4d N=2 SCFT and singularity theory Part I: Classification

TL;DR

The paper proposes a geometric program to classify 4d $\mathcal{N}=2$ SCFTs by mapping them to isolated three-fold singularities with a good $\mathbb{C}^*$ action. The Seiberg-Witten solution is realized through mini-versal deformations of these singularities, with Coulomb-branch data read from the Jacobi algebra and central charges computed from spectra and Milnor data; the monodromy and vanishing cycles encode the BPS quiver. A complete classification of isolated hypersurface singularities under the $\mathbb{C}^*$ constraint is presented, organized into 19 Type I–Type XIX families, with explicit infinite and sporadic solutions and their dualities, while extensions beyond hypersurfaces (ICIS, quotient by finite groups, and non-isolated cases) are outlined. The work provides a systematic geometric bridge between singularity theory and 4d SCFTs, enabling computation of SW geometry, central charges, BPS spectra, and RG networks, and sets the stage for broader classifications and connections to class $\mathcal S$ and other 2d-4d correspondences.

Abstract

This is the first of a series of papers in which we systematically use singularity theory to study four dimensional N=2 superconformal field theories. Our main focus in this paper is to identify what kind of singularity is needed to define a SCFT. The constraint for a hypersurface singularity has been found by Sharpere and Vafa, and here the complete set of solutions are listed using a related mathematical result of Stephen S. T. Yau and Yu. We also study other type of singularities such as the complete intersection, quotient of hypersurface singularity by a finite group and non-isolated singularity. We finally conjecture that any three dimensional rational Gorenstein graded isolated singularity should define a N=2 SCFT. We explain how to extract various interesting physical quantities such as Seiberg-Witten geometry, central charges, exact marginal deformations, BPS quiver, RG flow trajectory, etc from the properties of singularity.

Figures (7)

• Figure 1: S: the Coulomb branch moduli space for $\mathcal{N}=2$ theory. Here the curve on S means the discriminate locus; a is a point on which the Milnor fiber becomes singular, b is a point where the Milnor fiber is smooth.
• Figure 2: The discriminate locus of $A_2$ SCFT, here the parameters $\lambda_1, \lambda_2$ are taken to be real.
• Figure 3: Up: The three cycle becomes vanishing cycle in approaching the singular point $s_i$. Bottom: A path around the singular point on the moduli space.
• Figure 4: A set of loops around various $A_1$ singularities, and these generate the $\pi_1$ of the base of the fibration.
• Figure 5: Braiding operation which changes the basis of vanishing cycle.