Soutenance de Bastien Duboeuf

String Theories are our most promising candidates in the goal of unifying all interactions in nature. Gravity is naturally encapsulated in all String Theories, which are ultraviolet-free from divergences. A general feature of String Theories is the number of spacetime dimensions required. This number is either 26 or 10. In any case, this requires some further modifications of the theories since the world we are living in is a priori a 4-dimensional spacetime space. One way to tackle this problem is via so-called Kaluza-Klein compactifications. The latter is a process in which some of the dimensions of the total space are made compact and small. The idea is that at our macroscopic scale and daily energy level, the effects of those "extra" dimensions are sufficiently small so we can ignore them.

In this thesis we will be interested in Supergravities, obtained from a certain regime of String Theories. In those theories, additional symmetries show up in dimensional compactifications, which allow us to reformulate Supergravities into so-called Exceptional Field Theories. Let us finally remark that Supergravities are particularly important in the context of the AdS/CFT correspondence, stating the equivalence between a theory of gravity, here a Supergravity defined on an Anti-de-Sitter (AdS) background, and a Quantum Field Theory, here a Conformal Field Theory living on the boundary of the AdS previous spacetime.

In the first part of the thesis, we will make general comments and use it to introduce various concepts needed throughout the thesis.

In the second part of the thesis, we will show how to use Exceptional Field Theory tech- niques to compute spectra of solutions of 11-dimensional Supergravity on AdS4 × Σ7 . After reviewing the state-of-the-art techniques for Kaluza-Klein spectroscopy, we will show how Exceptional Field Theory techniques can be extended to spaces with the condition of Generalized parallelizability. We will illustrate how this technology works in the case of AdS4 × S7squashed solution of 11- dimensional Supergravity, for which we give a complete answer for the spectrum. We will then show a concrete realization of the AdS4/CFT3 correspondence, by computing the domain-wall solution of the Supergravity equations interpolating between the round and the squashed seven- sphere, which is dual to a Renormalization Group flow on the Conformal Field Theory side. This demonstrates that not only Exceptional Field Theory techniques can be used to compute spectra around Supergravities with AdS background, but it also allows us to compute quadratic couplings of Kaluza-Klein fluctuations around a domain-wall solution of Supergravity.

Finally, in the last section of the thesis, we will be interested in n-point couplings in Super- gravities. In a first part, we will show how to compute cubic couplings for specific fields on AdS5 background of IIB 10-dimensional Supergravity via a brute force calculation. After discussing the achievements made using these techniques, we will show how we can use Exceptional Field Theory techniques to efficiently compute these couplings. Not only will this prove more efficient, but it will also allow us to write the same couplings in a more compact form, to derive formulas that apply to any vacua that are Leibniz parallelizable, and also to reveal Exceptional Field Theory structures that prove long-standing conjectures. We will illustrate the power of these techniques on the example of AdS5 background of IIB 10-dimensional Supergravity.