Matrix model superpotentials and ADE singularities

TL;DR

This paper connects matrix models and Calabi-Yau geometries to explain ADE classifications of certain superconformal theories, revealing new singular geometries and providing methods for resolving complex singularities.

Contribution

It demonstrates the exact match between ADE superpotentials and matrix model superpotentials from Calabi-Yau singularities, and introduces new geometries with non-isolated singularities.

Findings

ADE superpotentials match matrix model superpotentials

Discovery of new `hat' geometries with non-isolated singularities

Development of an algorithm for resolving Gorenstein threefold singularities

Abstract

We use F. Ferrari's methods relating matrix models to Calabi-Yau spaces in order to explain much of Intriligator and Wecht's ADE classification of $\N=1$ superconformal theories which arise as RG fixed points of $\N = 1$ SQCD theories with adjoints. We find that ADE superpotentials in the Intriligator-Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yaus with corresponding ADE singularities. Moreover, in the additional $\Hat{O}, \Hat{A}, \Hat{D}$ and $\Hat{E}$ cases we find new singular geometries. These `hat' geometries are closely related to their ADE counterparts, but feature non-isolated singularities. As a byproduct, we give simple descriptions for small resolutions of Gorenstein threefold singularities in terms of transition functions between just two coordinate charts. To obtain these results, we develop an algorithm for blowing down exceptional $\PP^1$s, described in the appendix.