The supersymmetric tensor hierarchy of N = 1 , d = 4 supergravity

In this paper we construct the supersymmetric tensor hierarchy of N=1, d=4 supergravity. We find some differences with the general bosonic construction of 4dimensional gauged supergravities. The global symmetry group of N = 1, d = 4 supergravity consists of three factors: the scalar manifold isometry group, the invariance group of the complex vector kinetic matrix and the U(1) R-symmetry group. In contrast to (half)-maximal supergravities, the latter two symmetries are not embedded into the isometry group of the scalar manifold. We identify some components of the embedding tensor with Fayet-Iliopoulos terms and we find that supersymmetry implies that the inclusion of R-symmetry as a factor of the global symmetry group requires a non-trivial extension of the standard p-form hierarchy. This extension involves additional 3and 4-forms. One additional 3-form is dual to the superpotential (seen as a deformation of the simplest theory). We study the closure of the supersymmetry algebra on all the bosonic p-form fields of the hierarchy up to duality relations. In order to close the supersymmetry algebra without the use of duality relations one must construct the hierarchy in terms of supermultiplets. Such a construction requires fermionic duality relations among the hierarchy’s fermions and these turn out to be local.