Supergravity in the Geometric Approach and its Hidden Graded Lie Algebra

Abstract

In this contribution, we present the geometric approach to supergravity. In the first part, we discuss in some detail the peculiarities of the approach and apply the formalism to the case of pure supergravity in four space-time dimensions. In the second part, we extend the discussion to theories in higher dimensions, which include antisymmetric tensors of degree higher than one, focussing on the case of eleven dimensional space-time. Here, we report the formulation first introduced in 1981 by R. D'Auria and P. Fr\`e, corresponding to a generalization of a Chevalley-Eilenberg Lie algebra, together with some more recent results, pointing out the relation of the formalism with the mathematical framework of $L_\infty$ algebras.