Argyres-Douglas Theories, Macdonald Indices and Arc Space of Zhu Algebra

TL;DR

This paper connects the Macdonald index of 4d $\\ ext{N}=2$ SCFTs with the arc space Hilbert series of Zhu algebras, proposing a formula for $(A_1,D_{2n+1})$ Argyres-Douglas theories and verifying it through identities and RG flow checks.

Contribution

It introduces a novel relation between Macdonald indices and arc space Hilbert series, and conjectures a new formula for specific Argyres-Douglas theories with supporting proofs.

Findings

Derived a conjectural formula for the Macdonald index of $(A_1,D_{2n+1})$ theories.

Proved new $q$-series identities to match the Schur limit.

Validated the formula through consistency checks with known limits and RG flows.

Abstract

In this paper, we relate the MacDonald index of a 4d $\mathcal{N}=2$ SCFT with the Hilbert series of the arc space of the Zhu algebra of the corresponding Schur VOA. Using this, we conjecture a simple formula for the MacDonald index of $(A_1,D_{2n+1})$ Argyres-Douglas theory. We perform checks of the formula against the known Schur limits and RG flows. To match the Schur limit, we prove new $q$-series identities.