Radiative corrections to the vacua of broken extended supergravity

When effective theories in String Theory are considered, only tree-level analysis of mechanisms of vacuum selection and supersymmetry breaking is taken into account. It is important, instead, to consider quantum corrections to the potential in order to understand if vacua are stable or not, especially if the vacuum breaks completely the supersymmetry. A recent analysis of the maximally supersymmetric theory in four dimensions has shown that one-loop corrections of any supergravity model, which admits a Minkowski vacuum that breaks supersymmetry completely, are finite. All the analysed models exhibit negative corrections which correspond to classically stable Anti-de Sitter vacua. Following the Weak Gravity Conjecture, this should lead to the non-perturbative instability, which undermines the existence of such vacua. Therefore, it is interesting to reanalyse these topics in theories with less supersymmetries in order to find if these results and these problems remain. The thesis is about the analysis of one-loop corrections in N=4 supergravity theories with arbitrary gaugings and different couplings of matter.