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

by Dan Xie, Wenbin Yan

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Submission summary

Authors (as registered SciPost users): Wenbin Yan
Submission information
Preprint Link:  (pdf)
Date accepted: 2021-03-26
Date submitted: 2021-03-09 07:39
Submitted by: Yan, Wenbin
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


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.

Reports on this Submission

Anonymous Report 2 on 2021-3-18 (Invited Report)


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

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

Anonymous Report 1 on 2021-3-10 (Invited Report)


In my opinion the paper can be published as is.

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

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Anonymous on 2021-03-10  [id 1295]

In my opinion the paper can be published as is.

Anonymous on 2021-03-10  [id 1294]

In my opinion the paper can be published as is.