Determine the four-dimensional charge map and test the Macdonald specialization

Determine directly in four dimensions whether the charge \(R+r\) in the Macdonald index of the \((A_2,D_{3n-2})\) Argyres–Douglas theories maps to either \(-J_{\bm v^{(1)}}\) or \(-J_{\bm v^{(2)}}\) in the proposed \(R\)-twisted three-dimensional reductions, and prove or refute the resulting Macdonald-index specialization for \(n>2\).

Background

The paper conjectures that the four-dimensional grading $R+r$ is identified with one of two Weyl-related three-dimensional topological charges. This conjecture is supported by the known $n=2$ Macdonald-index matching and by the uniform charge structure of the proposed three-dimensional theories.

For $n>2$, the paper has no independent four-dimensional calculation of the charge map. A direct computation of the four-dimensional superconformal index, particularly using proposed $\mathcal N=1$ Lagrangian descriptions, would test both the charge identification and the predicted refined fermionic character.