Compositio Mathematica ( IF 1.8 ) Pub Date : 2021-02-10 , DOI: 10.1112/s0010437x20007514 Andrew R. Linshaw
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.
中文翻译:
𝒲(2,3,…,N)类型的通用两参数𝒲∞代数和顶点代数
我们证明了长期存在的物理学猜想,即存在一个独特的两参数$ {\ mathcal {W}} _ {\ infty} $-代数,该代数可以自由生成$ {\ mathcal {W}}(2,3,\ ldots)$,并由权重$ 2 $和$ 3 $字段生成。受到一些适度的约束,对于某些$ N $,所有类型为$ {\ mathcal {W}}(2,3,\ ldots,N)$的顶点代数都可以作为该通用代数的商。作为应用程序,我们显示了对于$ n \ geq 3 $,主体$ {\ mathcal {W}} $-代数$ {\ mathcal {W}} ^ k({\ mathfrak s} {\ mathfrak l} _n,f _ {\ text {prin}})$是$ķ$和$ n $的,我们简单的商之间的所有的巧合分类$ {\ mathcal {白}} _ K({\ mathfrak S} {\ mathfrak L} _n,F _ {\文本{首席}})$的$ n \ geq 2 $。我们还获得了$ {\ mathcal {W}} _ k({\ mathfrak s} {\ mathfrak l} _n,f _ {\ text {prin}})$和其他$ {\ mathcal { W}}(2,3,\ ldots,N)$作为仿射顶点代数或非主$ {\ mathcal {W}} $-代数的同集而出现。