This master thesis is framed in the striking correspondence between gravity theories in Anti-de Sitter spacetime (AdS) and Conformal Field Theories (CFT). This is usually known as AdS/CFT duality and relates gravity theories in the bulk with CFTs that live in their conformal boundary. We start by presenting the notion of CFTs and some of the results and techniques that are widely used in this field. This includes conformal correlators for scalar and spin operators, the state-operator correspondence and the operator product expansion (OPE) of operators. The embedding formalism and the index-free notation to encode tensors in polynomials are also discussed and used throughout this work. The basic notions of AdS are outlined and CFT at finite temperature is then introduced. We include a review of thermal blocks and thermal coefficients for a thermal two-point function between scalar fields in mean field theory. We then analyse the thermal two-point function for conserved currents, which was not known in the literature. Finally, we start a study of its thermal blocks and thermal coefficients for the mean field theory application.

Hawking theorized in 1974 that black holes emit particles as a quantum effect. It follows from this fact that a black hole that emits particles while absorbing none ends up evaporating. The process of black hole evaporation studied from semiclassical gravity violates quantum mechanics leading to serious problems. This is the black hole information paradox, one of the most famous paradoxes in theoretical physics first pointed out by Hawking in 1975 and still unsolved today. Nowadays the widespread interpretation is that quantum mechanics cannot be violated and that the semiclassical gravity approach is not good enough. We need to go beyond semiclassical physics to understand this process. The paradox as originally stated by Hawking is that a pure state evolves into a mixed state, violating unitarity and losing information in the process. There is also an alternative way to state the paradox using the so-called Page curve, which involves working with entropies rather than states. In a unitary process, the entanglement entropy of the radiation will follow the Page curve. In 2019, it was shown explicitly using holographic tools that an evaporating black hole in an Anti-de Sitter spacetime follows the Page curve. Holography is a property of quantum gravity stating that a spatial region can be described by its area rather than its volume. These recent developments also involve the famous island rule as the formula that reproduces the Page curve. This master thesis reviews the current understanding of the paradox, exploring the original paradox as well as the recent developments in the field.

<p>For the first time in the history of physics we stand in front of a theory that might actually serve as a unification of it all - string theory. It provides a self-consistent framework for gravity and quantum mechanics, which naturally incorporates matter and gauge interactions of the type seen in the standard model. Unfortunately, at the moment we do not know of any principle that selects the vacuum of the theory, so predictions about our four-dimensional world are still absent. However, the introduction of extended objects opens up an intricate new arena of physics, which is non-trivial and challenging to map out, even at a basic level.</p><p>A key concept of quantum gravity is holography; this is realised in string theory by the AdS/CFT correspondence, which relates string theory to a field theory living in a lower dimensional space. In this thesis we discuss two limits of the correspondence, namely the BMN limit, giving rise to a plane wave geometry, and the tensionless limit, exhibiting massless higher spin interactions. We also study a limit of string theory in a background electric field, where the theory is described by open strings and positively wound closed strings only.</p><p>We begin with a brief review of the theory, focusing on an intuitive understanding of the basic aspects and serving as an introduction to the papers. In the first paper we calculate, from two different points of view, scattering amplitudes in the non-commutative open string limit. In the second paper we obtain the quadratic scalar field contributions to the stress-energy tensor in the minimal bosonic higher spin gauge theory in four dimensions. In the last paper we propose a way to avoid fermion doubling when discretizing the string in the BMN limit.</p>

<p>One of the still big problems in the Standard Model of particle physics is the problem of confinement. Quarks or other coloured particles have never been observed in isolation. Quarks are only observed in colour neutral bound states. The strong interactions are described using a Yang-Mills theory. These type of theories exhibits asymptotic freedom, i.e. the coupling is weak at high energies. This means that the theory is perturbative at high energies only. Understanding quark confinement requires knowledge of the non perturbative regime. One attempt has been to identify the proper order parameters for describing the low energy limit and then to write down effective actions in terms of these order parameters. We discuss one possible scenario for confinement and the effective models constructed with this as inspiration. Further we discuss solitons in these models and their properties.</p><p>Yang-Mills theory has also become important in the context of string theory. According to the AdS/CFT correspondence string theory in AdS₅×S⁵ is dual to four dimensional Yang-Mills with four supersymmetries. The duality relate the non perturbative regime of one of the theories to the perturbative regime of the other. This makes it in general hard to test this conjecture. For a special type of solutions it is however possible to use a perturbative expansion in both theories. We discuss this type of solutions and in particular we discuss a method, the Bethe ansatz, to find the solutions on the gauge theory side.</p>

