Building blocks for $W$-algebras of classical types

TL;DR

The paper constructs a new universal two-parameter vertex algebra ${\boldsymbol{\text{W}}}^{\frak{sp}}_{\infty}$ of type ${\rm W}(1^3,2,3^3,4,5^3,6,\dots)$ that serves as a classifying object for ${\rm W}$-algebras of classical types with a ${\frak sp}_2$-subalgebra. It identifies eight families of 1-parameter quotients ${\mathcal C}^{\psi}_{XY}(n,m)$ (type C) and additional quotients ${\mathcal C}^{\ell}(n)$, all realized as quotients of ${\mathcal W}^{\frak sp}_{\infty}$, and develops a reconstruction framework showing these quotients arise from extending affine and Virasoro data by modules transforming as $U=\mathbb{C}^2\otimes\rho_{rak a}$. The work demonstrates that many quotients are strongly rational, yielding new strongly rational ${\rm W}$-superalgebras, and provides a mechanism to explain type-B/C/D building blocks via these constructions, including diagonal cosets and tails. The Results illuminate how universal objects for classical types interact with reduction, cosets, and rationality, and establish new level-rank dualities and structural correspondences across the orthosymplectic families. The findings have significance for the classification of ${ m W}$-algebras, rational CFTs, and connections to geometric and representation-theoretic frameworks (e.g., Hamiltonian reduction, invariant theory, and coset realizations).

Abstract

The universal $2$-parameter vertex algebra $W_{\infty}$ of type $W(2,3,4,\dots)$ serves as a classifying object for vertex algebras of type $W(2,3,\dots,N)$ for some $N$ in the sense that under mild hypothesis, all such vertex algebras arise as quotients of $W_{\infty}$. There is an $\mathbb{N} \times \mathbb{N}$ family of such $1$-parameter vertex algebras which, after tensoring with a Heisenberg algebra, are known as $Y$-algebras. They were introduced by Gaiotto and Rapčák and are expected to be the building blocks for all $W$-algebras in type $A$, i.e., every $W$-(super) algebra in type $A$ is an extension of a tensor product of finitely many $Y$-algebras. Similarly, the orthosymplectic $Y$-algebras are $1$-parameter quotients of a universal $2$-parameter vertex algebra $W^{\text{ev}}_{\infty}$ of type $W(2,4,6,\dots)$, which is a classifying object for vertex algebras of type $W(2,4,\dots, 2N)$ for some $N$. Unlike type $A$, these algebras are not all the building blocks for $W$-algebras of types $B$, $C$, and $D$. In this paper, we construct a new universal $2$-parameter vertex algebra of type $W(1^3, 2, 3^3, 4, 5^3,6,\dots)$ which we denote by $W^{\mathfrak{sp}}_{\infty}$ since it contains a copy of the affine vertex algebra $V^k(\mathfrak{sp}_2)$. We identify $8$ infinite families of $1$-parameter quotients of $W^{\mathfrak{sp}}_{\infty}$ which are analogues of the $Y$-algebras. We regard $W^{\mathfrak{sp}}_{\infty}$ as a fundamental object on equal footing with $W_{\infty}$ and $W^{\text{ev}}_{\infty}$, and we give some heuristic reasons for why we expect the $1$-parameter quotients of these three objects to be the building blocks for all $W$-algebras of classical types. Finally, we prove that $W^{\mathfrak{sp}}_{\infty}$ has many quotients which are strongly rational. This yields new examples of strongly rational $W$-superalgebras.

Theorems & Definitions (120)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1: KW, Thm. 2.1
• Theorem 2.2
• Theorem 2.3: CL3, Thm 3.5 and Cor. 3.4
• Definition 2.1
• Conjecture 3.1
• Remark 3.1
• Definition 3.1
• Lemma 4.1