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

## TL;DR

This paper explores the structure of Argyres-Douglas theories through their associated $W$-algebras, deriving indices and uncovering algebraic connections that facilitate verification of dualities and properties.

## Contribution

It provides explicit formulas for the Schur and Macdonald indices from $W$-algebras and reveals a novel link between Zhu's $C_2$ algebra and hypersurface singularities.

## Key findings

- Schur index expressed via vacuum character of $W$-algebra
- Connection between Zhu's $C_2$ algebra and Jacobi algebra of hypersurface singularity
- Macdonald index derived from Kazhdan filtration

## 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.