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Schur sector of Argyres-Douglas theory and $W$-algebra

Dan Xie, Wenbin Yan

TL;DR

This work develops a VOA-based framework to probe protected sectors of 4d $\mathcal{N}=2$ Argyres-Douglas theories by leveraging their associated 2d $W$-algebras $W^{k'}(\mathfrak{g},f)$. The Schur index is computed from the $W$-algebra vacuum character and can be written in a compact plethystic-exponential form, enabling checks of level-rank dualities and S-duality conjectures; the Macdonald index is obtained via a Kazhdan filtration. The Zhu's $C_2$ algebra is identified as the ring governing the Schur sector, with a surprising connection to Jacobi algebras of hypersurface singularities in certain cases. Together, these results illuminate deep links between 4d protected spectra, 2d vertex algebras, and singularity theory, and they provide explicit indices for several $(A_1,G)$ theories. The approach promises further insights into 4d/2d correspondences and the structure of protected sectors in strongly coupled SCFTs.

Abstract

We study the Schur index, the Zhu's $C_2$ algebra, and the Macdonald index of a four dimensional $\mathcal{N}=2$ Argyres-Douglas (AD) theories from the structure of the associated two dimensional $W$-algebra. The Schur index is derived from the vacuum character of the corresponding $W$-algebra and can be rewritten in a very simple form, which can be easily used to verify properties like level-rank dualities, collapsing levels, and S-duality conjectures. The Zhu's $C_2$ algebra can be regarded as a ring associated with the Schur sector, and a surprising connection between certain Zhu's $C_2$ algebra and the Jacobi algebra of a hypersurface singularity is discovered. Finally, the Macdonald index is computed from the Kazhdan filtration of the $W$-algebra.

Schur sector of Argyres-Douglas theory and $W$-algebra

TL;DR

This work develops a VOA-based framework to probe protected sectors of 4d Argyres-Douglas theories by leveraging their associated 2d -algebras . The Schur index is computed from the -algebra vacuum character and can be written in a compact plethystic-exponential form, enabling checks of level-rank dualities and S-duality conjectures; the Macdonald index is obtained via a Kazhdan filtration. The Zhu's algebra is identified as the ring governing the Schur sector, with a surprising connection to Jacobi algebras of hypersurface singularities in certain cases. Together, these results illuminate deep links between 4d protected spectra, 2d vertex algebras, and singularity theory, and they provide explicit indices for several theories. The approach promises further insights into 4d/2d correspondences and the structure of protected sectors in strongly coupled SCFTs.

Abstract

We study the Schur index, the Zhu's algebra, and the Macdonald index of a four dimensional Argyres-Douglas (AD) theories from the structure of the associated two dimensional -algebra. The Schur index is derived from the vacuum character of the corresponding -algebra and can be rewritten in a very simple form, which can be easily used to verify properties like level-rank dualities, collapsing levels, and S-duality conjectures. The Zhu's algebra can be regarded as a ring associated with the Schur sector, and a surprising connection between certain Zhu's algebra and the Jacobi algebra of a hypersurface singularity is discovered. Finally, the Macdonald index is computed from the Kazhdan filtration of the -algebra.

Paper Structure

This paper contains 22 sections, 107 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Schur sector and VOA
  3. Quasi-lisse VOA
  4. 4d/2d correspondence
  5. Comments on constraints of 2d VOAs corresponding to 4d SCFTs
  6. Argyres-Douglas theories and $W$-algebras
  7. AD theories correspond to $W^{k'}( \mathfrak{g},f)$ algebras
  8. The character of $W$-algebra and the Schur index
  9. The $W$-algebra from the qDS reduction
  10. Character of $W$-algebra modules
  11. The simplified form
  12. Applications
  13. Explicit indices of $(A_1, G)$ theories
  14. $\tau \rightarrow 0$ limit and $a_{4d}-c_{4d}$
  15. Zhu's $C_2$ algebra and the ring of the Schur sector
  16. ...and 7 more sections

Figures (3)

  • Figure 1: A mapping of a 6d $(2,0)$ configuration to a 2d VOA, here $\mathfrak{g}$ is a simple Lie algebra, $\Phi$ is an irregular singularity, and $f$ represents a regular singularity. If $\Phi$ is of principal nilpotent type with type $h^\vee$ and an integer label $k$Xie:2012hsWang:2015mraWang:2018gvb, the VOA is just $W$-algebra $W^{k'}(\mathfrak{g},f)$ with $k'=-h^\vee+{h^\vee \over h^\vee+k}$.
  • Figure 2: Three Duality frames of theory \ref{['t1']}.
  • Figure 3: Duality frames of the theory constructed by gluing two $D_2[SU(3)]$ theory and one hypermultiplet.