For the first time in the history of physics we stand in front of a theory that might actually serve as a unification of it all - string theory. It provides a self-consistent framework for gravity and quantum mechanics, which naturally incorporates matter and gauge interactions of the type seen in the standard model. Unfortunately, at the moment we do not know of any principle that selects the vacuum of the theory, so predictions about our four-dimensional world are still absent. However, the introduction of extended objects opens up an intricate new arena of physics, which is non-trivial and challenging to map out, even at a basic level. A key concept of quantum gravity is holography; this is realised in string theory by the AdS/CFT correspondence, which relates string theory to a field theory living in a lower dimensional space. In this thesis we discuss two limits of the correspondence, namely the BMN limit, giving rise to a plane wave geometry, and the tensionless limit, exhibiting massless higher spin interactions. We also study a limit of string theory in a background electric field, where the theory is described by open strings and positively wound closed strings only. We begin with a brief review of the theory, focusing on an intuitive understanding of the basic aspects and serving as an introduction to the papers. In the first paper we calculate, from two different points of view, scattering amplitudes in the non-commutative open string limit. In the second paper we obtain the quadratic scalar field contributions to the stress-energy tensor in the minimal bosonic higher spin gauge theory in four dimensions. In the last paper we propose a way to avoid fermion doubling when discretizing the string in the BMN limit.

One of the still big problems in the Standard Model of particle physics is the problem of confinement. Quarks or other coloured particles have never been observed in isolation. Quarks are only observed in colour neutral bound states. The strong interactions are described using a Yang-Mills theory. These type of theories exhibits asymptotic freedom, i.e. the coupling is weak at high energies. This means that the theory is perturbative at high energies only. Understanding quark confinement requires knowledge of the non perturbative regime. One attempt has been to identify the proper order parameters for describing the low energy limit and then to write down effective actions in terms of these order parameters. We discuss one possible scenario for confinement and the effective models constructed with this as inspiration. Further we discuss solitons in these models and their properties. Yang-Mills theory has also become important in the context of string theory. According to the AdS/CFT correspondence string theory in AdS5×S5 is dual to four dimensional Yang-Mills with four supersymmetries. The duality relate the non perturbative regime of one of the theories to the perturbative regime of the other. This makes it in general hard to test this conjecture. For a special type of solutions it is however possible to use a perturbative expansion in both theories. We discuss this type of solutions and in particular we discuss a method, the Bethe ansatz, to find the solutions on the gauge theory side.

Cette thèse est consacrée à l'étude de systèmes quantiques intégrables tels des chaînes de spins, des théories de champs à 1+1 dimensions, et la dualité AdS/CFT. Cette dualité AdS/CFT est une conjecture, émise à la fin du siècle dernier, qui relie notamment le régime non-perturbatif d'une théorie de jauge superconforme (nommée N =4 super Yang- Mills) au régime perturbatif d'une théorie de cordes dans un espace à 10 dimensions (de géométrie AdS5×S5). Ce manuscrit explore les similarités entre des chaînes de spins intégrables et des théories de champs intégrables, tels Super Yang Mills. Il commence par une étude ap- profondie des chaînes de spins intégrables pour y construire explicitement un "flot de Bäcklund" et des "opérateurs Q" polynômiaux, qui permettent de diagonaliser le Hamil- tonien. Des théories de champs intégrables sont ensuite étudiées et des "fonctions Q" sont obtenues, qui sont l'analogue des opérateurs Q construits pour les chaînes de spins. Il apparaît que de nombreuses informations sont contenue dans les propriétés analytiques des fonctions Q. Cela permet d'aboutir, dans le cadre de l'ansatz de Bethe thermody- namique, à un nombre fini d'équations non-linéaires intégrales qui encode le spectre des niveaux d'énergie de la théorie considérée (en taille finie). Ce système d'équations est équivalent au système infini d'équations, connu sous le nom de système Y, qui dans le cas de la dualité AdS/CFT avait été conjecturé assez récemment.

