Argyres-Douglas Theories, the Macdonald Index, and an RG Inequality

TL;DR

This paper proposes explicit formulas for the Macdonald indices of specific Argyres-Douglas theories, revealing new symmetries, operator relations, and connections to Higgs branch Hilbert series, with implications for understanding N=2 superconformal theories.

Contribution

It introduces conjectured closed-form expressions for the Macdonald indices of (A_1, A_{2n-3}) and (A_1, D_{2n}) AD theories using deformed Macdonald polynomials, supported by multiple non-trivial checks.

Findings

Compatibility with S-dualities

Symmetry enhancement at special n values

Equivalence of Hall-Littlewood limits and Higgs branch Hilbert series

Abstract

We conjecture closed-form expressions for the Macdonald limits of the superconformal indices of the (A_1, A_{2n-3}) and (A_1, D_{2n}) Argyres-Douglas (AD) theories in terms of certain simple deformations of Macdonald polynomials. As checks of our conjectures, we demonstrate compatibility with two S-dualities, we show symmetry enhancement for special values of n, and we argue that our expressions encode a non-trivial set of renormalization group flows. Moreover, we demonstrate that, for certain values of n, our conjectures imply simple operator relations involving composites built out of the SU(2)_R currents and flavor symmetry moment maps, and we find a consistent picture in which these relations give rise to certain null states in the corresponding chiral algebras. In addition, we show that the Hall-Littlewood limits of our indices are equivalent to the corresponding Higgs branch Hilbert series. We explain this fact by considering the S^1 reductions of our theories and showing that the equivalence follows from an inequality on monopole quantum numbers whose coefficients are fixed by data of the four-dimensional parent theories. Finally, we comment on the implications of our work for more general N=2 superconformal field theories.