# Exploring new Boundary Conditions for $\mathcal{N}=(1,1)$ Extended Higher Spin $AdS_3$ Supergravity

## TL;DR

This paper investigates boundary conditions in $ ext{AdS}_3$ supergravity with extended higher spin symmetry, revealing how different boundary conditions influence the asymptotic symmetry algebra, including supersymmetric extensions.

## Contribution

It introduces a candidate for $ ext{AdS}_3$ supergravity with $ ext{N}=(1,1)$ extended higher spin symmetry under general boundary conditions and analyzes the resulting asymptotic symmetry algebras.

## Key findings

- Asymptotic symmetry algebra includes two copies of $ ext{osp}(3|2)_k$ with general boundary conditions.
- Restrictions on gauge fields lead to supersymmetric Brown-Henneaux boundary conditions.
- Reduced algebra is two copies of $ ext{SW}(rac{3}{2},2)$ for $ ext{N}=(1,1)$ supergravity.

## Abstract

In this paper, we present a candidate for $\mathcal{N}=(1,1)$ extended higher - spin $AdS_3$ supergravity with the most general boundary conditions discussed by Grumiller and Riegler recently. We show that the asymptotic symmetry algebra consists of two copies of the $\mathfrak{osp}(3|2)_k$ affine algebra in the presence of the most general boundary conditions.Furthermore, we impose some certain restrictions on gauge fields on the most general boundary conditions and that leads us to the supersymmetric extension of the Brown - Henneaux boundary conditions. We eventually see that the asymptotic symmetry algebra reduces to two copies of the $\mathcal{SW}(\frac{3}{2},2)$ algebra for $\mathcal{N}=(1,1)$ extended higher - spin supergravity.