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.

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.

L’objet de cette thèse est l’étude de la correspondance fluide/gravité, réalisation macroscopique de la dualité AdS/CFT, à la limite où la constante cosmologique tend vers zéro (limite plate). La jauge de Fefferman-Graham, habituellement utilisée dans le dictionnaire holographique, est singulière à la limite plate. Ici, en passant par la formulation hydrodynamique de la théorie dubord, nous construisons une jauge, appelée jauge du développement en série dérivatif, où cette limite est bien définie. Sur la géométrie du bord, elle correspond en fait à faire tendre vers zéro la vitesse de la lumière, situation connue comme limite carrollienne. Un fluide relativiste admet une telle lim-ite, qui donne lieu à l’hydrodynamique carrollienne, étudiée ici en dimension arbitraire, parallèlement à son homologue galiléen. Ensuite, nous montrons spécifiquement en dimensions 4 et 3 du bulk qu’il est possible de construire des solutions asymptotiquement plates des équations d’Einstein partant de systèmes hydrodynamiques conformes carrolliens du bord, qui est ici l’hypersurface degenre lumière à l’infini. En 4 dimensions nous introduisons des conditions d’intégrabilité permettant de resommer la série dérivative sous formefermée. En 3 dimensions toute configuration fluide du bord aboutit à une solution exacte des équations d’Einstein. Les solutions de Bañados sont un sous-ensemble des solutions obtenues et identifiées au moyen de leurs charges de surface. Nous accordons une attention particulière au rôle du repère hydrodynamique, trop souvent ignoré en holographie. Pour terminer, nous nous concentrons sur la formulation de la AdS/CFT dans laquelle la symétrie de Weyl est explicite. Quoique cette symétrie soit un ingrédient incontournable de la correspondance fluide/gravité, elle n’est pas codée dans la formulation habituelle de l’holographie. Nous introduisons une nouvelle jauge et analysonsses conséquences. / In this thesis we discuss the limit of vanishing cosmological constant (flat limit) of the fluid/gravity correspondence, which is a macroscopic realization of the AdS/CFT. The holographic dictionary is usually implemented in a gauge(Fefferman-Graham), which does not admit a flat limit. In the hydrodynamic formulation of the boundary theory, we introduce a gauge, dubbed derivative expansion, where such a limit turns out to be smooth. In the boundary we show that this corresponds to a Carrollian limit, i.e. a limit where the speed of light vanishes. We present Carrollian hydrodynamics, together with its dual Galilean counterpart. Then, for 4 and 3 bulk dimensions, we exhibit a resummed line element, which provides an asymptotically flat bulk solution of Einsteinequations starting only from boundary (i.e. null infinity) conformal Carrollian hydrodynamic data. In 4 dimensions we exploit specific integrability conditions, which restrict the achievable class of solutions in the bulk. In 3 dimensions every boundary fluid configuration leads to an exact solution of Einstein’s equations. Bañados solutions are a subset of the solutions reached in this way. They are rigorously identified with their surface charges and the corresponding algebra. We emphasize the choice of hydrodynamic frame, often sidesteppedin holography. Finally, we focus on the formulation of AdS/CFT to encompass Weyl symmetry. This symmetry is a key ingredient of fluid/gravity but it is not naturally encoded in the usual formulation of holography. We introduce an appropriate gauge for realizing it, and analyze its far-reaching consequences.

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