The Characteristic Dimension of Four-Dimensional $${\mathcal {N}}$$ = 2 SCFTs

In this paper we introduce the characteristic dimension of a four dimensional $\mathcal{N}=2$ superconformal field theory, which is an extraordinary simple invariant determined by scaling dimensions its Coulomb branch operators. We prove that only nine values are allowed, $-\infty$, 1 ,6/5, 4/3, 3/2, 2, 3, 4, and 6, thus giving new organizing principle to vast landscape 4d $\mathcal N=2$ SCFTs. Whenever differs from or very constrained special K\"ahler geometries (i.e. isotrivial, diagonal rigid) compatible with corresponding set extremely special, maximally strongly coupled, BPS spectra allowed for theories realize them. Our discussion applies arbitrary rank, i.e. branches any complex dimension. Along way, predict existence $\mathcal{N}=3$ rank two non-trivial one-form symmetries.