More on the hidden symmetries of 11D supergravity

In this paper we clarify the relations occurring among the osp(1|32) algebra, the M-algebra and the hidden superalgebra underlying the Free Differential Algebra of D=11 supergravity (to which we will refer as DF-algebra), that was introduced in the literature by D'Auria and Frè in 1981 and is actually a (Lorentz valued) central extension of the M-algebra including a nilpotent spinor generator, Q'. We focus in particular on the 4-form cohomology in 11D superspace of the supergravity theory, strictly related to the presence in the theory of a 3-form A^(3). Once formulated in terms of its hidden superalgebra of 1-forms, we find that A^(3) can be decomposed into the sum of two parts having different group-theoretical meaning: One of them allows to reproduce the FDA of the 11D Supergravity due to non trivial contributions to the 4-form cohomology in superspace, while the second one does not contribute to the 4-form cohomology, being a closed 3-form in the vacuum, defining however a one parameter family of trilinear forms invariant under a symmetry algebra strictly related to osp(1|32). We further discuss about the crucial role played by the 1-form spinor eta (dual to the nilpotent generator Q') for the 4-form cohomology of the eleven dimensional theory on superspace.