N = 4 Superconformal Algebras and Diagonal Cosets
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 1st and 3rd 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 ${{\mathfrak{p}}}{{\mathfrak{s}}}{{\mathfrak{l}}}(2|2)$, respectively. From these realizations, one finds a remarkable connection between the large $N=4$ algebra and the diagonal coset $C^{k_1, k_2} = \textrm{Com}(V^{k_1+k_2}({{\mathfrak{s}}}{{\mathfrak{l}}}_2), V^{k_1}({{\mathfrak{s}}}{{\mathfrak{l}}}_2) \otimes V^{k_2}({{\mathfrak{s}}}{{\mathfrak{l}}}_2))$, 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.