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

TL;DR

The paper establishes a robust VOA framework for a broad class of Argyres-Douglas theories built from M5-branes, identifying the associated chiral algebras as W-algebras ${\cal W}^{k_{2d}}(J,Y)$ with $k_{2d}=-h+\frac{b}{b+k}$. It demonstrates that the Schur index of these 4d theories coincides with the vacuum character of the corresponding VOA and that the Hall-Littlewood index reproduces the Higgs-branch Hilbert series, including compact closed-form expressions when $b=h$. By leveraging the M5-brane construction and the TQFT structure of the index, the work also verifies that the associated variety of the VOA matches the Higgs branch in many cases, with additional cross-checks from 3d mirrors. The results unify several known AD theories and provide explicit character/HL-index formulas for both general and puncture-reduced theories, highlighting deep ties between 4d ${\cal N}=2$ dynamics and 2d VOA structure. The findings offer a powerful toolkit for computing protected sectors, Higgs-branch data, and dualities across a wide landscape of AD theories, and point to extensions to non-admissible levels and more refined indices.

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.

Figures (1)

• Figure 1: 3d Mirror descriptions for some Argyres-Douglas theories considered in this paper, whose flavor symmetry is $SU$. Here, the number $n$ assigned to the edges mean that there are $n$ copies of the bifundamental hypermultiplet. As usual, the overall $U(1)$ factor has to be modded out. $A$ is the general structure of a quiver whose flavor symmetry would be a simple $SU$ group. $B-F$ are examples. We also list the flavor central charge here, and the corresponding theory in terms of the label $(J^b[k],F)$.

Theorems & Definitions (2)

• Conjecture 1
• Conjecture 2