Tesis sobre el tema "Dualité AdS/CFT"

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.

Cette thèse est consacrée à l'étude de systèmes quantiques intégrables tels des chaines de spin, des théories de champs à 1+1 dimensions, et la dualité AdS/CFT. Cette dualité AdS/CFT est une conjecture, emmise 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 AdS₅xS⁵). 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 approfondie des chaînes de spins intégrables pour y construire explicitement un "flot de Bäcklund" et des "opérateurs Q" polynomiaux, qui permettent de diagonaliser le Hamiltonien. 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 thermodynamique, à un nombre fini d'équations non-linéaires intégrales qui encode le spectre des niveaux d'energie de la théorie considérée (en taille finie). Ce système d'équation est équivalent au système infini d'équation, connu sous le nom de système Y, qui dans le cas de la dualité AdS/CFT avait été conjecturé assez récemment
This thesis is devoted to the study of integrable quantum systems such as spin chains, bidimentional field theories and the AdS/CFT duality. This AdS/CFT duality is a conjecture, stated in the end of the last century, which relates (for instance) the non-perturbative regime of a superconformal gauge theory (called N=4 super Yang-Mills) and the perturbative regime of a string theory on a 10-dimensioonal space with the geometry AdS₅xS⁵. This thesis explores the similarities between integrable spin chains and quantum field theories, such as Super Yang Mills. We first study integrable spin chains and build explicitely a polynomial "Bäcklund flow" and polynomial "Q-operators", which allow to diagonalize the Hamiltonian. We then study integrable field theories et show how to obtain "Q-functions", analogous to the Q-operators built for spin chains. It turns out that several important informations are contained in the analytic properties of these Q-functions. That allows to obtain, in the framework of the thermodynamic Bethe Ansatz, a finite number of non-linear integral equations encoding the spectrum of the theory which we study. This system of equations is equivalent to an infinite system of equations, known as "Y-system", which had been quite recently conjectured in the case of the AdS/CFT duality

Dans le cadre de la dualité holographique entre une vaste famille de vides 1/2-maximalement supersymétriques Anti-de Sitter à quatre dimensions (AdS₄) et des théories des champs superconformes N=4 supersymétriques à trois dimensions (sCFT₃), nous étudions des questions théoriques majeures de gravité quantique et de théories de jauge. Ce travail a deux directions principales : La premiere partie est consacrée aux mécanismes par lesquels le graviton AdS₄ peut acquérir une petite masse, tandis que la seconde partie concerne la cartographie de la variété superconforme des sCFT₃ considérées. En ce qui concerne la question du mecanisme de Higgs pour le graviton d’AdS₄, nous proposons un nouveau mécanisme qui repose sur le couplage ”faible” de deux sCFT₃s, initialement découplées, en jaugent une symmétrie globale commune. Les deux tenseurs de stress initialement conservés se mélangent et le résultat est une combinaison conservée et une combinaison orthogonale, dont la dimension acquiert une petite dimension anormale. Holographiquement, cette configuration correspond à la connexion de deux univers AdS₄ initialement découplés via un AdS₅ × S⁵ fin, autrement appelé une “gorge” de Janus. Le résultat est une théorie AdS₄-bimétrique, avec un graviton sans masse et un graviton massif, dont la petite masse correspond à la dimension anormale de la combinaison duale de tenseurs de stress. Nous calculons la masse du graviton, qui est exprimée en termes de données géométriques de la ”gorge” de Janus et de l’univers AdS₄ vers zéro, résulte en une théorie de gravité massive dans AdS₄. En ce qui concerne la deuxième direction de ce travail, les déformations superconformes des sCFT₃s considérées qui génèrent la variété superconformale sont des déformations préservant N = 2 supersymétrie, générées par des opérateurs exactement marginaux. Nous présentons comment tous ces opérateurs peuvent être systématiquement extraits de l’index superconforme. Les opérateurs de branche de Coulomb et de Higgs sont pris en compte, tandis qu’une attention particuli ère est accordée aux opérateurs mixtes. On montre que les modules de branches mixtes de ces théories sont des opérateurs à double-corde qui se transforment dans la représentation (Adj, Adj) des groupes de saveursélectriques et magnétiques, modulo un surcomptage pour les quivers avec des noeuds de jauge abéliens. Enfin, nous commentons sur l’interprétation holographique des résultats, en affirmant que les supergravités mesurées peuvent capturer l’espace des modules tout entier si, outre les paramètres de la solution d’arri ère-plan, les modules de quantification des conditions aux limites sontégalement pris en compte
Based on the holographic duality between a large class of half-maximally supersymmetric four-dimensional Anti-de Sitter (AdS₄) vacua and three-dimensional N = 4 superconformal field theories (sCFT₃), we study quantum gravitational and gauge theoretic questions. This work has two main directions: The first part is devoted to the mechanisms through which the low-lying AdS₄-graviton can acquire a small mass whereas the second part regards the mapping of the superconformal manifold of the considered sCFT₃s. Regarding the question of the graviton Higgsing in AdS₄, we propose a new mechanism which relies on ”weakly” coupling two initially decoupled sCFT₃s, by gauging a common global symmetry. The two initially conserved stress tensors mix and the result of this mixing is a conserved combination and an orthogonal combination, the scaling dimension of which acquires a small anomalous dimension. Holographically, this setup is dual to connecting two initially decoupled AdS₄ universes via a thin AdS₅ × S⁵ or Janus ”throat”. The result is an AdS₄- bimetric theory, with one massless and one massive graviton, the small mass of which corresponds to the anomalous dimension of the dual stress tensor combination. We compute the mass of the graviton, which is expressed in terms of the geometric data of the Janus ”throat” and of the considered AdS₄ universe. A special decoupling limit of this theory, where the effective four-dimensional gravitational coupling of one of the two universes vanishes, results to an AdS₄-Massive gravity theory. Regarding the second direction of this work, superconformal deformations of the considered sCFT3s which generate the superconformal manifold, are N = 2 supersymmetry preserving deformations, generated by exactly marginal operators. We present how all these operators can be consistently extracted from the superconformal index. Coulomb and Higgs branch operators are considered, while particular attention is payed to mixedbranch operators. It is shown that the mixed-branch moduli of these theories are double-string operators transforming in the (Adj,Adj) representation of the electric and magnetic flavour groups, up to overcounting for quivers with abelian gauge nodes. Finally, we comment on the holographic interpretation of the results, arguing that gauged supergravities can capture the entire moduli space if, in addition to the parameters of the background solution, quantization moduli of boundary conditions are also taken into account

