Classification of rank-2 four-dimensional $N=2$ superconformal theories
In plain words
Four-dimensional theories with extra supersymmetry (a symmetry pairing each force particle with a matter particle) can be sorted by the shape of their space of vacua. The sorting is complete only for the simplest case, with one complex dimension.
Precise statement
Classify 4D $N=2$ SCFTs whose Coulomb branch has complex dimension 2 by their scale-invariant special Kahler geometry (fibrations by 2-dimensional abelian varieties, their stratification, and allowed Coulomb branch scaling dimensions), extending the rank-1 classification of Argyres, Lotito, Lu and Martone (2015-2016), and determine which geometries correspond to consistent SCFTs.
What would settle it
A proven complete list of rank-2 Coulomb branch geometries with each entry matched to an SCFT or excluded.
Status in the literature
Rank 1 is classified (2016); rank-2 lists (Martone 2021; Kaidi, Martone, Rastelli, Weaver 2022) are not proven complete.