Description
In recent work, with M. J. Kang and J. Song, we conjecture that the Macdonald index (thus the Schur index) and the Higgs branch of a 4d N=2 SCFT can be obtained from an algebro-geometric object: a bifiltered affine scheme. This scheme, in practice a by-word for a collection of polynomial equations, is determined from nilpotency/decoupling relations in the operator product expansion of the SCFT. We propose such a scheme for a variety of theories, with particular focus on Argyres--Douglas theories where the Higgs branch is a point, and demonstrate how to recover the physical data. Although the associated scheme typically admits continuous deformations, we find that a geometric extremization principle uniquely fixes these moduli, thereby providing a possible geometric route toward a classification of 4d N=2 SCFTs.