Scientific abstract In this dissertation we perform a series of study concerning dualities and integrability properties underlying the AdS3/CFT2 and AdS2/CFT1 correspondences. These are particularly interesting because symmetry do not constrain the dynamics of AdS3 and AdS3 superstrings in the same way as in the higher dimensional instances of AdS/CFT, allowing for novel phenomena such as the presence of massless worldsheet modes or non-coset fermions. We will investigate the self-duality of Green–Schwarz supercoset sigma models on AdSd⇥Sd⇥Sd (d = 2, 3), whose isometry supergroups are (d − 1) copies of the exceptional Lie supergroup D(2, 1; ↵). Our main finding is that additional complex T-dualities along one of the spheres Sd are needed to map the superstring action to itself. Importantly, this proves dual superconformal symmetry of their CFT duals via AdS/CFT. Dual superconformal symmetry is strictly related to integrability, which we study in depth for both AdS3 and AdS2 superstrings. Indeed, we will derive the exact S-matrix conjectured to be related to the massive modes of type IIB AdS2 ⇥ S2 ⇥ T6 superstrings. This S-matrix is psuc(1|1) invariant and it was found to be in perfect agreement with the tree-level result following from string perturbation theory. We also unveil the Yangian algebra ensuring the integrability of the AdS2⇥S2⇥T6 superstring in the planar limit, Y[psu(1|1)c], as well as its secret symmetries. By using the RTT realisation, we provide two di↵erent representations of the Hopf algebra: one is reminiscent of the Yangian underlying AdS5/CFT4, but it is not of evaluation type. The other representation, obtained from co-commutativity, is instead of evaluation type. We explore two limits of the S-matrix for AdS2/CFT1: one is the classical r-matrix, which is the first non-trivial order in the 1/g expansion, with g being the e↵ective tension in the AdS2 ⇥ S2 ⇥ T6 superstring action. In this limit, corresponding to classical strings, we found that secret symmetries not only are present, but also essential to formulate the classical rmatrix in a universal, representation independent form. On the other hand, the limit g ! 0, corresponding to the weakly coupled CFT1, shows that the dual integrable model is described by an e↵ective theory of free fermions on a periodic spin-chain if g = 0, while one obtains a non-trivial spin chain of XYZ type if g 6= 0. Finally, we investigate Yangian and secret symmetries for AdS3 type IIB superstring backgrounds, verifying the persistence of such structures in AdS3/CFT2. Especially, we find that the antipode map, related to crossing symmetry, exchanges in a non-trivial way left and right generators of Y[psu(1|1)2c], the Yangian underlying the integrability of AdS3 superstrings.

Motivated by the AdS/CFT correspondence, new supersymmetric solutions to Type IIB and Type IIA supergravity are presented. These solutions contain $AdS_5$ or $AdS_4$ factors and are generated using T-duality symmetries of supergravity. The technique used to generate these solutions consists of performing a series of non-Abelian and Abelian T-dualities, sometimes with coordinate shifts in-between, to Freund-Rubin type seed backgrounds. An added bonus of the gauge fixing procedure inherent in non-Abelian T-Duality is the freedom to generate backgrounds with extra free parameters, some examples of which are presented. Aspects of the dual field theories of these new solutions are analyzed using holography techniques. The supersymmetry of these new backgrounds is also discussed. In addition to supergravity backgrounds with AdS, the study of generalized Calabi-Yau manifolds in the context of flux compactifications is briefly reviewed. The particular case of the resolved cone over $Y^{p,q}$ and its admission of generalized SU(3) structure solutions is examined. Contrary to geometries with $AdS$ factors, whose field theory duals are conformal field theories, these types of geometries can be phenomenologically interesting to study, as their gauge theory duals are minimally supersymmetric and confining, thus they could someday help aid our understanding of strongly-coupled QCD (Quantum Chromodynamics).

Motivated by the holographic principle, within the context of the AdS/CFT Correspondence in the large t'Hooft limit, we investigate how the geometry of certain highly symmetric bulk spacetimes can be recovered given information of physical quantities in the dual boundary CFT. In particular, we use the existence of bulk-cone singularities (relating the location of operator insertion points of singular boundary correlation functions to the endpoints of boundary-to boundary null geodesics in the bulk spacetime) and the holographic entanglement entropy proposal (relating the entanglement entropy of certain subsystems on the boundary to the area of static minimal surfaces) to recover the bulk metric. Using null and zero-energy spacelike boundary-to-boundary geodesic probes, we show that for classes of static, spherically symmetric, asymptotically AdS spacetimes, one can find analytic expressions for extracting the metric functions given boundary data. We find that if the spacetime admits null circular orbits, the bulk geometry can only be recovered from the boundary, down to the radius of null circular orbits. We illustrate this for various analytic and numerical boundary functions of endpoint separation of null and spacelike geodesics. We then extend our analysis to higher dimensional minimal surface probes within a class of static, planar symmetric, asymptotically AdS spacetimes. We again find analytic and perturbative expressions for the metric function in terms of the entanglement entropy of straight belt and circular disk subsystems of the boundary theory respectively. Finally, we discuss how such extractions can be generalised.

