Argyres-Douglas theories, the Macdonald index, and an RG inequality

A bstract We conjecture closed-form expressions for the Macdonald limits of the super-conformal indices of the ( A 1 , A 2 n − 3 ) and ( A 1 , D 2 n ) 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.