New universal vertex algebras as glueings of the basic ones
Thomas Creutzig, Vladimir Kovalchuk, Andrew R. Linshaw
TL;DR
The paper develops a framework in which classical ${\mathcal{W}}$-algebras are organized by universal two-parameter vertex algebras built as glueings of basic universals: ${\mathcal{W}}_{\\infty}$ in type A and ${\mathcal{W}}^{ev}_{\\infty}$ and ${\mathcal{W}}^{\\mathfrak{sp}}_{\\infty}$ in types B, C, D. It constructs the first nontrivial universal object, ${\mathcal{W}}^{\mathfrak{so}_2}_{\\infty}$, as a glueing of two copies of ${\mathcal{W}}^{ev}_{\\infty}$, and shows that its eight one-parameter quotients and various GKO/diagonal cosets arise from quotients of this universal object. A reconstruction theorem establishes that the full operator algebra of these ${\mathfrak{so}_2}$-rectangular W-algebras with tails is determined by a small set of data (weights, parity, and zero-mode action), allowing explicit conformal embeddings and rationality results to be deduced. The framework connects to shifted (twisted) Yangians and truncation curves, providing a principled route to classify building blocks and to prove strong rationality for a broad family of ${\mathcal{W}}$-algebras, with potential applications to conformal embeddings and orbifold constructions in orthosymplectic types. Altogether, the paper offers a structured, universality-driven approach to understanding and constructing ${\mathcal{W}}$-algebras across classical Lie types.
Abstract
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 $\mathcal{W}^{\text{ev}}_{\infty}$.