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 = δ p,q . With some plausible restrictions, this means that the Coulomb branch chiral ring [InlineMediaObject not available: see fulltext.] is a graded polynomial ring generated by global holomorphic functions u i of dimension Δ i . The coarse-grained classification of the CSG consists in listing the (finitely many) dimension k-tuples Δ 1 , Δ 2 , ⋯ , Δ 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 Δ 1 , ⋯ , Δ 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 Erdös-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large kN(k)=2ζ(2)ζ(3)ζ(6)k2+o(k2). 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 Δ 1 , ⋯ , Δ 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.