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

Abstract

This study presents comprehensive examples of $\mathfrak{osp}(\mathcal{N}|2)$ 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 $\mathfrak{osp}(3|2{)}_k$ 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 $Ad{S}_3$.

28 pages,LaTeX

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