Universal two-parameter $\mathcal{W}_{\infty}$-algebra and vertex algebras of type $\mathcal{W}(2,3,\dots, N)$

TL;DR

This paper proves the existence of a unique universal two-parameter $ ext{W}_ ext{infty}$-algebra generated by weights 2 and 3, and classifies coincidences among principal $ ext{W}$-algebras and their quotients.

Contribution

It establishes the universal $ ext{W}_ ext{infty}$-algebra and describes how all $ ext{W}(2,3, ext{...},N)$ vertex algebras are quotients, also analyzing structure constants and coincidences.

Findings

Existence of a unique two-parameter $ ext{W}_ ext{infty}$-algebra.

Rationality of structure constants for principal $ ext{W}$-algebras.

Classification of coincidences among simple quotients of $ ext{W}$-algebras.

Abstract

We prove the longstanding physics conjecture that there exists a unique two-parameter $\mathcal{W}_{\infty}$-algebra which is freely generated of type $\mathcal{W}(2,3,\dots)$, and generated by the weights $2$ and $3$ fields. Subject to some mild constraints, all vertex algebras of type $\mathcal{W}(2,3,\dots, N)$ 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 $\mathcal{W}(2,3,\dots, N)$ which arise as cosets of affine vertex algebras or nonprincipal $\mathcal{W}$-algebras