$N=4$ superconformal algebras and diagonal cosets

TL;DR

This paper constructs coset realizations of the large and small $N=4$ superconformal algebras, revealing new connections with diagonal cosets and providing examples of rational $ ext{W}$-algebras at special parameters.

Contribution

It introduces coset constructions for the $N=4$ superconformal algebras and uncovers their relation to diagonal cosets, also identifying new rational $ ext{W}$-algebras at specific levels.

Findings

The large $N=4$ algebra is realized as a coset related to diagonal $ ext{sl}_2$-based vertex algebras.

At special parameters, cosets are isomorphic to known $ ext{W}$-algebras.

New strongly rational $ ext{W}$-algebras of type C are identified at degenerate admissible levels.

Abstract

Coset constructions of $\mathcal{W}$-algebras have many applications, and were recently given for principal $\mathcal{W}$-algebras of $A$, $D$, and $E$ types by Arakawa together with the first and third authors. In this paper, we give coset constructions of the large and small $N=4$ superconformal algebras, which are the minimal $\mathcal{W}$-algebras of $\mathfrak{d}(2,1;a)$ and $\mathfrak{psl}(2|2)$, respectively. From these realizations, one finds a remarkable connection between the large $N=4$ algebra and the diagonal coset $C^{k_1, k_2} = \text{Com}(V^{k_1+k_2}(\mathfrak{sl}_2), V^{k_1}(\mathfrak{sl}_2) \otimes V^{k_2}(\mathfrak{sl}_2))$, namely, as two-parameter vertex algebras, $C^{k_1, k_2}$ coincides with the coset of the large $N=4$ algebra by its affine subalgebra. We also show that at special points in the parameter space, the simple quotients of these cosets are isomorphic to various $\mathcal{W}$-algebras. As a corollary, we give new examples of strongly rational principal $\mathcal{W}$-algebras of type $C$ at degenerate admissible levels.