Hidden Gauge Structure of Supersymmetric Free Differential Algebras

TL;DR

This paper explores the hidden superalgebra structure underlying D=11 supergravity's free differential algebra, emphasizing the role of nilpotent fermionic generators in ensuring correct gauge transformations and extending the framework to other supergravity theories.

Contribution

It clarifies the role of nilpotent fermionic generators in the hidden superalgebra of supergravity and constructs explicit examples for N=2, D=7 supergravity.

Findings

Nilpotent fermionic generators are essential for correct gauge transformations.

Hidden superalgebras can be constructed for various supergravity theories.

The approach may extend to theories in enlarged superspaces like double field theory.

Abstract

The aim of this paper is to clarify the role of the nilpotent fermionic generator Q' introduced in Ref. [3] 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 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.