CFT de sección circular rellenadas con HAR. La formulación para el diseño de las columnas CFT de sección circular, cuantifica el aumento de la resistencia que se produce en éstas por el efecto del confinamiento pasivo. Dicho confinamiento pasivo se produce al disponer en el exterior del núcleo de hormigón una camisa de acero que coarta las deformaciones laterales que en él se producen. Como consecuencia de ello, la resistencia que es capaz de soportar el hormigón aumenta significativamente, aumentando también la resistencia global de la columna CFT. Para reproducir adecuadamente la evolución de las deformaciones laterales del núcleo de hormigón durante el proceso de carga (aspecto clave en el estudio del confinamiento), se desarrolló un modelo de hormigón que tiene en cuenta las deformaciones laterales tanto elásticas, a través del coeficiente de Poisson, como plásticas, a través del ángulo de dilatancia. Para ello, fue necesario obtener las curvas de definición del ángulo de dilatancia que regula las deformaciones laterales plásticas en función de la presión lateral de confinamiento a la que está sometido el núcleo de hormigón y la resistencia de éste, así como unas nuevas curvas que regulan las relaciones entre tensión y deformación longitudinal adaptadas para hormigones confinados pasivamente. Una vez se dispuso del modelo, calibrado mediante un proceso iterativo en el que se utilizaron ensayos experimentales, se llevó a cabo una validación de éste comparándolo con numerosos casos experimentales de diferentes características y geometrías. Con el modelo numérico definido se realizó un estudio paramétrico que permitió proponer una formulación que completa a la que actualmente existe, abarcando hormigones de hasta 100 MPa. / Lacuesta Carrión, CD. (2013). ESTUDIO DE LA RESISTENCIA DE PILARES TUBULARES CIRCULARES RELLENOS DE HORMIGÓN CON UN MODELO NUMÉRICO DE CONFINAMIENTO PASIVO VARIABLE [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/28587

In this thesis I explore the interplay of geometry and quantum information theory via the holographic principle, with a specific focus on geometric phases in quantum systems like two interacting qubits, and how they relate to entanglement measures and Hilbert space factorisation. I establish geometric phases as an indicator for Hilbert space factorsiation, both in an abstract sense using von Neumann operator algebras as well as applied to the eternal black hole within the AdS/CFT correspondence. For the latter case I show that geometric phases allow to diagnose non-factorisation from a boundary point of view. I also introduce geometric quantum discord as a second geometric measure for non-factorisation and reveals its potential implications for the study of black hole microstates. / In dieser Arbeit untersuche ich das Zusammenspiel von Geometrie und Quanteninformation mit Hilfe des holografischen Prinzips. Dabei konzentriere ich mich besonders auf geometrische Phasen in Quantensystemen wie zwei wechselwirkenden Qubits und darauf, wie sie mit Verschränkungsmaßen und Hilbert-Raum-Faktorisierung zusammenhängen. Ich führe geometrische Phasen als Indikator für die Faktorisierung des Hilbert-Raums ein, sowohl in einem abstrakten Sinne unter Verwendung von von Neumann-Operator-Algebren als auch angewandt auf das ewige Schwarze Loch im Rahmen der AdS/CFT-Korrespondenz. im zweiten Fall zeige ich, dass geometrische Phasen es erlauben, die Nicht-Faktorisierung von der Randperspektive aus zu diagnostizieren. Außerdem führe ich die geometrische Quantendiskordanz als zweites geometrisches Maß für die Nicht-Faktorisierung ein und zeige ihre möglichen Auswirkungen auf die Untersuchung von Mikrozuständen Schwarzer Löcher auf.

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In this thesis we explore CFTs and related nonperturbative phenomena through several various techniques. First we construct a novel way to latticize AdS via the triangle group. We characterize aspects of this tessellation in 2d as well extend the construction to 3d, where we find a bulk critical point for scalar 𝜙⁴ theory. We then study 2d minimal model CFTs in a bulk AdS₂ through powerful boundary CFT and JT/Schwarzian techniques. Finally, we use an algebraic formulation of a conformal interface separating two minimal models -- a RG brane -- to compute correlators in the presence of the brane to better understand the relation between different minimal model CFT theories.

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)

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.