The aim of this paper is to clarify the role of the nilpotent fermionic generator Q^′ introduced in [6] and appearing in the hidden supergroup underlying the free differential algebra (FDA) of D=11 supergravity. We give a physical explanation of its role by looking at the gauge properties of the theory. We find that its presence is necessary, in order that the extra 1-forms of the hidden supergroup give rise to the correct gauge transformations of the p-forms of the FDA. This interpretation is actually valid for any supergravity containing antisymmetric tensor fields, and any supersymmetric FDA can always be traded for a hidden Lie superalgebra containing extra fermionic nilpotent generators. As an interesting example we construct the hidden superalgebra associated with the FDA of \( \mathcal{N} \) = 2, D = 7 supergravity. In this case we are able to parametrize the mutually non local 2- and 3-form B⁽²⁾ and B⁽³⁾ in terms of hidden 1-forms and find that supersymmetry and gauge invariance require in general the presence of two nilpotent fermionic generators in the hidden algebra. We propose that our approach, where all the invariances of the FDA are expressed as Lie derivatives of the p-forms in the hidden supergroup manifold, could be an appropriate framework to discuss theories defined in enlarged versions of superspace recently considered in the literature, such us double field theory and its generalizations.

The purpose of this paper is to show that the so-called Maxwell superalgebra in four dimensions, which naturally involves the presence of a nilpotent fermionic generator, can be interpreted as a...

We study a metric-affine gravitational theory given by the Einstein–Hilbert (EH) action plus a parity violating contribution (which we will refer to as the Hojman term, also known as Holst term) in vacuum. We find out that for a certain value of the Barbero–Immirzi (BI) parameter the total action possesses a remarkable invariance under particular transformations of the affine connection. We prove that in all cases, with appropriate gauge choices, the connection reduces to the Levi-Civita one and that the theory turns out to be equivalent to general relativity (GR) in vacuum. Subsequently, we generalize our discussion and analyze the case of metric-affine f ( R ) gravity plus the Hojman term. In particular, we show that for f ′( R ) ≠ constant the theory results to be on-shell equivalent to a metric-compatible torsionless scalar–tensor model with a propagating pseudo-scalar. Matter coupling of the aforementioned models is also discussed, together with explicit examples and applications.

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.Comment: Title and abstract changed to better represent the content, some points clarified, mainly in Section 2 and in the concluding section, references added. Version accepted for publication on Physics Letters