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 ( $\mathcal{N}$ \( \mathcal{N} \) = 1, 2, 4), motivated by the prospective of possibly finite amplitudes. In the process, we develop some of the representation theory of OSp(2|2, ℝ) and PSU(1, 1|2), relevant for the $\mathcal{N}$ \( \mathcal{N} \) = 2 and $\mathcal{N}$ \( \mathcal{N} \) = 4 cases.