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

TL;DR

This paper classifies the types of singularities needed to define four-dimensional N=2 superconformal field theories, linking singularity theory with physical quantities like Seiberg-Witten geometry and central charges.

Contribution

It systematically identifies singularity conditions for SCFTs and proposes a conjecture relating rational Gorenstein singularities to N=2 SCFTs, expanding the mathematical framework for physical theories.

Findings

Complete set of hypersurface singularities for SCFTs listed

Conjecture that rational Gorenstein singularities define N=2 SCFTs

Methods to extract physical quantities from singularity properties

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.