Localizzazione supersimmetrica e olografia

The study of strongly coupled gauge quantum field theories is a very rich field of research. One of the most interesting approaches to this subject is the AdS/CFT duality. In its weak form, this correspondence relates some operators of a strongly coupled gauge theory to the classical action of a supergravity theory defined on a space which is asymptotically anti de Sitter. Another extremely powerful technique allowing the investigation of strongly coupled regimes is supersymmetric localization. This approach relies on the phenomenon of equivariant localization, that is a generalization of the saddle point method. Namely, supersymmetric localization relies on supersymmetry to prove that the path integral of certain observables only depends on the contributions of the fixed points of the supersymmetry. Usually the localization approach is implemented in calculations of the partition function and the vev of some protected operators of extended superconformal field theories defined on curved manifolds of various dimensions. Following a recent approach to localization, in this thesis we investigate this phenomenon applied to the gravity theory side of the AdS/CFT correspondence. Namely we are interested in localizing the action of a N=2 hypermultiplet defined on AdS5 to an effective action on a segment AdS1. In order to do so we study a map that relates a localizing supercharge on the gauge theory side of the correspondence to a suitable Killing spinor in its holographic dual. Then we focus on a bottom-up approach for the formulation of the Lagrangian and supersymmetry variations of a N=2 hypermultiplet in AdS5. Since we are interested in localization of the action, hence to the level of equation of motion, we are interested in both on-shell and off-shell formulations of the said hypermultiplet. If for the on-shell case we could find an example in literature, we could not do the same for the off-shell case. Thus we suggested a possible way to define the Lagrangian and the supersymmetry variations of an off-shell N=2 hypermultiplet on AdS_5 as the analytical continuation of the formulae defined for the five-sphere.