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

### Submission summary

 As Contributors: Wenbin Yan Arxiv Link: https://arxiv.org/abs/1904.09094v2 (pdf) Date accepted: 2021-03-26 Date submitted: 2021-03-09 07:39 Submitted by: Yan, Wenbin Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### 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.

Published as SciPost Phys. 10, 080 (2021)

### List of changes

Changes following suggestions of referee 1
1. corrected to quasi-lisse VOA
2. corrected all the typos including the ones pointed out by the referee
3. References are added accordingly. We also add an appendix explaining different names for generalized AD theories
4. Briefly explained the relation to 1903.07624. Actually more is discussed in our subsequent paper arXiv:1910.02281

Changes following suggestions of referee 2
1. corrections are made to make our results clear. For example, we stated clearly the triple (g,f,k') studied are for any simple Lie algebra g, any nilpotent orbit f, and boundary admissible level k'(which is also defined in the introduction)
2. We emphasis that although our PE formula is equivalent to the formula by mathematicians. Our form has the advantage that dualities are manifest in the PE form, and one can easily read off generators and singular vectors from the PE formula
3. added more references to math literature.
4. there are indeed typos and confusions in that part, and we corrected accordingly.
5. We made clear that right now the C2 algebra does not have a sharp 4d meaning. However, we still think it's worth study because not many 4d theories have no Higgs branch, but Schur sector is always there. Unfortunately we also don't have good description how C2 algebra behaves under quantum Hamiltonian reduction.

### Submission & Refereeing History

Resubmission 1904.09094v2 on 9 March 2021
Submission 1904.09094v1 on 12 January 2021

## Reports on this Submission

### Report

The authors addressed the points in the previous report. I recommend the paper for publication.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Report

In my opinion the paper can be published as is.

• validity: good
• significance: good
• originality: good
• clarity: ok
• formatting: good
• grammar: good