Branes in JT (super)gravity from group theory

In this work, we revisit the end-of-the-world (EOW) brane amplitudes in JT gravity from a BF gauge theoretic perspective. Observing and identifying the correct group theoretic ingredient for a closed EOW brane as a discrete series character, we use the group theory framework as a guide towards formulating the analogous supersymmetric problem. We compute these amplitudes explicitly in the supersymmetric generalizations of JT gravity ($\mathcal{N}=1,2,4$), motivated by the prospective of possibly finite amplitudes. In the process, we develop some of the representation theory of OSp$(2\mathrm{\mid }2,\mathbb{R})$ and PSU($1,1\mathrm{\mid }2)$, relevant for the $\mathcal{N}=2$ and $\mathcal{N}=4$ cases.