On the N=3 and N=4 superconformal holographic dictionary

This study presents comprehensive examples of osp(N|2) Chern$\,-\,$Simons supergravity on AdS₃ for N>2. These formulations, which include the most general boundary conditions, represent extensions of previously discovered works (Ozer and Filiz,Eur Phys J C 82(5):472, 2022) for 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 osp(3|2)ₖ and 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 N=3 and N=4 superconformal algebras for N=(3,3) and N=(4,4) extended higher$\,-\,$spin supergravity theory in AdS₃.