N = 4 Superconformal Algebras and Diagonal Cosets

Coset constructions of W-algebras have many applications and were recently given for principal 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 W-algebras of d(2, 1; α) and psl(2|2), respectively. From these realizations, one finds a remarkable connection between the large N = 4 algebra and the diagonal coset Ck1,k2 = Com(Vk1+k2 (sl2),Vk1 (sl2) ⊗ Vk2 (sl2)), namely, as two-parameter vertex algebras, Ck1,k2 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 Walgebras. As a corollary, we give new examples of strongly rational principalW-algebras of type C at degenerate admissible levels.