Recently, many geometric aspects of $\mathcal{N}$-extended anti–de Sitter supergravity in chiral variables have been encountered and clarified. In particular, if the theory is supposed to be invariant under supersymmetry transformations also on boundaries, the boundary term has to be the action of an $\mathrm{OSp}(\mathcal{N}|2{)}_{\mathbb{C}}$ super Chern-Simons theory, and particular boundary conditions must be met. Based on this, we propose a way to calculate an entropy $S$ for surfaces, presumably including black hole horizons, in the supersymmetric version of loop quantum gravity for the minimal case $\mathcal{N}=1$. It proceeds in analogy to the nonsupersymmetric theory, by calculating dimensions of quantum state spaces of the super Chern-Simons theory with punctures, for a fixed quantum (super) area of the surface. We find $S=a_H/4$ for large areas and determine the subleading correction. Because of the noncompactness of $\mathrm{OSp}(1|2{)}_{\mathbb{C}}$ and the corresponding difficulties with the Chern-Simons quantum theory, we use analytic continuation from the Verlinde formula for a compact real form, $\mathrm{UOSp}(1|2)$, in analogy to work by Noui et al. This also entails studying some properties of $\mathrm{OSp}(1|2{)}_{\mathbb{C}}$ representations that we have not found elsewhere in the literature.

• Received 26 April 2022
• Accepted 22 June 2022

DOI:https://doi.org/10.1103/PhysRevD.106.026001