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

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 G2 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.