Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

4d N=2 SCFT and singularity theory Part II: Complete intersection

Bingyi Chen, Dan Xie, Shing-Tung Yau, Stephen S. -T. Yau, Huaiqing Zuo

TL;DR

The paper classifies 3-dimensional weighted homogeneous rational ICIS defined by two polynomials in five variables, establishing a complete list of 303 weight types and infinite families. By computing Milnor numbers $\mu$ and the monomial bases of the miniversal deformations, it furnishes the Coulomb branch spectra and Seiberg-Witten data for the resulting 4d $\mathcal{N}=2$ SCFTs. The connection to gauge theories is illustrated (e.g., affine $D_5$ quivers), and the authors prove an embedding-dimension bound $N\le 5$ to constrain the classification. This work provides a concrete, computable framework linking singularity theory to the landscape of 4d SCFTs via the Coulomb branch, enhancing our ability to enumerate and analyze these theories through their associated ICIS.

Abstract

We classify three dimensional isolated weighted homogeneous rational complete intersection singularities, which define many new four dimensional N=2 superconformal field theories. We also determine the mini-versal deformation of these singularities, and therefore solve the Coulomb branch spectrum and Seiberg-Witten solution.

4d N=2 SCFT and singularity theory Part II: Complete intersection

TL;DR

The paper classifies 3-dimensional weighted homogeneous rational ICIS defined by two polynomials in five variables, establishing a complete list of 303 weight types and infinite families. By computing Milnor numbers and the monomial bases of the miniversal deformations, it furnishes the Coulomb branch spectra and Seiberg-Witten data for the resulting 4d SCFTs. The connection to gauge theories is illustrated (e.g., affine quivers), and the authors prove an embedding-dimension bound to constrain the classification. This work provides a concrete, computable framework linking singularity theory to the landscape of 4d SCFTs via the Coulomb branch, enhancing our ability to enumerate and analyze these theories through their associated ICIS.

Abstract

We classify three dimensional isolated weighted homogeneous rational complete intersection singularities, which define many new four dimensional N=2 superconformal field theories. We also determine the mini-versal deformation of these singularities, and therefore solve the Coulomb branch spectrum and Seiberg-Witten solution.

Paper Structure

This paper contains 6 sections, 17 theorems, 343 equations, 3 figures.

Table of Contents

  1. Introduction
  2. ICIS and 4d $\mathcal{N}=2$ SCFT
  3. Preliminaries
  4. Deformation of singularities
  5. Homogeneous isolated complete intersection singularity
  6. Classification of three dimensional weighted homogeneous rational isolated complete intersection singularity

Key Result

Proposition 3.1

(Corollary 3.10, Ha) $(X, x)$ is called Cohen-Macaulay iff $H^1_{\{x\}}(\mathcal{O}_{X})=\cdots=H^{n-1}_{\{x\}}(\mathcal{O}_{X})=0$. $(X, x)$ is normal iff $H^1_{\{x\}}(\mathcal{O}_{X})=0$.

Figures (3)

  • Figure 1: 4d $\mathcal{N}=2$ SCFT described by the singularity $(f_1, f_2)=(z_1^2+z_2^2+z_3^2+z_4^2+z_5^{2N},z_1^2+2z_2^2+3z_3^2+4z_4^2+5z_5^{2N})$. Here the gauge group is $SU(2N)$.
  • Figure 2:
  • Figure :

Theorems & Definitions (36)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Theorem 3.1
  • proof
  • ...and 26 more