Testing Macdonald Index as a Refined Character of Chiral Algebra

TL;DR

This paper tests the proposal that the Macdonald index serves as a refined character of the dual chiral algebra in certain Argyres-Douglas theories, extending analysis to higher ranks and surface operators.

Contribution

It extends the analysis of Macdonald indices as refined characters to higher rank theories and surface operators, providing evidence for Song's proposal in specific modules.

Findings

Refined characters in higher rank minimal models are established from dual theories.

Evidence supports Song's proposal in some simple modules at finite parameters.

Discussion of mismatches observed in the approach.

Abstract

We test in $(A_{n-1},A_{m-1})$ Argyres-Douglas theories with $\mathrm{gcd}(n,m)=1$ the proposal of Song's in arXiv:1612.08956 that the Macdonald index gives a refined character of the dual chiral algebra. In particular, we extend the analysis to higher rank theories and Macdonald indices with surface operator, via the TQFT picture and Gaiotto-Rastelli-Razamat's Higgsing method. We establish the prescription for refined characters in higher rank minimal models from the dual $(A_{n-1},A_{m-1})$ theories in the large $m$ limit, and then provide evidence for Song's proposal to hold (at least) in some simple modules (including the vacuum module) at finite $m$. We also discuss some observed mismatch in our approach.