# More on the Hidden Symmetries of 11D Supergravity

## TL;DR

This paper explores the complex algebraic structures underlying 11D supergravity, clarifying the relations among various superalgebras and their role in the theory's 4-form cohomology, revealing new insights into its hidden symmetries.

## Contribution

It identifies how the 3-form in 11D supergravity decomposes into parts with distinct group-theoretical meanings, linking the FDA to hidden superalgebras and cohomology.

## Key findings

- Decomposition of the 3-form into two parts with different algebraic roles
- Identification of a closed 3-form defining a family of invariant trilinear forms
- Clarification of the role of the nilpotent spinor generator in 4-form cohomology

## Abstract

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 Fr\'e 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 related to osp(1|32) by redefining the spin connection and adding a new Maurer-Cartan equation. 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.