Doppia riduzione dimensionale alla Scherk-Schwarz: nuovi vuoti supersimmetrici e gauging dionici

Gauged supergravities are the most promising framework to find the effective theory of the superstring theories and the M-theory. Being the theory corre- spondence well known only in higher spacetime dimensions these theories look quite far from realistic ones. According to this, the N = 8, D = 4 supergrav- ities gain a specific relevance to overcome the problem. The main benefit of considering these models is that, via the Exceptional Field Theory framework, certain lower-dimensional models can be directly embedded in superstring the- ory. In this sense, the study of supergravity looks like the most convenient way to discover new string vacua. This thesis analyzes and develops an idea first proposed by deWit, Samtleben, and Trigiante: the double Scherk-Schwarz reduction. This allows obtaining D = 4 gauged supergravity with mass deformations and Minkowski vacua fea- turing spontaneous supersymmetry breaking. This new kind of gauging needs to be completed and deepened, and we do it through the software Wolfram Math- ematica, an essential tool to carry out the huge calculations we need to deal with in the maximal theory. Initially, we identify the generators of the algebra of G = SO(5, 5)×SL(2, R)× SO(1, 1) (the group characterizing the new frame) as a subgroup of E7(7) (the global symmetry group in four dimensions). The construction of the embed- ding tensor Θ strictly depends on the quadratic constraint analysis, the only independent constraint in the maximal theory; in the present case, this re- veals a new stringent condition on the choice of the two generators leading the compactification: [T1, T2] = 0. The core of the code provides the explicit con- struction of Θ combining the Scherk-Schwarz ansatz and the X-tensor definition X MNK = Θα M (tα)KN and tests both linear and quadratic constraints for these ob- jects. It turns out that choosing both generators compact the theory exhibits a supersymmetric vacuum at the origin of the scalar manifold Mscal. This al- lows extending the original work of deWit, Samtleben, and Trigiante with the calculation of the (bosonic) mass spectrum of the model. These masses are ex- plicitly calculated as functions of the R-symmetry weights, the SO(5) × SO(5) weights, and the Higgs mechanism, giving mass to the 24 vectors, is verified in the pure electric case. A general formula for the squared masses in the double Scherk-Schwarz reduction is given, while a proper multiple reduction version is proposed. The final section is dedicated to exploring a new idea: the dyonic gaugings from the double Scherk-Schwarz reduction. Here Θ can exhibit non-vanishing components also in the dual magnetic representation (45, 2)+2 of the pure elec- tric case (45, 2)−2. We start over from the quadratic constraint analysis and a new consistency condition arises: the product of the spinorial representation of the electric generator T1, and the magnetic one T’ 2 must vanish. Solving in detail such a constraint, the consistency of Θ is again verified and even in this context a compact-compact choice reveals a Minkowski vacuum at the origin of Mscal. Here again, the mass spectrum is calculated and a brief comparison with the previous case is done: the masses only depend on 2 free parameters in contrast to the 8 observed in the pure electric case. This loss of ”freedom” suggests these frames can’t be connected by an E7(7) transformation, i.e. it seems we are dealing with a new independent gauging.