Closed form fermionic expressions for the Macdonald index


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, A2k ) 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].

A preprint version of the article is available at ArXiv.


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Foda, O., Zhu, R. Closed form fermionic expressions for the Macdonald index. J. High Energ. Phys. 2020, 157 (2020).

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