# Vertex operator algebras of Argyres-Douglas theories from M5-branes

## TL;DR

This paper explores the connection between Argyres-Douglas theories engineered from M5-branes and their associated vertex operator algebras, verifying conjectures through index computations and uncovering their algebraic and geometric structures.

## Contribution

It proposes and verifies that the VOAs for certain AD theories are W-algebras, linking indices to VOA characters and relating the associated variety to the Higgs branch.

## Key findings

- Schur index matches the vacuum character of the VOA.
- Hall-Littlewood index computes the Higgs branch Hilbert series.
- Closed-form expressions for indices when b=h.

## Abstract

We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d N=(2, 0) theory of type $J$ on a punctured sphere. We denote the AD theories as $(J^b[k],Y)$, where $J^b[k]$ and $Y$ represent an irregular and a regular singularity respectively. We restrict to the `minimal' case where $J^b[k]$ has no associated mass parameters, and the theory does not admit any exactly marginal deformations. The VOA corresponding to the AD theory is conjectured to be the W-algebra $\mathcal{W}^{k_{2d}}(J,Y)$, where $k_{2d}=-h+ \frac{b}{b+k}$ with $h$ being the dual Coxeter number of $J$. We verify this conjecture by showing that the Schur index of the AD theory is identical to the vacuum character of the corresponding VOA, and the Hall-Littlewood index computes the Hilbert series of the Higgs branch. We also find that the Schur and Hall-Littlewood index for the AD theory can be written in a simple closed form for $b=h$. We also test the conjecture that the associated variety of such VOA is identical to the Higgs branch. The M5-brane construction of these theories and the corresponding TQFT structure of the index play a crucial role in our computations.