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The sorting is complete only for the simplest case, with one complex dimension.","posed_since":"","precise":"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.","problem_ref":null,"references":"","settled_by":"A proven complete list of rank-2 Coulomb branch geometries with each entry matched to an SCFT or excluded.","status_note":"Rank 1 is classified (2016); rank-2 lists (Martone 2021; Kaidi, Martone, Rastelli, Weaver 2022) are not proven complete.","title":"Classification of rank-2 four-dimensional $N=2$ superconformal 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