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 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.

The Grassmannian VOA

“…The coset description has a manifest Z 2 symmetry exchanging the two parameters µ 1 and µ 2 . The first surprising feature of the algebra is that the Z 2 duality symmetry manifest in coset description (1.1) is enhanced to an S 3 × Z 2 symmetry, see also [35]. One way to see this enhancement is to observe that if we perform the coset (2.1) in two steps, the intermediate algebra is the matrix-valued W 1+∞ algebra (or the affine gl(M ) Yangian), which was studied previously in [36][37][38][39].…”

The Grassmannian VOA

“…The coset description has a manifest Z 2 symmetry exchanging the two parameters µ 1 and µ 2 . The first surprising feature of the algebra is that the Z 2 duality symmetry manifest in coset description (1.1) is enhanced to an S 3 × Z 2 symmetry, see also [35]. One way to see this enhancement is to observe that if we perform the coset (2.1) in two steps, the intermediate algebra is the matrix-valued W 1+∞ algebra (or the affine gl(M) Yangian), which was studied previously in [36][37][38][39].…”

Trialities of $\mathcal{W}$-algebras

“…Relative semi-infinite Lie algebra cohomology acts on modules M of an affine vertex algebra at level −2h ∨ [72,13], and we use Section 2.5 of [39] as background. Most importantly, it satisfies…”

“…We recall relative semi-infinite Lie algebra cohomology [72], and for this we use Section 2.5 of [39]. Let g be a simple Lie algebra with basis B and dual basis B .…”