This study presents comprehensive examples of $\mathfrak{ osp}(\mathcal{ N}|2)$ Chern$\,-\,$Simons supergravity on $AdS_3$ for $\mathcal{ N}>2$. These formulations, which include the most general boundary conditions, represent extensions of previously discovered works $(\textit{ Ozer and Filiz$,$Eur Phys J C 82(5):472, 2022})$ for $\mathcal{ N}<3$. In our work, we show that under the loosest set of boundary conditions, the asymptotic symmetry algebras consist of two copies of the $\mathfrak{ osp}(3|2)_k$ and $\mathfrak{ osp}(4|2)_k$ algebras. We subsequently restrict the gauge fields upon the boundary conditions to achieve supersymmetric extensions of the Brown$\,-\,$Henneaux boundary conditions. Based on these results, we finally find that the asymptotic symmetry algebras are two copies of the $\mathcal{ N}=3$ and $\mathcal{ N}=4$ superconformal algebras for $\mathcal{ N}=(3,3)$ and $\mathcal{ N}=(4,4)$ extended higher$\,-\,$spin supergravity theory in $AdS_3$.
Categories: hep-th