Classification of N=2 Superconformal Field Theories with Two-Dimensional Coulomb Branches

TL;DR

The paper advances a systematic framework to classify scale-invariant rigid special Kahler geometries in two complex dimensions, demonstrating that the problem reduces to solving a finite algebraic system derived from central-charge integrability, holomorphy, and single-valuedness. By analyzing genus-2 curves and exploiting reparametrization and scaling, the authors identify 31 candidate curves and impose a stringent $Z$-consistency condition that leaves 12 physically viable two-dimensional RSK geometries, some corresponding to known SCFTs (including a G$_2$-based theory and an SU(3) theory with a marginal coupling). The results substantiate the conjecture that scale-invariant RSK geometries describe the Coulomb branches of 4D $N=2$ SCFTs and provide a concrete, calculable map between geometric degenerations and massless BPS spectra, with implications for extending the classification to higher dimensions and for exploring deformations away from scale invariance. The study also highlights concrete connections to known SCFTs and suggests systematic avenues for completing the full two-dimensional classification and its extensions. Overall, the work significantly deepens the geometric understanding of 4D $N=2$ SCFTs and their Coulomb branches.

Abstract

We study the classification of 2-dimensional scale-invariant rigid special Kahler (RSK) geometries, which potentially describe the Coulomb branches of N=2 supersymmetric field theories in four dimensions. We show that this classification is equivalent to the solution of a set of polynomial equations by using an integrability condition for the central charge, scale invariance, constraints coming from demanding single-valuedness of physical quantities on the Coulomb branch, and properties of massless BPS states at singularities. We find solutions corresponding to lagrangian scale invariant theories--including the scale invariant G_2 theory not found before in the literature--as well as many new isolated solutions (having no marginal deformations). All our scale-invariant RSK geometries are consistent with an interpretation as effective theories of N=2 superconformal field theories, and, where we can check, turn out to exist as quantum field theories.