On the $\mathcal{N}=3$ and $\mathcal{N}=4$ superconformal holographic dictionary

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 $AdS_3$.