Macdonald Index From Refined Kontsevich-Soibelman Operator

TL;DR

This paper introduces a refined Kontsevich-Soibelman operator for certain 4d N=2 superconformal theories and conjectures its trace relates to the Macdonald index, providing explicit formulas for Argyres-Douglas theories.

Contribution

It proposes a new refined operator framework and conjectures a direct link to Macdonald indices, including explicit formulas for a class of Argyres-Douglas theories.

Findings

Conjectured closed-form expressions for Macdonald indices of (A_1, g) Argyres-Douglas theories.

Evidence supporting the relation between the refined operator's trace and the Macdonald index.

Identification of special conditions on BPS quivers for the refinement to apply.

Abstract

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 $\mathfrak{g}$.