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O objetivo desse trabalho concentra-se em estudar uma nova metodologia para resolução de problemas em matéria condensada, nos quais as partículas estão fortemente acopladas, não permitindo um tratamento perturbativo padrão. Esse método, originário da dualidade AdS/CFT (Anti-de Sitter/Conformal Field Theory), está sendo conhecido na comunidade de matéria condensada como Dualidade Holográfica. Como aplicação desta teoria, o comportamento de supercondutores foi escolhido. A metodologia da dualidade holográfica utiliza uma teoria de campos em um espaço-tempo curvo. Partindo da ação associada é possível obter as equações de movimento das partículas envolvidas. A seguir foi empregado o uso do software Mathematica-Wolfram, para a resolução das Equações Diferenciais através do Método de Frobenius. Os resultados obtidos expressam que os valores esperados de operadores, as chamadas funções de correlação, são bem similares à curva de intervalo de energia previsto pela teoria BCS em função da temperatura. Também infere-se a partir da fórmula de Kubo, que os resultados numéricos da condutividade elétrica, claramente indicam uma condutividade infinita abaixo de uma determinada temperatura crítica Tc, resultando em um supercondutor.
The objective of this work is to study a new methodology for solving problems in condensed matter, in which the particles are strongly coupled, not allowing a standard perturbative treatment. This method, originating from the AdS / CFT (Anti-Sitter / Conformal Field Theory) duality, is being known in the condensed matter community as Holographic Duality. As an application of this theory, the behavior of superconductors was chosen. The holographic duality methodology uses a field theory in a curved space-time. From the associated action it is possible to obtain the equations of motion of the particles involved. Next, the use of the Mathematica-Wolfram software was used to solve the Differential Equations by the Frobenius Method. The obtained results express that the expected values of operators, so-called correlation functions, are very similar to the energy interval curve predicted by the BCS theory as a function of temperature. It is also inferred from the Kubo formula that the numerical results of electrical conductivity clearly indicate an infinite conductivity below a given critical temperature Tc, resulting in a superconductor.
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This thesis consists in a self-contained study of gauge/gravity dualities in the line of the Klebanov-Witten model. Here we explore first the known Maldacena duality that relates N=4 SYM theory in four dimensions to type IIB supergravity on AdS_5×S^5 in reasonable detail, after some necessary preliminaries on supersymmetric gauge theories, where we display in detail the supersymmetry algebra and representations for N 1 supersymmetry. There we also construct the so-called superfields that will be helpful to write invariant lagrangians for gauge theoriesmreadily, and then useful to construct the gauge theory side of the Klebanov-Witten model. In the original AdS/CFT correspondence and its phenomenologically interesting extensions, Dp-branes as solutions of supergravity and nonperturbative objects in string theory where gauge theory lives are crucial. So, to preserve the self-contained nature of this work, we include a brief review of superstring theory addressed to understand the need to include this higher-dimensional objects by T-duality and, at low-energy limit of the string theory, as solutions of the Einstein equations. The first climax of this work occurs when we use all we learned to establish the Maldacena conjecture, N=4 SU(Nc) SYM theory we study in the supersymmetry chapter, living on the four-dimensional worldvolume of a stack of Nc D3-branes in a flat-space, corresponds exactly to type IIB supergravity on AdS_5×S^5 .In order to prove it, we match symmetries and operators with states in both sides. But actually this corresponds to the weak form of the correspondence, because it is not possible to handle neither string theory or gauge theory at strong coupling. The focus and main motive to have to learn the first hundred of pages here will be to extend the dual gauge theory we studied in AdS/CFT towards more realistic gauge theories as duals of some supergravity theory. The Klebanov-Witten model, consists in replacing the five-sphere in the gravity background of type IIB for a more interesting Einstein manifold X5 , a coset space called T^1,1 .The resulting dual gauge theory is expected to be less supersymmetric, and it is indeed N = 1 superconformal field theory with matter content in the bifundamental representation of the gauge group SU(N)×SU(N), and a quartic superpotential that exhibits SU(2)×SU(2)×U(1) global symmetry, which is precisely the symmetry of the coset space in the gravity side. This is not the end of the story, the Klebanov-Witten model extended the Maldacena correspondence and found as a dual gauge theory a less supersymmetric but still conformal theory. Breaking of the conformal theory, proposed by Klebanov, Nekrasov and Tseytlin, is achieved by introducing fractional M D3-branes in addition to the N regular D3-branes. The resulting theory is an SU(N+M)×SU(N) gauge theory with N = 1 supersymmetry, no longer conformal and then a little more interesting as a part of the crusade to find a QCD-like theory. This is still not the end, the last model suffers from a singularity in the deep IR, rendering the gravitational description invalid in that regime. It was conjectured that the strong dynamics of the gauge theory should somehow resolve this problem. Klebanov, again, and Strassler showed that this conjecture was correct, and argue that the RG flow is in fact an infinite series of Seiberg duality transformations- a cascade - in which the number of colors repeatedly drops from N NM , so the gaugegroup changes from SU(N+M)×SU(N) to SU(NM) ×SU(N). This process can be repeated until the IR limit where the gauge group simply becomes SU(M). So, at the end we get a N=1 SU(M) gauge theory, a QCD-like theory. We say that the standard model itself may lie at the base of a duality cascade.
Essa tese consiste num estudo autocontido das dualidades calibre/gravidade na linha do modelo do Klebanov-Witten. Aqui nos exploramos primeiro de um jeito razoavelmente detalhado,a conhecida dualidade do Maldacena que relaciona a teoria N=4 SYM em quatro dimensões com as supercordas tipo IIB no espaço AdS_5×S^5, depois de alguns preliminares necessários sobre teorias supersimétricas de calibre, onde nós mostramos em detalhe à algebra supersimétrica e as representações para N 1 supersimetria. Nós também construímos os conhecidos supercampos que são úteis para escrever lagrangianas invariantes para teorias de calibre facilmente, e então serão úteis para construir a teoría de calibre do modelo de Klebanov-Witten. Na correspondência AdS/CFT original e as suas extensões fenomenologicamente interessantes, as Dp-branas, como soluções de supergravidade e objetos não perturbativos na teoria de cordas onde as teorias de calibre moram, são essenciais. Assim ,a fim de preservar a natureza autocontida desse trabalho, nós incluímos uma breve revisão sobre teoria de supercordas dirigida a entender a necessidade de incluir esses objetos extra-dimensionais usando dualidade-T e, no limite de baixa-energia da teoria de cordas, como soluções das equações de Einstein. O primeiro clímax desse trabalho ocorre quando nós usamos tudo o que aprendemos para estabelecer a conjectura do Maldacena, a teoria de calibre N=4 SYM que nós estudamos no capítulo de supersimetria, morando no volume de mundo quadridimensional de uma pilha de Nc D3-branas (sim, o subscrito c significa cor!) em espaço plano, corresponde exatamente à teoria de supergravidade tipo IIB no espaço AdS_5×S^5 . A fim de testar ela, nós identificamos simetrias e operadores com estados em ambos lados da dualidade. Mas na verdade isto corresponde à forma fraca da correspondência, porque não é possível lidar nem com a teoria de cordas nem com a teoria de calibre no limite de acoplamento forte. O foco e motivo principal de porque nós temos que aprender as primeiras cem páginas aqui, será estender a teoria de calibre dual que estudamos em AdS/CFT, para teorias de calibre mais realisticas como duais de alguma teoria de supergravidade. O modelo do Klebanov-Witten, consiste em substituir a esfera de cinco dimensões no fundo de supergravidade da teoria de supercordas tipo IIB por um espaço que é mais interessante X5, um espaço coset chamado T^1,1. Nós esperamos que a teoria de calibre dual que resulta é menos supersimetrica, e na verdade é N =1 superconforme com um conteúdo de matéria na representação bifundamental do grupo de calibre SU(N)×SU(N), e um superpotencial quártico que tem simetria global SU(2)×SU(2)×U(1), que é precisamente a simetria do espaço coset no lado da gravidade. Mas isso não é tudo, o modelo do Klebanov-Witten estendeu a correspondência do Maldacena e encontrou como teoria dual uma teoria menos supersimetrica mas ainda conforme. A quebra da simetria conforme, proposta pelo Klebanov, Nekrasov e Tseytlin, é obtida introduzindo M D3-branas fracionais além das N D3-branas regulares. A teoria resultante é uma teoria de calibre SU(N+M)×SU(N) com N = 1 supersimetria, não mais conforme e então um pouco mais interessante como parte da nossa cruzada para encontrar uma teoria tipo-QCD. Isso ainda não é o final, o modelo anterior sofre de uma singularidade no IR profundo, tornando inválido a descrição gravitacional. Foi conjeturado então que a dinâmica do acoplamento forte na teoria de gauge deveria de algum jeito resolver esse problema. Klebanov, de novo, e Strassler mostraram que essa conjetura foi correta, e argumentaram que o fluxo do GR é de fato uma serie infinita de transformações de dualidade de Seiberg - uma cascata - onde o numero de cores cai repetidamente de NNM, e o grupo de calibre muda de SU(N+M)×SU(N) a SU(NM)×SU(N). O processo pode ser repetido até o limite IV onde o grupo de calibre simplesmente torna-se SU(M). Então, no final nós obtemos uma N = 1 teoria de calibre SU (M ), ou seja uma teoria tipo-QCD. Então, nós dissemos que o modelo padrão mesmo pode se situar na base da cascata de dualidade.

