Extended supersymmetry in AdS 3 higher spin theories

We determine the asymptotic symmetry algebra (for fields of low spin) of the M × M matrix extended Vasiliev theories on AdS 3 and find that it agrees with the W $$ \mathcal{W} $$ -algebra of their proposed coset duals. Previously it was noticed that for M = 2 the supersymmetry increases from N = 2 $$ \mathcal{N}=2 $$ to N = 4 $$ \mathcal{N}=4 $$ . We study more systematically this type of supersymmetry enhancements and find that, although the higher spin algebra has extended supersymmetry for all M ≥ 2, the corresponding asymptotic symmetry algebra fails to be superconformal except for M = 2, when it has large N = 4 $$ \mathcal{N}=4 $$ superconformal symmetry. Moreover, we find that the Vasiliev theories based on s h s E N | 2 , ℝ $$ \mathfrak{s}\mathfrak{h}{\mathfrak{s}}^E\left(\mathcal{N}\Big|2,\mathrm{\mathbb{R}}\right) $$ are special cases of the matrix extended higher spin theories, and hence have the same supersymmetry properties