Holomorphic matrix models

Abstract

This is a study of holomorphic matrix models, the matrix models which underlie the conjecture of Dijkgraaf and Vafa. I first give a systematic description of the holomorphic one-matrix model. After discussing its convergence sectors, I show that certain puzzles related to its perturbative expansion admit a simple resolution in the holomorphic set-up. Constructing a `complex' microcanonical ensemble, I check that the basic requirements of the conjecture (in particular, the special geometry relations involving chemical potentials) hold in the absence of the hermicity constraint. I also show that planar solutions of the holomorphic model probe the entire moduli space of the associated algebraic curve. Finally, I give a brief discussion of holomorphic $ADE$ models, focusing on the example of the $A_2$ quiver, for which I extract explicitly the relevant Riemann surface. In this case, use of the holomorphic model is crucial, since the Hermitian approach and its attending regularization would lead to a singular algebraic curve, thus contradicting the requirements of the conjecture. In particular, I show how an appropriate regularization of the holomorphic $A_2$ model produces the desired smooth Riemann surface in the limit when the regulator is removed, and that this limit can be described as a statistical ensemble of `reduced' holomorphic models.

45 pages, reference added

References in corpus (13)

• Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
• Matrix Models, Topological Strings, and Supersymmetric Gauge Theories
• Chiral Rings and Anomalies in Supersymmetric Gauge Theory
• On Geometry and Matrix Models
• A Large N Duality via a Geometric Transition
• A Geometric Unification of Dualities
• Gravitational corrections in supersymmetric gauge theory and matrix models
• Phases of N=1 Supersymmetric Gauge Theories and Matrices
• Matrix models vs. Seiberg-Witten/Whitham theories
• Gauge Invariant Matrix Model for the Â-D-Ê Closed Strings
• Adding Fundamental Matter to ``Chiral Rings and Anomalies in Supersymmetric Gauge Theory''
• Unoriented Strings, Loop Equations, and N=1 Superpotentials from Matrix Models
• Konishi anomaly and N=1 effective superpotentials from matrix models

Cited by in corpus (38)

• Method of Generating q-Expansion Coefficients for Conformal Block and N=2 Nekrasov Function by beta-Deformed Matrix Model
• An A_r threesome: Matrix models, 2d CFTs and 4d N=2 gauge theories
• The Spectral Curve of the Lens Space Matrix Model
• Intersecting brane solutions in string and M-theory
• Delayed Deconfinement and the Hawking-Page Transition
• (Anti)symmetric matter and superpotentials from IIB orientifolds
• Geometric transitions and integrable systems
• Large N Duality, Lens Spaces and the Chern-Simons Matrix Model
• Holomorphic matrix integrals
• Non-Perturbative Quantum Geometry II
• Non-perturbative effects and wall-crossing from topological strings
• Cubic curves from matrix models and generalized Konishi anomalies
• Complex saddles in the Gross-Witten-Wadia matrix model
• Matrix Model Description of Laughlin Hall States
• Determination of S-curves with applications to the theory of nonhermitian orthogonal polynomials
• Higher order loop equations for A_r and D_r quiver matrix models
• Chiral field theories from conifolds
• Phases and geometry of the N=1 A_2 quiver gauge theory and matrix models
• Chiral field theories, Konishi anomalies and matrix models
• Special geometry of local Calabi-Yau manifolds and superpotentials from holomorphic matrix models
• Octonions, G_2 Symmetry, Generalized Self-Duality and Supersymmetries in Dimensions D \le 8
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• Fine structure in the large n limit of the non-hermitian Penner matrix model
• Quasi BPS Wilson loops, localization of loop equation by homology and exact beta function in the large N limit of SU(N) Yang-Mills theory
• Matrix Models, Argyres-Douglas singularities and double scaling limits
• Multiple phases and meromorphic deformations of unitary matrix models
• Rolling G_2 Moduli
• Phase transitions and Wilson loops in antisymmetric representations in Chern-Simons-matter theory
• Quivers via anomaly chains
• Puzzles for Matrix Models of Chiral Field Theories
• Remarks on the analytic structure of supersymmetric effective actions
• Exact beta function from the holographic loop equation of large-N QCD_4
• Pure Spinor Integration from the Collating Formula
• Konishi Anomalies and Curves without Adjoints
• Spectral curves in gauge/string dualities: integrability, singular sectors and regularization
• Relating prepotentials and quantum vacua of N=1 gauge theories with different tree-level superpotentials
• Quantum Geometries of $A_2$
• Unitary Matrix Model of a Chiral [SU(N)]^K Gauge Theory