Macdonald Index From Refined Kontsevich-Soibelman Operator

TL;DR

The paper proposes a refined KS operator $O(q,T)$ for a class of $4d\,\mathcal{N}=2$ SCFTs with source/sink Coulomb-branch chambers and conjectures that the Macdonald index is given by $I_M(q,T,z_1,...,z_{N_f})=(q)_\infty^{r}(qT)_\infty^{r}\mathrm{Tr}\,O(q,T)$, aligning with the Schur limit as $T\to1$. It develops the refined operator using $E_{q,T}$ on a BPS quiver and confirms the conjecture through explicit calculations for $(A_1,A_k)$, $(A_1,D_k)$, and $(A_1,E_n)$ theories, obtaining closed forms or series that match known Macdonald indices and uncover nontrivial identities. The results provide a concrete bridge between refined wall-crossing data and protected subsector counting, enabling Macdonald-index computations in a broad class of Argyres-Douglas theories and suggesting chamber-independent, mutation-resilient structures. The work motivates further checks, chamber-independence formalisms, and extensions to a wider landscape of 4d $\mathcal{N}=2$ SCFTs.

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 $\mathfrak{g}$.

Figures (7)

• Figure 1: BPS quiver of the $(A_1,A_{2n})$ theory in the source/sink chamber. Figure adapted from Cordova:2015nma.
• Figure 2: BPS quiver of the $(A_1,A_{2n+1})$ theory in the source/sink chamber. Figure adapted from Cordova:2015nma.
• Figure 3: BPS quiver of the $(A_1,D_{2n+1})$ theory in the source/sink chamber. Figure adapted from Cordova:2015nma.
• Figure 4: BPS quiver of the $(A_1,D_{2n+2})$ theory in the source/sink chamber. Figure adapted from Cordova:2015nma.
• Figure 5: BPS quiver of the $(A_1,E_6)$ theory in the source/sink chamber. Figure adapted from Cordova:2015nma.