S-Expansion with Ideal Subtraction and Solutions in Extended Supergravities

This thesis is about a new methd to perform the S-expansion procedure and studies in extended supergravites. In the present work, the concept of zero-reduction has been extended reproducing a generalized Inönü-Wigner contraction. This involves an innite abelian semigroup and the removal of an ideal subalgebra. We refer about the use of theorem VII.2 of the reference [20] which serves to construct the topological invariant of a respective S-expanded algebra from another of which the bilinear form is known, therefore is an extension and generalization of the mentioned theorem. This procedure allows to develop the dynamics and construct the Lagrangians of several theories. This work reproduces the results already presented in the literature, concerning the generalized Inönü-Wigner contraction, and also gives some new features. Moreover, it gives a connection between the contraction processes and the expansion methods introduced in [17], which was an open question already mentioned in [12]. Also is shows one of the interesting applications, which is to obtain a particular hidden Maxwell superalgebra underlying supergravity in four dimensions. Thus we have written the hidden Maxwell superalgebra in the Maurer-Cartan formalism, and then, we have considered the parametrization of the 3-form A(3) in terms of 1-forms, in order to show the way in which the trivial boundary contribution in four dimensions, dA(3), can be naturally extended by considering particular contributions to the structure of the extra fermionic generator, appearing in the hidden Maxwell superalgebra. These extensions contain terms which involve the cosmological constant. Interestingly, the presence of these terms depends strictly on the form of the extra fermionic generators appearing in this hidden extension of D = 4 supergravity. Besides, we have reviewed some concepts of the S-expansion. We show especially, how S-expansion procedure aects the geometry of a Lie group: was found how the magnitude of a vector change and the angle between two vectors. About the kind of algebra, after apply an S-expansion, it is a non-simple Lie algebra. Then, considering resonance and reduction, we built an analytic method able to give us the multiplication table(s) of the set(s) involved in an Sexpansion process for reaching a target Lie (super)algebra from a starting one, after having chosen properly the partitions over subspaces of the considered (super)algebras. Furthermore, we study in the context of ungauged supergravities, the symmetry under the kinetic part in the action and the realization to a global symmetry group G. Moreover, we show the use of the solution generating technique. As a rst step in the unnished research we work with the AdS metric with the global group G2(2),which it is the global symmetry group of the 3D description of N = 2 supergravity coupled to a vector multiplet [13], where we found the charges. Considering that a solution with cosmological constant is always gauged in extended supergravities [36], it is planned to investigate the applicability of the solution generating technique on gauged theories asymptotically at, i.e AdS metric.