Invariant subalgebras of the small $\mathcal{N}=4$ superconformal algebra

TL;DR

This paper explores orbifolds and cosets of the small  superconformal algebra, identifying minimal generators, connections to geometric structures, and discovering a new level-rank duality involving Grassmannian supercosets.

Contribution

It determines minimal strong generators for various levels and uncovers novel dualities and coincidences with geometric and algebraic structures.

Findings

Vertex algebra of global sections on Enriques surfaces identified.

Orbifolds of cosets related to  superconformal algebra characterized.

New level-rank duality involving Grassmannian supercosets discovered.

Abstract

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.