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

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.

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