Branched Matrix Models and the Scales of Supersymmetric Gauge Theories

Abstract

In the framework of the matrix model/gauge theory correspondence, we consider supersymmetric U(N) gauge theory with $U(1{)}^N$ symmetry breaking pattern. Due to the presence of the Veneziano--Yankielowicz effective superpotential, in order to satisfy the $F$--term condition ${\sum }_i{S}_i=0$, we are forced to introduce additional terms in the free energy of the corresponding matrix model with respect to the usual formulation. This leads to a matrix model formulation with a cubic potential which is free of parameters and displays a branched structure. In this way we naturally solve the usual problem of the identification between dimensionful and dimensionless quantities. Furthermore, we need not introduce the $\mathbb{N}=1$ scale by hand in the matrix model. These facts are related to remarkable coincidences which arise at the critical point and lead to a branched bare coupling constant. The latter plays the role of the $\mathbb{N}=1$ and $\mathbb{N}=2$ scale tuning parameter. We then show that a suitable rescaling leads to the correct identification of the $\mathbb{N}=2$ variables. Finally, by means of the the mentioned coincidences, we provide a direct expression for the $\mathbb{N}=2$ prepotential, including the gravitational corrections, in terms of the free energy. This suggests that the matrix model provides a triangulation of the istanton moduli space.

1+18 pages, harvmac. Added discussion on the CSW relative shifts of theta vacua and the odd phases at the critical point. References added and typos corrected

References in corpus (20)

• Chiral Rings and Anomalies in Supersymmetric Gauge Theory
• A Perturbative Window into Non-Perturbative Physics
• Multi-Instanton Calculus and Equivariant Cohomology
• Gravitational corrections in supersymmetric gauge theory and matrix models
• Perturbative Computation of Glueball Superpotentials
• Phases of N=1 Supersymmetric Gauge Theories and Matrices
• Perturbative Analysis of Gauged Matrix Models
• Matrix models vs. Seiberg-Witten/Whitham theories
• The Dijkgraaf-Vafa prepotential in the context of general Seiberg-Witten theory
• N=1 Supersymmetry, Deconstruction, and Bosonic Gauge Theories
• Matrix model approach to the N=2 U(N) gauge theory with matter in the fundamental representation
• Seiberg-Witten prepotential from instanton counting
• The N=2 gauge theory prepotential and periods from a perturbative matrix model calculation
• On the Matter of the Dijkgraaf--Vafa Conjecture
• Effective Superpotentials via Konishi Anomaly
• Loop equations, matrix models, and N=1 supersymmetric gauge theories
• Seiberg--Witten Duality in Dijkgraaf--Vafa Theory
• On the property of Cachazo-Intriligator-Vafa prepotential at the extremum of the superpotential
• Chiral Ring Of Sp(N) And SO(N) Supersymmetric Gauge Theory In Four Dimensions
• The Glueball Superpotential

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