New universal vertex algebras as glueings of the basic ones

There are three universal $2$-parameter vertex algebras $\mathcal{W}_{\infty}$, $\mathcal{W}^{\text{ev}}_{\infty}$, and $\mathcal{W}^{\mathfrak{sp}}_{\infty}$ 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 $\mathcal{W}^{\text{ev}}_{\infty}$ and $\mathcal{W}^{\mathfrak{sp}}_{\infty}$ in types $B$, $C$, and $D$. We denote these universal objects by $\mathcal{W}^{X,S,M}_{\infty}$, 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 $n_0> n_1 > \cdots > n_t \geq 2$, $M = \{m_0,\dots, m_t\}$ 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 $\mathcal{W}^{\mathfrak{so}_2}_{\infty}:=\mathcal{W}^{BD, \emptyset, \{2\}}_{\infty}$, which is a glueing of two copies of $\mathcal{W}^{\text{ev}}_{\infty}$.