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Universal two-parameter 𝒲∞-algebra and vertex algebras of type 𝒲(2, 3, …, N)

Part of: Lie algebras and Lie superalgebras Groups and algebras in quantum theory

Published online by Cambridge University Press:  10 February 2021

Andrew R. Linshaw
Andrew R. Linshaw*
Affiliation:
Department of Mathematics, University of Denver, 2390 S. York St., Denver, CO80208, USAandrew.linshaw@du.edu
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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.

Keywords

vertex algebra𝒲-algebranonlinear Lie conformal algebracoset construction

MSC classification

Primary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B68: Virasoro and related algebras 17B69: Vertex operators; vertex operator algebras and related structures 81R10: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current algebras and their representations
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 1 , January 2021 , pp. 12 - 82
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Copyright
© The Author(s) 2021

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Footnotes

This work was partially supported by grants $\#$318755 and $\#$635650 from the Simons Foundation.

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