Macdonald Index From Refined Kontsevich-Soibelman Operator

We propose a refinement of the Kontsevich-Soibelman operator for a class of ``special'' 4d $\mathcal{N}=2$ superconformal field theories characterized by the following conditions: (1) their Coulomb branch admits a source/sink chamber, i.e., a chamber in which the BPS quiver consists of only source and sink nodes, (2) The nodes with valency greater than 2 of the BPS quiver in a source/sink chamber are either all sources or all sinks. We present strong evidence that the trace of this refined operator is related to the Macdonald index of the theory. In particular, we conjecture closed form expressions for the Macdonald indices of the $(A_1,\mathfrak{g})$ Argyres-Douglas theories for any simply-laced Lie algebra $\mathfrak{g}$.