$N=4$ 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 $\mathfrak{psl}(2|2)$, 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.

34 pages

References in corpus (11)

• Notes on the K3 Surface and the Mathieu group M_24
• W-algebras as coset vertex algebras
• Braided tensor categories of admissible modules for affine Lie algebras
• Even spin minimal model holography
• Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1|2)$ minimal models
• Tensor categories of affine Lie algebras beyond admissible levels
• Universal two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra
• S-duality for the large $N=4$ superconformal algebra
• Unitary and non-unitary $N=2$ minimal models
• Symmetry Algebras of Stringy Cosets
• Invariant subalgebras of the small $\mathcal{N}=4$ superconformal algebra

Cited by in corpus (9)

• Trialities of $\mathcal{W}$-algebras
• Duality of subregular W-algebras and principal W-superalgebras
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• Higher rank FZZ-dualities
• Invariant subalgebras of the small $\mathcal{N}=4$ superconformal algebra
• FZZ-triality and large $\mathcal{N}=4$ super Liouville theory
• Duality via convolution of W-algebras