Supersymmetric AdS Solutions in Supergravity and G-Structures

In this work we study supersymmetric solutions of supergravity theories with particular focus on the AdS/CFT correspondence. Since the 2000s remarkable progress has been made in the classification of these solutions, using constraints arising from the geometry underlying the theory. The link between differential geometry and supersymmetric solutions is evident from the simplest situations i.e. the bosonic solutions with vanishing fluxes. Here, a supersymmetric solutions is specified by a metric with special holonomy. In the seminal work by Martelli et al. the geometry of G-structures allowed the discovery of a non-vanishing fluxes generalization of this fact. The proposed method, relying on the fact that the Killing spinor ϵ defines a G-structure by means of its isotropy group, is known to provide tipically the most general local form of the supersymmetric solution of a given supergravity theory. Extremely interesting solutions to consider in this context are supersymmetric solutions with an AdS factor, the most relevant in order to investigate the AdS/CFT conjecture formulated by Maldacena in 1997. In five-dimensional minimal gauged supergravity recent results have been obtained by Ferrero et al.. A different class of geometrical objects is involved in these works. In particular the proposed solution is a Lorentzian variety, obtained as a product of an AdS_{3} factor and a two-dimensional orbifold known as spindle (Σ). This solution is belived to be dual to a two-dimensional superconformal field theory that is interpreted as the IR limit of an RG flow arising from the wrapping process of D3 branes on the spindle. What is still lacking is an interpolating supergravity solution between AdS_{3} x Σ and the corresponding AlAdS_{5} space in the UV. This fact is the motivation of the whole thesis work. In the first part, we review the theoretical background of the research: fundamental aspects of classical field theories, supersymmetry, supergravity and its links with superstring theory and M-theory. Then, we conclude this section reproducing the computations of some notable supersymmetric solutions of supergravity theories: D3 branes and the more general AdS_{3} x M_{7} supersymmetric solutions are some examples. The main emphasis of this section is on algebraic and geometrical aspects of these concepts. In the second part, we study both established and new aspects of the recent spindle solution proposed by Ferrero et al.. We present for the first time the manifestly supersymmetric form of the solution, accordingly to the classification of Gauntlett and Gutowski. After this classification, we move in the direction of an interpolating solution between AdS_{3} x Σ and AlAdS5. We find a master equation (a PDE) that hopefully can encode the interpolation, and we try some higly non-trivial ansatz. We restrict the space of allowed functional forms of the ansatz, formulating a conjecture - verified for both trivial and non trivial situations - that forbids polynomial behaviour of the solution to the master equation. We finally try to rationalize these results and we set the ground for future investigations. ​