Universal twoparameter even spin $\mathcal{W}_{\infty}$algebra
Abstract
We construct the unique twoparameter 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 arXiv:
 arXiv:1805.11031
 Bibcode:
 2018arXiv180511031K
 Keywords:

 Mathematics  Representation Theory;
 High Energy Physics  Theory;
 Mathematics  Quantum Algebra
 EPrint:
 Final version. New examples of lisse, rational Walgebras 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