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

Matteo Caorsi, Sergio Cecotti
2018 Journal of High Energy Physics  
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 R is a graded polynomial ring generated by global holomorphic functions u_i of dimension Δ_i.
more » ... 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 k 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 {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. We illustrate the various aspects with several examples and perform a number of explicit checks. We include tables of dimensions for the first few k's.
doi:10.1007/jhep07(2018)138 fatcat:2quy4tbwirhh5hcqpktn7qljii