Immirzi parameter and $\theta$ ambiguity in de Sitter MacDowell-Mansouri supergravity

In this paper we revisit a gauge theory of supergravity, where the tetrad, the spin $3/2$ field and the spin connection are identified with components of the gauge field. This theory naturally incorporates a cosmological constant term. Furthermore, when adding the topological $\theta$-term, as in the Yang-Mills theory, an analogue of the Immirzi term for $\mathcal{N}=1$ supergravity is obtained.