Geometric classification of 4d $N=2$ SCFTs

The classification of 4d $N=2$ SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG). Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected $Q$-factorial log-Fano variety with Hodge numbers $h^{p,q}={\delta }_{p,q}$. With some plausible restrictions, this means that the Coulomb branch chiral ring $R$ is a graded polynomial ring generated by global holomorphic functions $u_i$ of dimension ${\Delta }_i$. The coarse-grained classification of the CSG consists in listing the (finitely many) dimension $k$-tuples $\{{\Delta }_1,{\Delta }_2,\cdots ,{\Delta }_k\}$ which are realized as Coulomb branch dimensions of some rank-$k$ CSG: this is the problem we address in this paper. Our sheaf-theoretical analysis leads to an Universal Dimension Formula for the possible $\{{\Delta }_1,\cdots ,{\Delta }_k\}$'s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory. The number $N\left(k\right)$ of dimensions allowed in rank $k$ is given by a certain sum of the Erdös-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large $k$$$N\left(k\right)=\frac{2\zeta \left(2\right)\zeta \left(3\right)}{\zeta \left(6\right)}{k}^2+o\left(k^2\right).$$ In the special case $k=2$ our dimension formula reproduces a recent result by Argyres et al. Class Field Theory implies a subtlety: certain dimension $k$-tuples $\{{\Delta }_1,\cdots ,{\Delta }_k\}$ are consistent only if supplemented by additional selection rules on the electro-magnetic charges, that is, for a SCFT with these Coulomb dimensions not all charges/fluxes consistent with Dirac quantization are permitted. Since the arguments tend to be abstract, we illustrate the various aspects with several concrete examples and perform a number of explicit checks. We include detailed tables of dimensions for the first few $k$'s.