Universal two-parameter 𝒲∞-algebra and vertex algebras of type 𝒲(2, 3, …, N). (10th February 2021) - Linshaw, Andrew R.
Abstract : We prove the longstanding physics conjecture that there exists a unique two-parameter ${\mathcal {W}}_{∞ }$ -algebra which is freely generated of type ${\mathcal {W}}(2, 3, … )$, and generated by the weights $2$ and $3$ fields. Subject to some mild constraints, all vertex algebras of type ${\mathcal {W}}(2, 3, …, N)$ for some $N$ can be obtained as quotients of this universal algebra. As an application, we show that for $n≥ 3$, 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 $n≥ 2$ . We also obtain many new coincidences between ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ and other vertex algebras of type ${\mathcal {W}}(2, 3, …, N)$ which arise as cosets of affine vertex algebras or nonprincipal ${\mathcal {W}}$ -algebras.