Geometric pure and matter coupled supergravity theories

Abstract

This thesis deals with the formulation of pure and matter coupled supergravity theories in three and four dimensions. Different supergravity Lagrangians are constructed in geometrical terms, by using the useful properties of the abelian semigroup expansion method. Furthermore, a supergravity model with partial breaking of N = 2 to N = 1 supersymmetry which, in the low energy limit, gives rise to a rigid supersymmetric theory, is presented. In Chapter 1, we briefly review General Relativity in both, Einstein and Cartan formalism. It is also revised a natural extension of Einstein theory to D-dimensions, namely the Lanczos-Lovelock theory. Then, we study the Maxwell type algebras, and we show that standard General Relativity can be obtained in a certain limit of Chern-Simons and Born-Infeld theories, invariant under these algebras. Chapter 2 deals with the supersymmetric extension of gravity. We mainly study the MacDowell-Mansouri supergravity and the AdS Chern-Simons supergravity. In Chapters 3, 4, 5, 6 and 7, we present our main results, which are based on five articles written during the doctoral research. First, we present supersymmetric extensions of the Maxwell type algebras. We show that considering different choices of semigroups, inequivalent Maxwell superalgebras are obtained, when using the S-expansion procedure. Then, we construct the N = 1 supergravity action a la MacDowell-Mansouri from the minimal Maxwell superalgebra. Interestingly, the action describes pure supergravity. Based on the AdS-Lorentz superalgebra, we also build the minimal D = 4 supergravity action which includes a generalized supersymmetric cosmological constant term. The construction of the Chern-Simons supergravity action from a generalized minimal Maxwell superalgebra is also presented. Eventually, in Chapter 7 we present the multi-vector generalization of a rigid, partially broken N = 2 supersymmetric theory as a rigid limit of a gauged N = 2 supergravity with electric and magnetic charges