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 kNk=2ζ2ζ3ζ6k2+ok2.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\boldsymbol{N}(k)=\\frac{2\\zeta (2)\\zeta (3)}{\\zeta (6)}{k}^2+o\\left({k}^2\\right). $$\\end{document}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.
Highlights
The coarsegrained classification of the conic special geometries (CSG) consists in listing the dimension k-tuples {∆1, ∆2, · · ·, ∆k} which are realized as Coulomb branch dimensions of some rank-k CSG: this is the problem we address in this paper
The present paper belongs to this second line of thought: it is meant to be a contribution to the geometric classification of CSG with applications to N = 2 superconformal field theories (SCFT)
In the physical applications we are mainly interested in Kahler cones with quasi-regular Sasaki bases B, that is, with compact Reeb vector orbits, so that R generates a compact group of isometries U(1)R which we identify with the R-symmetry group which must be compact on physical grounds
Summary
The coarse-grained classification of CSG aims to list the allowed Coulomb branch dimension k-tuples {∆1, · · · , ∆k} for each rank k ∈ N. An even less ambitious program is to list the finitelymany real numbers ∆ which may be the dimension of a Coulomb branch generator in a N = 2 SCFT of rank at most k. After the submission of the present paper, the article [30] appeared in the arXiv where examples of N = 2 theories with non-free chiral rings are constructed. Those examples are in line with our geometric discussion being related to the free ring case by a finite quotient (gauging). Is roughly independent of k up to a few percent modulation, see e.g. table 2
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