On the role of torsion and higher forms in off-shell supergravity

We elaborate on the presence of a nonvanishing totally antisymmetric (super)torsion, equivalent to an axial vector, and higher forms in the "new minimal" and "old minimal" off-shell formulations of $\mathcal{N}=1$, $D=4$ supergravity. We adopt the geometric superspace approach and study both the geometric Lagrangian and the off-shell closure of the Bianchi identities in this framework, showing how the aforementioned axial vector torsion contributes to both the new and the old minimal set of auxiliary fields. In particular, to reproduce the old minimal set within the geometric setup, we also introduce two real auxiliary 3-form potentials.