This thesis investigates applications of the gauge/gravity duality to strongly coupled quantum field theories. After a review of the duality and of correlators and transport in quantum systems, we present our results on second-order non-conformal hydrodynamics. We derive new Kubo formulae for five second-order transport coefficients in non-conformal relativistic fluids. We then apply these Kubo formulae to a class of non-conformal holographic fluids at infinite coupling. We find strong evidence that the Haack-Yarom identity, known to relate second-order coefficients in conformal holographic fluids at infinite coupling, continues to hold in holographic fluids without conformal symmetry: Within our class of models, we prove that it still holds when leading non-conformal corrections are taken into account, and we show numerically that it is also obeyed beyond leading order. This provides further evidence that the identity may be a universal feature of strongly coupled fluids. Next, we present our results on magnetic spin impurities in strongly correlated systems. We build a holographic two-impurity Kondo model, identifying the inter-impurity interaction as double-trace deformation. Our numerical results for the phase diagram suggest a quantum phase transition between a trivial phase with uncorrelated spins and no Kondo screening, and a non-trivial phase with anti-ferromagnetic correlations and simultaneous Kondo screening. Computing the spectrum in the single-impurity case, we observe Fano resonances, which at low temperatures we identify with the Kondo resonance.

The gauge/string duality, a.k.a. the holographic principle is a profound assertion that emerged from string theory. It relates strongly-coupled gauge theories to weakly coupled string theories living in a higher-dimensional curved geometry. Nevertheless, it is a conjecture, and only a few instances of its more concrete form, the AdS/CFT correspondence, are well-understood. The most well-studied example is the duality between N=4 SYM, which is a CFT, and type IIB string theory in AdS5xS5 background. Generalization to less symmetric cases is a must, and the next logical step is to add a mass scale to N=4 SYM, therefore breaking its conformal symmetry and leading to N=2* SYM, the theory we study in this thesis. It is supersymmetric enough to employ the powerful localization method that reduces its partition function to a matrix model. We will see that the mass scale causes non-trivial phase structures in its vacuum configuration, visible in the holographic regime. We will probe them using Wilson loops in different representations of the gauge group. On the other hand, the dual supergravity background was derived by Pilch-Warner, making N=2* theory an explicitly testable non-conformal holographic case, which is a rare example. We will prove that the duality works for the dual observables (string action, D-branes) we managed to compute, even at a quantum-level.

