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 $(p,p^{\prime })=(2,2k+3)$ 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 $(p,p^{\prime })=(2,2k+3)$ 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.

arXiv comment30 pages, 15 figures; Definitions and references added in v3 (eulogy added in v4 for Prof. Omar Foda, who passed away on 4 May 2020)

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• Infinitely many 4d $\mathcal{N}=2$ SCFTs with $a=c$ and beyond
• Argyres-Douglas Theories in Class S Without Irregularity
• Exact Operator Map from Strong Coupling to Free Fields: Beyond Seiberg-Witten Theory
• A landscape of 4d N=1 SCFTs with a=c