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

TL;DR

This paper develops an $ ext{N}=(2,2)$ extended higher-spin supergravity theory in $AdS_3$ with new boundary conditions, revealing asymptotic symmetries described by two copies of the super $ ext{W}_3$ algebra, advancing understanding of supersymmetric higher-spin gravities.

Contribution

It constructs the first $ ext{N}=(2,2)$ extended higher-spin $AdS_3$ supergravity with general boundary conditions, extending previous $ ext{N}=(1,1)$ work and identifying its asymptotic symmetry algebra.

Findings

Asymptotic symmetry algebra is two copies of super $ ext{W}_3$ algebra.

Most general boundary conditions lead to consistent asymptotic symmetries.

Restrictions on gauge fields produce supersymmetric extensions of Brown-Henneaux conditions.

Abstract

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 $AdS_3$.