Phases and geometry of the N=1 A_2 quiver gauge theory and matrix models

TL;DR

This paper investigates the phases and geometric structures of the N=1 A_2 quiver gauge theory using matrix models and anomaly equations, revealing connections to Seiberg-Witten curves and effective superpotentials.

Contribution

It introduces a geometric framework linking anomaly equations to the Seiberg-Witten curve via meromorphic forms and periods, advancing understanding of N=1 gauge theories.

Findings

Meromorphic one-form sigma(z)dz defined on the curve Sigma

Effective superpotential evaluated using Dijkgraaf-Vafa conjecture

Sigma(z)dz has integer periods, indicating a related meromorphic function

Abstract

We study the phases and geometry of the N=1 A_2 quiver gauge theory using matrix models and a generalized Konishi anomaly. We consider the theory both in the Coulomb and Higgs phases. Solving the anomaly equations, we find that a meromorphic one-form sigma(z)dz is naturally defined on the curve Sigma associated to the theory. Using the Dijkgraaf-Vafa conjecture, we evaluate the effective low-energy superpotential and demonstrate that its equations of motion can be translated into a geometric property of Sigma: sigma(z)dz has integer periods around all compact cycles. This ensures that there exists on Sigma a meromorphic function whose logarithm sigma(z)dz is the differential. We argue that the surface determined by this function is the N=2 Seiberg-Witten curve of the theory.