Various aspects of orbifolds and cosets of the small $\mathcal{ N}=4$ superconformal algebra are studied. First, we determine minimal strong generators for generic and specific levels. As a corollary, we obtain the vertex algebra of global sections of the chiral de Rham complex on any complex Enriques surface. We also identify orbifolds of cosets of the small $\mathcal{ N}=4$ superconformal algebra with $\text{ Com}(V^{ \ell}(\mathfrak{ sl}_2), V^{ \ell+1}(\mathfrak{ sl}_2) \otimes \mathcal{ W}_{ -5/2}(\mathfrak{ sl}_4, f_{ \text{ rect}}))$ and in addition at special levels with Grassmanian cosets and principal $\mathcal{ W}$-algebras of type $A$ at degenerate admissible levels. These coincidences lead us to a novel level-rank duality involving Grassmannian supercosets.
Categories: math.RT hep-th math.QA