Extended supersymmetry in AdS_3 higher spin theories

TL;DR

This paper analyzes the asymptotic symmetry algebra of matrix extended Vasiliev higher spin theories in AdS3, revealing supersymmetry enhancements and their relation to superconformal symmetry, especially for the case M=2.

Contribution

It systematically studies supersymmetry enhancements in extended higher spin theories and clarifies their asymptotic symmetry algebras, connecting them to coset duals and superconformal symmetry.

Findings

For all M≥2, higher spin algebra has extended supersymmetry.

Only for M=2, the asymptotic symmetry algebra is superconformal with large N=4 supersymmetry.

Vasiliev theories based on shs^E(N|2, R) share the same supersymmetry properties as matrix extended theories.

Abstract

We determine the asymptotic symmetry algebra (for fields of low spin) of the $M\times M$ matrix extended Vasiliev theories on AdS$_3$ and find that it agrees with the $\mathcal{W}$-algebra of their proposed coset duals. Previously it was noticed that for $M=2$ the supersymmetry increases from $\mathcal{N}=2$ to $\mathcal{N}=4$. We study more systematically this type of supersymmetry enhancements and find that, although the higher spin algebra has extended supersymmetry for all $M\geq 2$, the corresponding asymptotic symmetry algebra fails to be superconformal except for $M=2$, when it has large $\mathcal{N}=4$ superconformal symmetry. Moreover, we find that the Vasiliev theories based on $shs^E\! \left( \mathcal{N} \vert 2, \mathbb{R} \right)$ are special cases of the matrix extended higher spin theories, and hence have the same supersymmetry properties.