Using the AdS/CFT correspondence, we study the anisotropic transport properties of both supersymmetric and non-supersymmetric matter fields on (2+1)-dimensional defects coupled to a (3+1)-dimensional N=4 SYM "heat bath". We address on the one hand the purely conformal defect where the only non-vanishing background field that we turn on is a "topological", parameter parametrizing the impact on the bulk. On the other hand we also address the case of a finite external background magnetic field, finite net charge density and finite mass. We find in the purely conformal limit that the system possesses a conduction threshold given by the wave number of the perturbation and that the charge transport arises from a quasiparticle spectrum which is consistent with an intuitive picture where the defect acquires a finite width in the direction of the SYM bulk. We also examine finite-coupling modifications arising from higher derivative interactions in the probe brane action. In the case of finite density, mass and magnetic field, our results generalize the conformal case. We discover at high frequencies a spectrum of quasiparticle resonances due to the magnetic field and finite density and at small frequencies a Drude-like expansion around the DC limit. Both of these regimes display many generic features and some features that we attribute to strong coupling, such as a minimum DC conductivity and an unusual behavior of the "cyclotron" and plasmon frequencies, which become correlated to the resonances found in the conformal case. We further study the hydrodynamic regime and the relaxation properties, in which the system displays a set of different possible transitions to the collisionless regime. The mass dependence can be cast in two regimes: a generic relativistic behavior dominated by the UV and a non-linear hydrodynamic behavior dominated by the IR. In the massless case, we also extend earlier results to find an interesting duality under the transformation of the conductivity and the exchange of density and magnetic field. Furthermore, we look at the thermodynamics and the phase diagram, which reproduces general features found earlier in 3+1 dimensional systems and demonstrates stability in the relevant phase.

Using the AdS/CFT correspondence, we study the anisotropic transport properties of both supersymmetric and non-supersymmetric matter fields on (2+1)-dimensional defects coupled to a (3+1)-dimensional N=4 SYM "heat bath". We address on the one hand the purely conformal defect where the only non-vanishing background field that we turn on is a "topological", parameter parametrizing the impact on the bulk. On the other hand we also address the case of a finite external background magnetic field, finite net charge density and finite mass. We find in the purely conformal limit that the system possesses a conduction threshold given by the wave number of the perturbation and that the charge transport arises from a quasiparticle spectrum which is consistent with an intuitive picture where the defect acquires a finite width in the direction of the SYM bulk. We also examine finite-coupling modifications arising from higher derivative interactions in the probe brane action. In the case of finite density, mass and magnetic field, our results generalize the conformal case. We discover at high frequencies a spectrum of quasiparticle resonances due to the magnetic field and finite density and at small frequencies a Drude-like expansion around the DC limit. Both of these regimes display many generic features and some features that we attribute to strong coupling, such as a minimum DC conductivity and an unusual behavior of the "cyclotron" and plasmon frequencies, which become correlated to the resonances found in the conformal case. We further study the hydrodynamic regime and the relaxation properties, in which the system displays a set of different possible transitions to the collisionless regime. The mass dependence can be cast in two regimes: a generic relativistic behavior dominated by the UV and a non-linear hydrodynamic behavior dominated by the IR. In the massless case, we also extend earlier results to find an interesting duality under the transformation of the conductivity and the exchange of density and magnetic field. Furthermore, we look at the thermodynamics and the phase diagram, which reproduces general features found earlier in 3+1 dimensional systems and demonstrates stability in the relevant phase.

Nous présentons dans cette thèse deux applications de la correspondance AdS/CFT. La première est l'analyse des propriétés de transport des modes fermioniques dans les théories des champs fortement couplées. Nous étudions en particulier les propriétés de la constante de diffusion du phonino dans N=4 SYM à densité nulle ou finie. Nous trouvons que la constante de diffusion dépend du potentiel chimique et par conséquent qu'elle n'a pas une propriété d'universalité similaire à celle de la viscosité de cisaillement. La deuxième application traite du comportement de l'entropie d'intrication dans des théories qui contiennent des degrés de liberté massifs. Pour identifier les contributions de la masse à l'entropie d'intrication nous l'évaluons dans un système comprenant des branes de saveur et nous identifions certains des termes dépendants de la masse. Nous trouvons que le coefficient du terme logarithmique est différent de celui calculé dans la théorie des champs libre, un résultat qui est qualitativement en accord avec de résultats holographique antérieurs. De plus nous calculons d'autres termes, prédits dans la théorie des champs, mais qui n'ont pas été identifiés auparavant dans un système holographique.