Universal two-parameter 𝒲∞-algebra and vertex algebras of type 𝒲(2, 3, …, N). (10th February 2021)

• Record Type: Journal Article
• Authors: Linshaw, Andrew R.
• Is Part Of: Compositio mathematica. Volume 157:Number 1(2021)
• Page Start: 12
• Page End: 82
• Publication Date: 2021-02-10
• Subjects: vertex algebra, -- 𝒲-algebra, -- nonlinear Lie conformal algebra, -- coset construction; 17B67, -- 17B68, -- 17B69, -- 81R10; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X20007514
• Languages: English
• ISSNs: 0010-437X

Abstract

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.