Closed form fermionic expressions for the Macdonald index

TL;DR

This paper derives explicit fermionic sum formulas for Macdonald indices of certain Argyres-Douglas and $ ext{W}_3$ theories, connecting 4D gauge theories, minimal models, and statistical mechanics paths.

Contribution

It provides the first closed-form fermionic expressions for Macdonald indices of specific Argyres-Douglas and $ ext{W}_3$ theories, linking them to minimal model characters and statistical mechanics.

Findings

Derived fermionic sum expressions for Macdonald indices.

Confirmed formulas match with 4D gauge theory computations.

Connected indices to paths in statistical mechanics models.

Abstract

We interpret aspects of the Schur indices, that were identified with characters of highest weight modules in Virasoro $(p,p')=(2,2k+3)$ minimal models for $k=1,2,\dots$, in terms of paths that first appeared in exact solutions in statistical mechanics. From that, we propose closed-form fermionic sum expressions, that is, $q, t$-series with manifestly non-negative coefficients, for two infinite-series of Macdonald indices of $(A_1,A_{2k})$ Argyres-Douglas theories that correspond to $t$-refinements of Virasoro $(p,p')=(2,2k+3)$ minimal model characters, and two rank-2 Macdonald indices that correspond to $t$-refinements of $\mathcal{W}_3$ non-unitary minimal model characters. Our proposals match with computations from 4D $\mathcal{N} = 2$ gauge theories \textit{via} the TQFT picture, based on the work of J Song arXiv:1509.06730.