The holographic principle has become an extraordinary tool in theoretical physics, most notably in the form of the Anti-deSitter Conformal Field Theory (AdS/CFT) correspondence, in which classical gravitational degrees of freedom in N-dimensions are related quantum field theory degrees of freedom in N − 1 dimensions in the limit of a large number of fields. Here we present an account of the AdS/CFT correspondence, also known as the gauge/gravity duality, from its origins in the large N 'tHooft expansion, up to Maldacena's proposal that type IIB string theory in the presences of D-branes at low energy is dual to an N = 4, d = 4, U(N) super Yang-Mills on AdS5 × S5 . We begin with an extensive review of (super)string theory including D-branes. We then present the general formulation of the AdS/CFT in the supergravity background of AdS5 × S5 , along with several examples of how it is used in terms of the identification of bulk fields with operators on the bound- ary of a CFT. We move on to discuss two applications of the gauge/gravity duality. The first is the application of the holographic gauge/gravity correspondence to the QCDk-string. The second applies the AdS/CFT formalism to a Kerr black hole solution embedded in 10-dimensional heterotic sting theory. These two applications of the holographic gauge/gravity duality comprise the original work presented here. We follow with summaries and discussions of the background material, the original work, and future investigations.

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Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
Nessa tese estudamos propriedades de soluções de supergravidade tipo II obtidas através da dualidade T abeliana e não abeliana. Também determinamos, através da conjectura gauge/gravidade, aspectos da teoria de campos dual a essas soluções obtidas por dualidade T. Consideramos três tipos distintos de soluções: duais a teorias de campos que confinam, duais a teoria de campos conformes e duais a teoria de campos não-relativistas. Concluimos essa tese com uma análise dos laços de Wilson em soluções com simetria não relativista.
In this thesis we study properties of type II supergravity solutions generated by abelian and nonabelian T-duality. Also we determine, through the gauge/gravity conjecture, some aspects of the field theory dual to the supergravity solutions obtained by T-dualization. We consider three distinct types of backgrounds solutions, namely, backgrounds that are dual to confining field theories, backgrounds dual to conformal field theories and those dual to nonrelativistic field theories. We conclude this thesis with an analysis of Wilson loops on backgrounds with nonrelativistic symmetries.
CNPq: 140588/2012-4

In this thesis we study gauge/gravity duals in the 5d/6d AdS/CFT correspondence. We start with field theories defined on squashed five-spheres with SU(3) × U(1) symmetry. These five-sphere backgrounds are continuously connected to the round sphere. We find a one-parameter family of 3/4 BPS deformations and a two-parameter family of (generically) 1/4 BPS deformations. The gravity duals are constructed in Euclidean Romans F(4) gauged supergravity in six dimensions, and uplift to massive type IIA supergravity. We holographically renormalize the Romans theory, and use our general result to compute the renormalized on-shell actions for the solutions. The results agree perfectly with the large N limit of the dual gauge theory partition function, which we compute using large N matrix model techniques. In addition we compute BPS Wilson loops in these backgrounds, both in supergravity and in the large N matrix model, again finding precise agreement. We conjecture a general formula for the partition function on any five-sphere background, which for fixed gauge theory depends only on a certain supersymmetric Killing vector. We then proceed to study Euclidean Romans supergravity in six dimensions with a non-trivial Abelian R-symmetry gauge field. We show that supersymmetric solutions are in one-to-one correspondence with solutions to a set of differential constraints on an SU(2) structure. As an application of our results we (i) show that this structure reduces at a conformal boundary to the five-dimensional rigid supersymmetric geometry previously studied, (ii) find a general expression for the holographic dual of the VEV of a BPS Wilson loop, matching an exact field theory computation, (iii) construct holographic duals to squashed Sasaki-Einstein backgrounds, again matching to a field theory computation, and (iv) find new analytic solutions to the squashed five-sphere background. We also analyse the classification of gravity duals with zero B-field.

