$\mathcal{N} = (2,2)$ extended $\mathfrak{sl}(3|2)$ Chern-Simons $AdS_3$ supergravity with new boundaries

We present the first example of $\mathcal{N}=(2,2)$ formulation for the extended higher-spin $AdS_3$ supergravity with the most general boundary conditions as an extension of the $\mathcal{N} =\,(1,1)$ work, discovered recently by us [1]. Using the method proposed by Grumiller and Riegler, we construct a consistent class of the most general boundary conditions to extend it. An important consequence of our method is that, for the loosest set of boundary conditions it ensures that their asymptotic symmetry algebras consist of two copies of the $\mathfrak{sl}(3|2)_k$. Moreover, we enjoin some certain restrictions on the gauge fields for the most general boundary conditions, leading to the supersymmetric extensions of the Brown and Henneaux boundary conditions. Based on these results, we finally find out that the asymptotic symmetry algebras are two copies of the super $\mathcal{W}_3$ algebra for $\mathcal{N} = (2,2)$ extended higher-spin supergravity theory in $AdS_3$.