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Towards a classification of graded unitary ${\mathcal W}_3$ algebras

Christopher Beem, Harshal Kulkarni

Abstract

We study constraints imposed by four-dimensional unitarity (formalised as graded unitarity in recent work by the first author) on possible ${\mathcal W}_3$ vertex algebras arising from four-dimensions via the SCFT/VOA correspondence. Under the assumption that the $\mathfrak{R}$-filtration is a weight-based filtration with respect to the usual strong generators of the vertex algebra, we demonstrate that all values of the central charge other than those of the $(3,q+4)$ minimal models are incompatible with four-dimensional unitarity. These algebras are precisely the ones that are realised by performing principal Drinfel'd--Sokolov reduction to boundary-admissible $\mathfrak{sl}_3$ affine current algebras; those affine algebras were singled out by a similar graded unitarity analysis in \cite{ArabiArdehali:2025fad}. Furthermore, these particular vertex algebras are known to be associated with the $(A_2,A_q)$ Argyres--Douglas theories.

Towards a classification of graded unitary ${\mathcal W}_3$ algebras

Abstract

We study constraints imposed by four-dimensional unitarity (formalised as graded unitarity in recent work by the first author) on possible vertex algebras arising from four-dimensions via the SCFT/VOA correspondence. Under the assumption that the -filtration is a weight-based filtration with respect to the usual strong generators of the vertex algebra, we demonstrate that all values of the central charge other than those of the minimal models are incompatible with four-dimensional unitarity. These algebras are precisely the ones that are realised by performing principal Drinfel'd--Sokolov reduction to boundary-admissible affine current algebras; those affine algebras were singled out by a similar graded unitarity analysis in \cite{ArabiArdehali:2025fad}. Furthermore, these particular vertex algebras are known to be associated with the Argyres--Douglas theories.
Paper Structure (16 sections, 93 equations, 2 figures, 1 table)

This paper contains 16 sections, 93 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Relevant aspects of the W3 algebra
  3. Verma modules and the vacuum module
  4. Submodules and singular vectors of M(0,0;c)
  5. Verma resolution of the vacuum module
  6. Jantzen filtration and a vacuum determinant formula
  7. Graded unitarity structures for the W3 algebra
  8. Constraints from graded unitarity
  9. Explicit computations of low-level vacuum Kac determinants
  10. Large negative central charge
  11. Analysis of zeros of the vacuum Kac determinant
  12. Zeros at $c = c_{p,q}$ with $p \geqslant 4$ and $(p,q) = 1$.
  13. The smallest zeros $c = c_{3,q}$.
  14. Central charge constraints
  15. Kac determinant computation details
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Diagram representing the embedding of singular vectors/submodules in the vacuum Verma module $M(0,0;c)$ at generic central charge. Arrows indicate the embedding of a singular vector (the highest weight vector of the source submodule) into the target submodule.
  • Figure 2: Central charge constraints at levels $N = 0 \text{ mod }3$ and $N = 1 \text{ mod } 3$. Dashed blue lines indicate regions excluded at earlier levels. Black lines and red dots indicate allowed central charges not ruled out by graded unitarity constraints at level $N$.

Theorems & Definitions (1)

  • Remark