La correspondance CFT (Anti-De Sitter) (AdS) / Théorie des champs conformes (CFT), également connue sous le nom de dualité holographique, constitue un lien remarquable entre la théorie des cordes (qui inclut la gravité) et les théories de jauge. Elle relie une CFT dans un espace-temps d-dimensionnel à une théorie de la gravité dans un espace-temps dimension supérieur, également appelé bloc. Ce dernier a une limite dans laquelle réside la théorie du champ conforme.Dans cette thèse, le sujet d'étude est la description holographique des flux de groupes de renormalisation (RG) des théories (de champ) sur les espaces-temps à symétrie maximale. Le cadre théorique que j'ai utilisé est la théorie d'Einstein-scalaire. L'inclusion du champ scalaire dynamique correspond à la rupture de l'invariance conforme aux limites. Dans ce travail, les limites et les tranches du bloc sont choisies pour être des espaces-temps à symétrie maximale et l'évolution des champs en bloc est étudiée. Il décrit les écoulements RG holographiques sur des variétés courbes. De plus, deux applications sont présentées dans cette thèse. La première application s'inscrit dans le contexte des théorèmes F et la seconde concerne un défaut incurvé dans les flux RG holographiques en masse.Les théorèmes F pour les théories de champs quantiques (QFT) définies dans des espaces-temps tridimensionnels exigent l'existence de fonctions dites F. Ce sont des fonctions décroissantes de façon monotone le long du flux RG. Dans ce travail, de nouvelles fonctions F pour les théories holographiques ont été découvertes. Elles sont construites à partir de l'action sur la parois d'une solution de flux holographique RG sur une sphère à 3-sphères. Ils permettent une interprétation entropique, fournissant ainsi un lien direct entre la formulation entropique du théorème F et sa définition en termes d’énergie libre.La deuxième application des flux RG holographiques explorée dans cette thèse se situe dans le contexte de modèles affichant un mécanisme d'auto-ajustement en tant que résolution proposée du problème de la constante cosmologique (CC). Dans ces modèles, notre univers à 4-dimensions est réalisé comme une brane intégrée dans un volume à 5-dimensions. Ce cadre permet des solutions où la géométrie de la brane est plate malgré la présence d'une énergie de vide non triviale sur son worldvolume. Ceci est appelé réglage automatique. De chaque côté de la brane, les solutions sont des flux RG holographiques. Le nouvel aspect introduit dans cette thèse consiste à utiliser les flux RG holographiques sur des variétés courbes, ce qui permet à son tour d’étudier des solutions à réglage automatique dans lesquelles la brane est également courbe
The Anti-de Sitter (AdS)/Conformal Field Theory (CFT) correspondence, also known as holographic duality, is a remarkable connection between string theory (which includes gravity) and gauge theories. It relates a CFT in a d-dimensional space-time to a gravity theory in higher dimensional space-time which is also referred to as the bulk. The latter has a boundary on which the conformal eld theory may be thought to reside. In this thesis, the subject of study is the holographic description of Renormalization Group (RG) fows of (field) theories on maximally symmetric space-times. The theoretical framework I used is Einstein-scalar theory. Inclusion of the dynamical scalar field corresponds to breaking boundary conformal invariance. In this work, both the boundary and bulk slices are chosen to be maximally symmetric space-times and the evolution of bulk fields is studied. It describes holographic RG flows on curved manifolds. Furthermore, two applications are presented in this thesis. The first application is in the context of F-theorems and the second is regarding a curved defect in the bulk holographic RG flows.F-theorems for Quantum Field Theories (QFT) defined on 3-dimensional space-times demand the existence of so-called F-functions. These are monotonically decreasing functions along the RG flow. In this work, new F-functions for holographic theories have been found which are constructed from the on-shell action of a holographic RG flow solution on a 3-sphere. They allow an entropic interpretation, therefore providing a direct connection between the entropic formulation of the F-theorem and its definition in terms of free energy. The second application of holographic RG flows explored in this thesis is in the context of models displaying a self-tuning mechanism as a proposed resolution of the cosmological constant (CC) problem. In these models, our 4-dimensional universe is realized as a brane embedded in a 5-dimensional bulk. This framework allows solutions where the brane geometry is flat despite of the presence of non-trivial vacuum energy on its worldvolume. This is referred to as self-tuning. On each side of the brane, the solutions are holographic RG flows. The new aspect introduced in this thesis is to use the holographic RG flows on curved manifolds, which in turn allows the study of self-tuning solutions where the brane is also curved

Hydrodynamics is an effective theory that is extremely successful in describing a wide range of physical phenomena in liquids, gases and plasmas. However, our understanding of the structure of the theory, its microscopic origins and its behaviour at strong coupling is far from complete. To understand how an effective theory of dissipative hydrodynamics could emerge from a closed microscopic system, we analyse the structure of effective Schwinger-Keldysh Closed-Time-Path theories. We use this structure and the action principle for open systems to derive the energy-momentum balance equation for a dissipative fluid from an effective CTP Goldstone action. Near hydrodynamical equilibrium, we construct the first-order dissipative stress-energy tensor and derive the Navier-Stokes equations. Shear viscosity is shown to vanish, while bulk viscosity and thermodynamical quantities are determined by the form of the effective action. The exploration of strongly interacting states of matter, particularly in the hydrodynamic regime, has been a major recent application of gauge/string duality. The strongly coupled theories involved are typically deformations of large-$N$ SUSY gauge theories with exotic matter that are unusual from a low-energy point of view. In order to better interpret holographic results, an understanding of the weak-coupling behaviour of such gauge theories is essential. We study the exact and SUSY-broken N=1 and N=2 super-QED with finite densities of electron number and R-charge, respectively. Despite the fact that fermionic fields couple to the chemical potentials, the strength of scalar-fermion interactions, fixed by SUSY, prevents a Fermi surface from forming. This is important for hydrodynamical excitations such as zero sound. Intriguingly, in the absence of a Fermi surface, the total charge need not be stored in the scalar condensates alone and fermions may contribute. Gauss-Bonnet gravity is a useful laboratory for non-perturbative studies of the higher derivative curvature effects on transport coefficients of conformal fluids with holographic duals. It was previously known that shear viscosity can be tuned to zero by adjusting the Gauss-Bonnet coupling, λ_GB, to its maximal critical value. To understand the behaviour of the fluid in this limit, we compute the second-order transport coefficients non-perturbatively in λ_GB and show that the fluid still produces entropy, while diffusion and sound attenuation are suppressed at all order in the hydrodynamic expansion. We also show that the theory violates a previously proposed universal relation between three of the second order transport coefficients. We further compute the only second-order coefficient thus far unknown, λ₂, in the N=4 super Yang-Mills theory with the leading-order 't Hooft coupling correction. Intriguingly, the universal relation is not violated by these leading-order perturbative corrections. Finally, by adding higher-derivative photon field terms to the action, we study charge diffusion and non-perturbative parameter regimes in which the charge diffusion constant vanishes.

