Holst-MacDowell-Mansouri action for (extended) supergravity with boundaries and super Chern-Simons theory

In this article, the Cartan geometric approach toward (extended) supergravity in the presence of boundaries will be discussed. In particular, based on new developments in this field, we will derive the Holst variant of the MacDowell-Mansouri action for $\mathcal{N}=1$ and $\mathcal{N}=2$ pure AdS supergravity in $D=4$ for arbitrary Barbero-Immirzi parameters. This action turns out to play a crucial role in context of boundaries in the framework of supergravity if one imposes supersymmetry invariance at the boundary. For the $\mathcal{N}=2$ case, it follows that this amounts to the introduction of a $\theta$-topological term to the Yang-Mills sector which explicitly depends on the Barbero-Immirzi parameter. This shows the close connection between this parameter and the $\theta$-ambiguity of gauge theory. We will also discuss the chiral limit of the theory, which turns out to possess some very special properties such as the manifest invariance of the resulting action under an enlarged gauge symmetry. Moreover, we will show that demanding supersymmetry invariance at the boundary yields a unique boundary term corresponding to a super Chern-Simons theory with $\mathrm{OSp}(\mathcal{N}|2)$ gauge group. In this context, we will also derive boundary conditions that couple boundary and bulk degrees of freedom and show equivalence to the results found in the D'Auria-Fr\'e approach in context of the non-chiral theory. These results provide a step towards of quantum description of supersymmetric black holes in the framework of loop quantum gravity.