Testing Macdonald Index as a Refined Character of Chiral Algebra

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.