Universal two-parameter 𝒲 <sub>∞</sub> -algebra and vertex algebras of type 𝒲(2, 3, …, <i>N</i> )
Andrew R. Linshaw
Abstract
We prove the longstanding physics conjecture that there exists a unique two-parameter ${\mathcal {W}}_{\infty }$ -algebra which is freely generated of type ${\mathcal {W}}(2,3,\ldots )$ , and generated by the weights $2$ and $3$ fields. Subject to some mild constraints, all vertex algebras of type ${\mathcal {W}}(2,3,\ldots , N)$ for some $N$ can be obtained as quotients of this universal algebra. As an application, we show that for $n\geq 3$ , the structure constants for the principal ${\mathcal {W}}$ -algebras ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ are rational functions of $k$ and $n$ , and we classify all coincidences among the simple quotients ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ for $n\geq 2$ . We also obtain many new coincidences between ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ and other vertex algebras of type ${\mathcal {W}}(2,3,\ldots , N)$ which arise as cosets of affine vertex algebras or nonprincipal ${\mathcal {W}}$ -algebras.