Abstract
The classification of 4d \( \mathcal{N}=2 \) 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 hp,q = δp,q. With some plausible restrictions, this means that the Coulomb branch chiral ring is a graded polynomial ring generated by global holomorphic functions ui of dimension Δi. The coarse-grained classification of the CSG consists in listing the (finitely many) dimension k-tuples {Δ1, Δ2, ⋯ , Δk} 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, ⋯ , Δk}’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
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, ⋯ , Δk} 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.
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Caorsi, M., Cecotti, S. Geometric classification of 4d \( \mathcal{N}=2 \) SCFTs. J. High Energ. Phys. 2018, 138 (2018). https://doi.org/10.1007/JHEP07(2018)138
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DOI: https://doi.org/10.1007/JHEP07(2018)138