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, 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].
Highlights
1.1 Schur and Macdonald indices in Argyres-Douglas theories as vacuum and t-refined vacuum WN charactersIn [10, 11, 28], Beem et al showed that the Schur indices in certain Argyres-Douglas theories are characters of irreducible highest-weight vacuum modules in a class of nonunitary WN minimal models
Due to the technical complication in the Higgsing method used in [85] to generate surface operators in Argyres-Douglas theories, only two infinite series of rank-2 Macdonald indices, the series that corresponds to the vacuum modules, and the series that corresponds to the next-to-vacuum modules of W3 characters, were conjectured
We show that (1) aspects of the W2 Schur index can be read directly from the paths, including the multiplicities of the Schur operators that contribute to the index, the composition of these operators in terms of Schur operators that are not derivatives of simpler ones and Schur operators that are derivatives of simpler ones, as well as the precise counting of the derivatives, and (2) that a refinement of these sum expressions in terms of a parameter t with a specific power that depends on the numbers of particles, gives a closed form expression for the corresponding Macdonald character
Summary
He shared the core idea of this paper with me back in 2017 It took me rather a long period to prepare and develop the tools we needed in this article, but I still feel lucky enough to finish this work with Omar. He first mentioned his illness to me in last September, when we started to summarize our results into an article. Omar Foda was one of the most important people for me during my early academic life He was a kind and active collaborator for me.
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