Dans cette thèse, nous étudions les fonctions de corrélation à deux points ou plus de la théorie N = 4 super Yang-Mills (SYM) et ses déformations. On pense que cette théorie et ses déformations sont intégrables dans la limite planaire pour toutes les valeurs de la constante de couplage, ce qui nous permet de calculer plus efficacement diverses grandeurs physiques. Nous commençons par passer en revue les bases de la théorie N = 4 SYM et les outils d’intégrabilité nécessaires. Dans la partie II, nous présentons le formalisme en hexagone pour le calcul des constantes de structure de la théorie N = 4 SYM. En utilisant l’ansatz de Bethe imbriqué, nous étudions la constante de structure de deux opérateurs BPS et d’un opérateur non BPS du sous-secteur su (1, 1 | 2) dans les limites de couplage faible et forte. Le résultat de couplage faible est appliqué à l’étude du développement en ondes partielles conformes des fonctions asymptotiques à quatre points. A fort couplage, le résultat est appliqué pour le calcul des constantes de structure Heavy-Heavy-Light. Dans la partie III, nous nous concentrons sur une limite de double échelle particulière de la théorie N = 4 SYM, la limite de fishnet. Le lagrangien est obtenu à partir du processus de déformation et nous en présentons les théorèmes de non-renormalisation. En outre, nous donnons la première preuve de la symétrie du Yangien pour une grande classe de diagrammes de fishnet, en utilisant la fameuse relation RTT de l’intégrabilité. En utilisant les techniques d'intégrabilité héritées de la théorie N = 4 SYM, nous trouvons également que la limite du continuum des diagrammes de Feynman en fishnet peut être décrit comme un modèle sigma AdS
In this thesis, we study two- and higher-point correlation functions of the N = 4 super Yang-Mills theory (SYM) and its deformations. This theory and its deformations are believed to be integrable in the planar limit for any values of the coupling constant, which allow us to compute various physical quantities more efficiently. The basics of N = 4 SYM and the necessary integrability tools are reviewed in the very beginning. In part II, we review the hexagon formalism for computing structure constant in N = 4 SYM. Using the nested Bethe ansatz techniques, we investigate the structure constant of two BPS operators and one non-BPS operator in the su(1, 1|2) subsector in both weak and strong coupling limit. The weak coupling result is applied to the study of the conformal partial wave expansion of asymptotic four-point functions. At strong coupling, the result is applied to the Heavy-Heavy-Light structure constants. In part III, we focus on a particular double scaling limit of N = 4 SYM, namely the fishnet limit. The Lagrangian is obtained from the deformation process, and we present the non-renormalization theorems about it. Besides, we give the first principle proof of the Yangian symmetry for a large class of fishnet Feynman diagrams, using the famous RTT-formulation of integrability. Using the integrability techniques inherited from N = 4 SYM, we also find that the continuum limit of fishnet Feynman diagrams admit an AdS sigma model description

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.

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.

In this thesis we study various aspects of integrable methods in AdS3/CFT2 holog- raphy. These techniques come from exploiting special form of gauge compatible with asymptotic boundary conditions on the bulk fields in Chern Simons formulation of 3d gravity and higher spin theories. The integrable systems emerging are so called Toda field theories. The rigid structure of these theories allows us to learn more about certain gravitational solutions - multi centered solutions. One of the most interesting applications of the formalism developed is in the problem of black hole deconstruction and information paradox where we can read of the information about the temperature from the Toda field theory charges and. Aside of the gravitational applications we also study a vast generalization of the WZW to Toda field theory reduction and its connection to the Closed Toda systems. 1

We aim to have a comprehensive understanding of holographic correspondence and to demonstrate how the holographic correspondence (or renormalization) can be applied. Thus this thesis is divided into two parts. The first part is devoted to the former purpose (chapters 1 to 4 including appendix A,B, and C), and the second part is dedicated for the latter purpose (chapter 5 to 7). In Part I, the structure of the AdS/CFT correspondence is analyzed, and the properties of the AdS spacetime is studied in the context of the AdS/CFT correspondence; Here, we investigate the isometry group, the conformal structure, and generation of asymptotic solution near the conformal boundary. This solution yields significant convenience for the process of holographic renormalization. Moreover the properties of the Minkowski spacetime are compared to those of the AdS spacetime. To develop a greater understanding of the Lifshitz/quantum critical theory correspondence, the quantum phase transition is studied. Furthermore The holographic renormalization is briefly reviewed. In part II, the holographic renormalization associated with the Mann-Marolf (MM) counterterm is investigated for the asymptotically Minkowski spacetime in (n+3) dimensions. As a boundary condition, the cylindrical coordinate is considered. The solution of the MM-counterterm is obtained by solving the given algebraic equation, and from the counterterm solution, the boundary stress tensor is calculated. It is proven that the formula for conserved quantities via the boundary stress tensor holds. Next, we investigate deformations of Lifshitz holography with the Gauss-Bonnet term in (n+1) dimensional spacetime. To admit the non-trivial solution of the sub-leading orders, a value of the dynamical critical exponent z is restricted by z= n-1-2(n-2)α̃, where is the (redefined) Gauss-Bonnet coupling constant. Such solution of sub-leading orders correspond to the marginally relevant modes for the massive vector field and are generated by Λ~0, at the asymptotic region. A generic black hole solution, which is characterized by the horizon flux of the vector field and α̃, is considered in the bulk. We explore its thermodynamic properties, which depend on temperature, by varying n and α̃. As a result, the contribution of the marginally relevant mode is found in a function of Λ^z/T, and the relation between the free energy density and the energy density is numerically recovered when the marginally relevant mode is turned off (Λ=0), is obtained.

