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Recently, many geometric aspects of $\mathcal{N}$-extended AdS supergravityin chiral variables have been encountered and clarified. In particular, if thetheory is supposed to be invariant under SUSY transformations also onboundaries, the boundary term has to be the action of a$\mathrm{OSp}(\mathcal{N}|2)_{\mathbb{C}}$ super Chern-Simons theory, andparticular boundary conditions must be met. Based on this, we propose a way tocalculate an entropy $S$ for surfaces, presumably including black holehorizons, in the supersymmetric version of loop quantum gravity for the minimalcase $\mathcal{N}=1$. It proceeds in analogy to the non-supersymmetric theory,by calculating dimensions of quantum state spaces of the super Chern-Simonstheory with punctures, for fixed quantum (super) area of the surface. We find$S = a_H/4$ for large areas and determine the subleading correction. Due to thenon-compactness of $\mathrm{OSp}(1|2)_{\mathbb{C}}$ and the correspondingdifficulties with the Chern-Simons quantum theory, we use analytic continuationfrom the Verlinde formula for a compact real form, $\mathrm{UOSp}(1|2)$, inanalogy to work by Noui et al. This also entails studying some properties of$\mathrm{OSp}(1|2)_{\mathbb{C}}$ representations that we have not foundelsewhere in the literature.

Towards black hole entropy in chiral loop quantum supergravity