Transgression Forms and Abelian Semigroups in Supergravity

Abstract

Two main themes populate this Thesis's pages: transgression forms as Lagrangians for gauge theories and the Abelian semigroup expansion of Lie algebras. A transgression form is a function of two gauge connections whose main property is its full invariance under gauge transformations. From this form a Lagrangian is built, and equations of motion, boundary conditions and associated Noether currents are derived. A subspace separation method, based on the extended Cartan homotopy formula, is proposed, which allows to (i) split the Lagrangian in 'bulk' and 'boundary' contributions and (ii) separate the bulk term in sublagrangians corresponding to the subspaces of the gauge algebra. Use is made of Abelian semigroups to develop an expansion method for Lie (super)algebras, based on the work by de Azcarraga, Izquierdo, Picon and Varela. The main idea consists in considering the direct product between an Abelian semigroup S and a Lie (super)algebra g. General conditions under which smaller algebras can be extracted from S \otimes g are given. It is shown how to recover the known expansion cases in this new context. Several d=11 superalgebras are obtained as examples of the application of the method. General theorems that allow to find an invariant tensor for the expanded algebra from an invariant tensor for the original algebra are formulated. Finally, a d=11 gauge theory for the M Algebra is considered by using the ideas developed in the Thesis. The dynamical properties of this theory are briefly analyzed.