Matrix Model Superpotentials and Calabi-Yau Spaces: an ADE Classification

TL;DR

This paper connects matrix models, Calabi-Yau geometries, and ADE classifications of superconformal theories, providing new insights into singular geometries and resolutions relevant to string theory and gauge theories.

Contribution

It demonstrates the ADE classification of superpotentials via Calabi-Yau singularities and introduces new non-isolated geometries, developing techniques for small resolutions and blow-downs.

Findings

ADE superpotentials match matrix model superpotentials from Calabi-Yau singularities.

Discovery of new non-isolated 'hat' geometries related to ADE singularities.

Development of algorithms for small resolutions and conjecture on deformations of matrix factorizations.

Abstract

We use F. Ferrari's methods relating matrix models to Calabi-Yau spaces in order to explain Intriligator and Wecht's ADE classification of $\N=1$ superconformal theories which arise as RG fixed points of $\N = 1$ SQCD theories with adjoints. The connection between matrix models and $\N = 1$ gauge theories can be seen as evidence for the Dijkgraaf--Vafa conjecture. We find that ADE superpotentials in the Intriligator--Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yau's 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 techniques for performing small resolutions and small blow-downs, including an algorithm for blowing down exceptional $\PP^1$'s. In particular, we conjecture that small resolutions for isolated Gorenstein threefold singularities can be obtained by deforming matrix factorizations for simple surface singularities -- and prove this in the length 1 and length 2 cases.