$N=4$ superconformal algebras and diagonal cosets
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 $\mathfrak{psl}(22)$, 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 twoparameter 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.01228
 Bibcode:
 2019arXiv191001228C
 Keywords:

 Mathematics  Representation Theory;
 High Energy Physics  Theory;
 Mathematics  Quantum Algebra
 EPrint:
 34 pages