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

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 $n\ge 3$, 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 $n\ge 2$. 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

Final version. Minor corrections, numbering of theorems has been changed to agree with published version. To appear in Compositio Mathematica

References in corpus (5)

• Field Algebras
• W-algebras as coset vertex algebras
• Webs of W-algebras
• Quantum Langlands duality of representations of W-algebras
• Cosets of the ${\mathcal{W}}^k({\mathfrak{sl}}_4,f_{\text{subreg}})$-algebra

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