New universal vertex algebras as glueings of the basic ones

Abstract

There are three universal $2$-parameter vertex algebras ${\mathcal{W}}_{\infty }$, ${\mathcal{W}}_{\infty }^{\text{ev}}$, and ${\mathcal{W}}_{\infty }^{\mathfrak{sp}}$ which are freely generated of types $\mathcal{W}(2,3,4,\dots )$, $\mathcal{W}(2,4,6,\dots )$, and $\mathcal{W}(1^3,2,3^3,4,\dots )$, respectively. They serve as classifying objects for vertex algebras with these generating types satisfying mild hypotheses. Their $1$-parameter quotients are expected to be the building blocks of all $\mathcal{W}$-algebras of classical Lie types. Furthermore, such $\mathcal{W}$-algebras are expected to be organized into families that are governed by new universal $2$-parameter vertex algebras, which are themselves glueings of copies of ${\mathcal{W}}_{\infty }$ in type $A$ (together with a Heisenberg algebra), and copies of ${\mathcal{W}}_{\infty }^{\text{ev}}$ and ${\mathcal{W}}_{\infty }^{\mathfrak{sp}}$ in types $B$, $C$, and $D$. We denote these universal objects by ${\mathcal{W}}_{\infty }^{X,S,M}$, where $X$ denotes the Lie type (either $A$, $C$, or $BD$ since types $B$ and $D$ can be treated uniformly), and $S$, $M$ are sets of positive integers that determine certain families of partitions. More precisely, for a partition $P=(n_0^{m_0},n_1^{m_1},\dots ,n_t^{m_t})$ of $N={\sum }_{i=0}^t{n}_i{m}_i$ consisting of $m_i$ parts of size $n_i$, where $n_0>n_1>\cdots >n_t\ge 2$, $M=\{m_0,\dots ,m_t\}$ is the set of multiplicities, and $S=\{d_1,\dots ,d_t\}$ is the set of height differences $d_{i+1}=n_i-n_{i+1}$. After introducing this general conjectural picture, we will construct the first nontrivial example ${\mathcal{W}}_{\infty }^{{\mathfrak{so}}_2}:={\mathcal{W}}_{\infty }^{BD,\varnothing ,\{2\}}$, which is a glueing of two copies of ${\mathcal{W}}_{\infty }^{\text{ev}}$.

60 pages, comments welcome