Geometric classification of 4d $$ \mathcal{N}=2 $$ SCFTs

Abstract

Abstract The classification of 4d $$ \mathcal{N}=2 $$ 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 "Image missing" 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 {Δ1 , Δ2 , ⋯ , 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 {Δ1 , ⋯ , 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 k $$ \boldsymbol{N}(k)=\frac{2\zeta (2)\zeta (3)}{\zeta (6)}{k}^2+o\left({k}^2\right). $$ N k = 2 ζ 2 ζ 3 ζ 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 {Δ1 , ⋯ , 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.