Remarks on Phase Transitions in Matrix Models and N=1 Supersymmetric Gauge Theory

TL;DR

This paper explores phase transitions in matrix models and their application to N=1 supersymmetric gauge theories, showing how matrix model solutions inform superpotential behavior and phase stability.

Contribution

It connects matrix model phase transitions to supersymmetric gauge theory superpotentials, demonstrating smoothness at critical points and phase preference.

Findings

Superpotential remains smooth at the phase transition point.

Two-cut phase is energetically favored below the critical scale.

U(1) coupling diverges due to massless monopoles, confirming Ferrari's formula.

Abstract

A hermitian one-matrix model with an even quartic potential exhibits a third-order phase transition when the cuts of the matrix model curve coalesce. We use the known solutions of this matrix model to compute effective superpotentials of an N=1, SU(N) supersymmetric Yang-Mills theory coupled to an adjoint superfield, following the techniques developed by Dijkgraaf and Vafa. These solutions automatically satisfy the quantum tracelessness condition and describe a breaking to SU(N/2) x SU(N/2) x U(1). We show that the value of the effective superpotential is smooth at the transition point, and that the two-cut (broken) phase is more favored than the one-cut (unbroken) phase below the critical scale. The U(1) coupling constant diverges due to the massless monopole, thereby demonstrating Ferrari's general formula. We also briefly discuss the implication of the Painleve II equation arising in the double scaling limit.