On the dualization of scalars into ( d − 2)-forms in supergravity. Momentum maps, R-symmetry and gauged supergravity

We review and investigate different aspects of scalar fields in supergravity theories both when they parametrize symmetric spaces and when they parametrize spaces of special holonomy which are not necessarily symmetric (kähler and quaternionic-kähler spaces): their rôle in the definition of derivatives of the fermions covariant under the r-symmetry group and (in gauged supergravities) under some gauge group, their dualization into (d − 2)-forms, their role in the supersymmetry transformation rules (via fermion shifts, for instance) etc. we find a general definition of momentum map that applies to any manifold admitting a killing vector and coincides with those of the holomorphic and tri-holomorphic momentum maps in kähler and quaternionic-kähler spaces and with an independent definition that can be given in symmetric spaces. we show how the momen-tum map occurs ubiquitously: in gauge-covariant derivatives of fermions, in fermion shifts, in the supersymmetry transformation rules of (d − 2)-forms etc. we also give the general structure of the noether-gaillard-zumino conserved currents in theories with fields of different ranks in any dimension.