In this Ph.D. thesis, we construct and study a number of new type IIB supergravity backgrounds that realize various flavored, finite temperature, and non-supersymmetric deformations of the resolved and deformed conifold geometries. We make heavy use of a U-duality solution generating procedure that allows us to begin with a modification of a family of solutions describing the backreaction of D5 branes wrapped on the S^2 of the resolved conifold, and generate new backgrounds related to the Klebanov-Strassler background. We first construct finite temperature backgrounds which describe a configuration of N_c D5 branes wrapped on the S^2 of the resolved conifold, in the presence of N_f flavor brane sources and their backreaction i.e. N_f/N_c ~ 1. In these solutions the dilaton does not blow up at infinity but stabilizes to a finite value. The U-duality procedure is then applied to these solutions to generate new ones with D5 and D3 charge. The resulting backgrounds are a non-extremal deformation of the resolved deformed conifold with D3 and D5 sources. It is tempting to interpret these solutions as gravity duals of finite temperature field theories exhibiting phenomena such as Seiberg dualities, Higgsing and confinement. However, a first necessary step in this direction is to investigate their stability. We study the specific heat of these new flavored backgrounds and find that they are thermodynamically unstable. Our results on the stability also apply to other non-extremal backgrounds with Klebanov-Strassler asymptotics found in the literature. In the second half of this thesis, we apply the U-duality procedure to generate another class of solutions which are zero temperature, non-supersymmetric deformations of the baryonic branch of Klebanov-Strassler. We interpret these in the dual field theory by the addition of a small gaugino mass. Using a combination of numerical and analytical methods, we construct the backgrounds explicitly, and calculate various observables of the field theory.
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Gauge-string duality has provided a powerful framework for the study of strongly coupled gauge theories and non-perturbative string models. This thesis analyzes the holographic description of non-local gauge theory operators and some aspects of the Bagger-Lambert theory. The latter, as a proposal for a multiple M2-branes effective theory, is conjectured to be the holographic dual of a compactification of M-theory. We show that all half-BPS Wilson loop operators in N=4 SYM - which are labeled by Young tableaus - have a gravitational dual description in terms of D5-branes or alternatively in terms of D3-branes in AdS5xS5. We prove that the insertion of a half-BPS Wilson loop operator in the N=4 SYM path integral is achieved by integrating out the degrees of freedom on a configuration of bulk D5-branes or alternatively on a configuration of bulk D3-branes. We construct a new class of supersymmetric surface operators in N=4 SYM and find the corresponding dual supergravity solutions. Consistency requires constructing N=4 SYM in the D7 supergravity background and not in flat space. This enlarges the class of holographic gauge theories dual to string theory backgrounds to gauge theories in non-trivial supergravity backgrounds. We write down a maximally supersymmetric one parameter deformation of the field theory action of Bagger and Lambert and we show that this theory on RxT2 is invariant under the superalgebra of the maximally supersymmetric Type IIB plane wave. It is argued that this theory holographically describes the Type IIB plane wave in the discrete light-cone quantization (DLCQ). Finally, we show by explicit computation that the Bagger-Lambert Lagrangian realizes the M2-brane superalgebra, including also two p-form central charges that encode the M-theory intersections involving M2-branes.

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.

Στη γραμμική εκδοχή της γενικής θεωρίας της σχετικότητας, θεωρεί κανείς τις διαταραχές κάποιας μετρικής γύρω από κάποιο χωροχρονικό υπόβαθρο. Κρατώντας όρους διαταραχών μέχρι και πρώτης τάξεως, οδηγείται κανείς στις γραμμικές εξισώσεις Einstein. Σε αυτό το πλαίσιο αποδεικνύεται μια σχέση δυισμού ανάμεσα στα διάφορα στοιχεία του τανυστή Weyl, αντίστοιχη με το δυισμό ανάμεσα στην ηλεκτρική και τη μαγνητική ροή της ηλεκτρομαγνητικής θεωρίας του Maxwell. Στην εργασία αυτή κάνουμε μία ανασκόπηση της έρευνας που έχει γίνει μέχρι τώρα αναφορικά με αυτές τις σχέσεις δυισμού. Πιο συγκεκριμένα, εξετάζουμε την ισχύ των σχέσεων στον Anti-de Sitter χωρόχρονο και επισημαίνουμε το τρόπο με τον οποίο κατασκευάζει κανείς δυικές δομές από τις εκφράσεις για τις διαταραχές. Επίσης, χρησιμοποιώντας τη τεχνική της ολογραφικής επανακανονικοποίησης, εξετάζουμε το σύμμορφο σύνορο του χωροχρόνου. Βρίσκουμε εκεί μια σχέση δυισμού ανάμεσα στα στοιχεία του τανυστή ενέργειας-ορμής και του τανυστή Cotton της αντίστοιχης Chern - Simons θεωρίας, η οποία αποδεικνύεται ότι είναι άμεση συνέπεια του δυισμού στο AdS υπόβαθρο. Τέλος, εφαρμόζουμε την ίδια συλλογιστική στο Schwarzschild - Anti-de Sitter υπόβαθρο, όπου η παρουσία της μελανής οπής διαφοροποιεί τις συνοριακές συνθήκες του προβλήματος. Λόγω αυτού του γεγονός δεν μπορεί να πει κανείς με σιγουριά εάν μπορούν να διατυπωθούν σχέσεις δυισμού σε αυτή τη περίπτωση. Παρόλα αυτά βρίσκουμε ότι ισχύουν σχέσεις δυισμού στο σύμμορφο σύνορο παρόμοιες με αυτές του AdS υποβάθρου, πράγμα που σημαίνει ότι στο σύστημα παραμένει κάποια συμμετρία από τη γραμμική θεωρία. Η εργασία καταλήγει σε σχόλια και μία εκτενή συζήτηση για τις πιθανές μελλοντικές κατευθύνσεις.
In the linear version of the general theory of relativity, one considers metric perturbations around a fixed background. Keeping terms up to first order of perturbation leads to the linearized Einstein equations. In this framework it has been proved that a duality between the various elements of the Weyl tensor holds. This duality is similar to the one between the electric and magnetic fluxes of Maxwell's electromagnetism. In the present work we review the status of these dualities for non trivial backgrounds. We examine the Anti-de Sitter background, where we point out the way to explicitly construct dual configurations using the metric perturbation expressions. Using the holographic renormalization technique, we examine the conformal boundary where a duality between the components of the energy-momentum tensor and the Cotton tensor of the corresponding Chern - Simons theory holds. It is then proved that this duality is a direct consequence of the electric/magnetic duality in the bulk, in the case of the AdS background. Finally, we apply same procedure to the Schwarzschild - Anti-de Sitter background, where the presence of the black hole changes the boundary conditions of the problem. This simple fact makes it impossible say whether such a duality exists in this case. Nevertheless, we find that a duality similar to that of the AdS background still holds for the conformal boundary, which means that there is a remnant of symmetry from the linear theory. We conclude with comments and a extensive discussion on possible future directions.

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.