In this thesis we present solutions of spontaneous compactifications of D=11, N=1 supergravity on parallelizable manifolds S^1, S^3 and S^7. In Freund-Rubin compactifications one usually obtains AdS vacua in 4D, these solutions usually sets the fermionic VEV's to zero. However giving them non zero VEV's allows us to define torsion given by the fermionic bilinears that essentially flattens the geometry giving us a vanishing cosmological constant on M_4. We further give an analysis of the consistent truncation of the bosonic sector of D=11 supergravity on a S^3 manifold and relate this to other known consistent truncation compactifications. We also consider the squashed S^7 where we check for surviving supersymmetries by analyzing the generalised holonomy, this compactification is of interest in phenomenology.

A complete non trivial supersymmetric deformation of the maximal supergravity in two dimensions is achieved by the gauging of a SO(9) group. The resulting theory describes the reduction of type IIA supergravity on an AdS_2 x S^8 background and is of first importance in the Domain-Wall / Quantum Field theory correspondence for the D0-brane case. To prepare the construction of the SO(9) gauged maximal supergravity, we focus on the eleven dimensional supergravity and the maximal supergravity in three dimensions since they give rise to important off-shell inequivalent formulations of the ungauged theory in two dimensions. The embedding tensor formalism is presented, allowing for a general desciption of the gaugings consistent with supersymmetry. The SO(9) supergravity is explicitly constructed and applications are considered. In particular, an embedding of the bosonic sector of the two-dimensional theory into type IIA supergravity is obtained. Hence, the Cartan truncation of the SO(9) supergravity is proved to be consistent. This motivated holographic applications. Therefore, correlation functions for operators in dual Matrix models are derived from the study of gravity side excitations around half BPS backgrounds. These results are fully discussed and outlooks are presented.

Nous avons réalisé une déformation non-triviale et complète de la théorie de supergravité maximale en dimension deux. Il s'agit de la supergravité maximale avec groupe de jauge SO(9). Cette théorie décrit de manière effective la supergravité de type IIA sur un espace-temps produit AdS_2 x S^8. Elle joue ainsi un rôle important dans la correspondance Gravité / Théorie de Jauge appliquée au cas de la D0-brane. Afin de préparer la construction de la supergravité maximale jaugée SO(9), nous nous intéressons aux supergravités maximales en dimension onze et trois, puisqu'elles donnent lieu à différentes formulations non équivalentes de la théorie bidimensionnelle non jaugée. Le formalisme d' « Embedding tensor » est ensuite présenté. Il permet de déterminer l'ensemble des groupes de jauges compatibles avec la supersymétrie maximale. La supergravité SO(9) est dès lors explicitement construite et ouvre la voie à deux applications importantes. P our commencer, nous avons réalisé l'inclusion d'un sous-secteur bosonique de la théorie SO(9), la troncature de Cartan, dans la supergravité de type IIA à dix dimensions d'espace-temps. Il s'agit d'une inclusion cohérente. Cela a motivé la deuxième application, de nature holographique. Ainsi, à partir du sous-secteur de Cartan de la supergravité SO(9), et en particulier de la découverte d'états fondamentaux de type « half-BPS », nous avons calculé un ensemble de fonctions de corrélation à un et deux points associées à des opérateurs de modèles de matrice duaux. Nous avons conclu en un résumé de nos travaux et en la présentation d'intéressantes perspectives. / A complete non trivial supersymmetric deformation of the maximal supergravity in two dimensions is achieved by the gauging of a SO(9) group. The resulting theory describes the reduction of type IIA supergravity on an AdS_2 x S^8 background and is of first importance in the Domain-Wall / Quantum Field theory correspondence for the D0-brane case. To prepare the construction of the SO(9) gauged maximal supergravity, we focus on the eleven dimensional supergravity and the maximal supergravity in three dimensions since they give rise to important off-shell inequivalent formulations of the ungauged theory in two dimensions. The embedding tensor formalism is presented, allowing for a general desciption of the gaugings consistent with supersymmetry. The SO(9) supergravity is explicitly constructed and applications are considered. In particular, an embedding of the bosonic sector of the two-dimensional theory into type IIA supergravity is obtained. Hence, the Cartan truncation of the SO(9) supergravity is proved to be consistent. This motivated holographic applications. Therefore, correlation functions for operators in dual Matrix models are derived from the study of gravity side excitations around half BPS backgrounds. These results are fully discussed and outlooks are presented.