Closed form fermionic expressions for the Macdonald index

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, . . . , 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 (A₁, 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 via the TQFT picture, based on the work of J Song [75].