Universal two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra

Abstract

We construct the unique two-parameter vertex algebra which is freely generated of type $\mathcal{W}(2,4,6,\dots )$, and generated by the weights $2$ and $4$ fields. Subject to some mild constraints, all vertex algebras of type $\mathcal{W}(2,4,\dots ,2N)$ for some $N$, can be obtained as quotients of this universal algebra. This includes the $B$ and $C$ type principal $\mathcal{W}$-algebras, the ${\mathbb{Z}}_2$-orbifolds of the $D$ type principal $\mathcal{W}$-algebras, and many others which arise as cosets of affine vertex algebras inside larger structures. As an application, we classify all coincidences among the simple quotients of the $B$ and $C$ type principal $\mathcal{W}$-algebras, as well as the ${\mathbb{Z}}_2$-orbifolds of the $D$ type principal $\mathcal{W}$-algebras. Finally, we use our classification to give new examples of principal $\mathcal{W}$-algebras of $B$, $C$, and $D$ types, which are lisse and rational.

Final version. New examples of lisse, rational W-algebras of B and C type are given, some details added in Section 9. Note: our main construction is similar to the one in arXiv:1710.02275 but there are some additional subtleties