Geometric classification of 4d $\mathcal{N}=2$ SCFTs
Matteo Caorsi, Sergio Cecotti
TL;DR
The paper develops a birational, geometry-driven framework to classify 4d ${\rm N}=2$ SCFTs via conical special geometries (CSG), showing their Coulomb branches are affine cones over log-Fano bases with trivial Hodge diamond and that the Coulomb chiral ring is typically a graded polynomial ring. It introduces a Universal Dimension Formula connecting Coulomb-dimension tuples to normal complex rays, monodromy, and the polarization data, and proves that the number of admissible dimensions grows quadratically with rank, $N(k)=\frac{2\zeta(2)\zeta(3)}{\zeta(6)}k^2+o(k^2)$. The analysis uses period maps, Siegel modular groups, and Class Field Theory to encode selection rules and principal vs non-principal polarizations, yielding precise constraints on allowed dimension sets and their interdependencies across rays and ranks. The results recover known rank-1 spectra, align with Springer theory for Lagrangian theories, and provide a structured, number-theoretically informed path toward tabulating admissible Coulomb-branch dimension tuples, including explicit tables for small ranks. The work thus offers a principled route to enumerate and check potential ${\cal N}=2$ SCFTs by geometric and arithmetic data, with implications for duality frames, conformal manifolds, and the landscape of non-Lagrangian theories.
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 $\mathbb{Q}$-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 $\mathscr{R}$ 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 $\{Δ_1,Δ_2,\cdots,Δ_k\}$ 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 $\{Δ_1,\cdots,Δ_k\}$'s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory. The number $\boldsymbol{N}(k)$ 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 $k$ $$ \boldsymbol{N}(k)=\frac{2\,ζ(2)\,ζ(3)}{ζ(6)}\,k^2+o(k^2). $$ 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 $\{Δ_1,\cdots,Δ_k\}$ 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.