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

This study presents comprehensive examples of $\mathfrak{o}\mathfrak{s}\mathfrak{p}(\mathcal{N}\mathrm{\mid }2)$ Chern$-$Simons supergravity on $Ad{S}_3$ 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 $\mathfrak{o}\mathfrak{s}\mathfrak{p}(3\mathrm{\mid }2{)}_k$ and $\mathfrak{o}\mathfrak{s}\mathfrak{p}(4\mathrm{\mid }2{)}_k$ 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 $Ad{S}_3$.