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), ..., 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), ..., Delta(k)}'s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory. The number N(k) of dimensions allowed in rank k is given by a certain sum of the Erdos-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large k N(k) = 2 zeta(2)zeta(3)/zeta(6) k(2) + o(k(2)). 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), ..., 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.