The work is divided into three main parts. In a first time (chapter 1) we define a “BTZ” black hole in anti de Sitter space in any dimension. That will be done by means of group theoretical and symmetric spaces considerations. A physical “good domain” is identified as an open orbit of a subgroup of the isometry group of anti de Sitter. Then (chapter 2) we show that the open orbit is in fact isomorphic to a group (we introduce the notion of globally group type manifold) for which a quantization exists. The quantization of the black hole is performed and its Dirac operator is computed. The third part (appendix A and B) exposes some previously known results. Appendix A is given in a pedagogical purpose: it exposes generalities about deformation quantization and careful examples with SL(2,R), and split extensions of Heisenberg algebras. Appendix B is devoted to some classical results about homogeneous spaces and Iwasawa decompositions. Explicit decompositions are given for every algebra that will be used in the thesis. It serves to make the whole text more self contained and to fix notations. Basics of quantization by group action are given in appendix A.4. One more chapter is inserted (chapter 3). It contains two small results which have no true interest by themselves but which raise questions and call for further development. We discuss a product on the half-plane or, equivalently, on the Iwasawa subgroup of SL(2,R), due to A. Unterberger. We show that the quantization by group action machinery can be applied to this product in order to deform the dual of the Lie algebra of that Iwasawa subgroup. Although this result seems promising, we show by two examples that the product is not universal in the sense that even the product of compactly supported functions cannot be defined on AdS2 by the quantization induced by Unterberger's product. Then we show that the Iwasawa subgroup of SO(2,n) (i.e. the group which defines the singularity) is a symplectic split extension of the Iwasawa subgroup of SU(1,1) by the Iwasawa subgroup of SU(1,n). A quantization of the two latter groups being known, a quantization of SO(2,n) is in principle possible using an extension lemma. Properties of this product and the resulting quantization of AdSl were not investigated because we found a more economical way to quantize AdS4 .

In this thesis we study the time evolution of Rényi and entanglement entropies of thermal states in Conformal Field Theory (CFT). These quantities are usually hard to compute but Ryu-Takayanagi (RT) and Hubeny-Rangamani-Takayanagi (HRT) proposals allow us to find the same quantities using calculations in general relativity. We will introduce main concepts of holography, quantum information and conformal field theory that will be used to derive the results of this thesis. In the first part of the thesis, we explicitly compute entanglement entropy of the rotating BTZ black hole by directly applying HRT proposal and finding lengths of spacelike geodesics. Rényi entropy of thermal state perturbed by a local quantum quench is computed by mapping correlators on two glued cylinders to the plane for field theory containing a single free boson and for 2d CFTs in the large c limit. We consider Thermofield Double State (TFD) which is an entangled state in direct product of two 2D CFTs. It is conjectured to be holographically equivalent to the eternal BTZ black hole. TFD state is perturbed by a local quench in one CFT and mutual information between two intervals in two CFTs is computed. We find when mutual information vanishes and interpret this as scrambling time, i.e. time scale required for the system to thermalize. This field theory result is modelled with a massive free falling particle in the BTZ black hole. We have computed the back-reaction of the particle on the metric of BTZ and used RT proposal to find holographic entanglement entropy. Finally, we generalize this calculation to the case of rotating BTZ with inner and outer horizons. It is dual to the CFT with different temperatures for left and right moving modes. We calculate mutual information and scrambling time and find exact agreement between results in the gravity and those in the CFT.