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

### Submission summary

 As Contributors: Wenbin Yan Arxiv Link: https://arxiv.org/abs/1904.09094v1 (pdf) Date submitted: 2021-01-12 23:57 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.

###### Current status:
Editor-in-charge assigned