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

TL;DR

This paper explores the asymptotic symmetry algebras of $ ext{osp}( ext{N}|2)$ Chern-Simons supergravity on $AdS_3$ for $ ext{N}>2$, revealing extensions to superconformal algebras for $ ext{N}=3$ and $ ext{N}=4$, with implications for higher-spin supergravity.

Contribution

It extends the supergravity boundary conditions and symmetry analysis to $ ext{N}=3$ and $ ext{N}=4$, providing new superconformal holographic dictionaries for these cases.

Findings

Asymptotic symmetry algebras are two copies of $ ext{osp}(3|2)_k$ and $ ext{osp}(4|2)_k$.

Boundary conditions lead to supersymmetric extensions of Brown-Henneaux conditions.

Resulting algebras are two copies of $ ext{N}=3$ and $ ext{N}=4$ superconformal algebras.

Abstract

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 $AdS_3$.