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

TL;DR

This work advances the classification of rank-2 N=2 SCFTs by analyzing 2D RS K geometries for the $y^2=x^5$ degeneration of genus-2 Seiberg-Witten curves. By imposing scale invariance and the integrability equation, the authors derive 28 candidate solutions parameterized by $s$, and, after enforcing the $Z$-consistency condition across all singularity channels, isolate 13 physically viable curves. Among these, one has a marginal coupling corresponding to the known $Sp(2)$ conformal theory, while the remaining 12 are new isolated fixed points, enriching the landscape of 4D $ obreak{N=2}$ SCFTs described by genus-2 Coulomb branches. The results demonstrate a concrete algebraic-path approach to cataloging interacting fixed points via maximal degeneration geometries and provide a framework for identifying additional SCFTs through moduli-tuning in related theories.

Abstract

We continue the classification of 2-dimensional scale-invariant rigid special Kahler (RSK) geometries. This classification was begun in [hep-th/0504070] where singularities corresponding to curves of the form y^2=x^6 with a fixed canonical basis of holomorphic one forms were analyzed. Here we perform the analysis for the y^2=x^5 type singularities. (The final maximal singularity type, y^2=x^3(x-1)^3, will be analyzed in a later paper.) These singularities potentially describe the Coulomb branches of N=2 supersymmetric field theories in four dimensions. We show that there are only 13 solutions satisfying the integrability condition (enforcing the RSK geometry of the Coulomb branch) and the Z-consistency condition (requiring massless charged states at singularities). Of these solutions, one has a marginal deformation, and corresponds to the known solution for certain Sp(2) gauge theories, while the rest correspond to isolated strongly interacting conformal field theories.