Matrix Models, Complex Geometry and Integrable Systems. I

Abstract

We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric properties of planar matrix models, demonstrating that they are universally described in terms of integrable systems directly related to the theory of complex curves. We study the main ingredients of this geometric picture, suggesting that it can be generalized beyond one complex dimension, and formulate them in terms of the quasiclassical integrable systems, solved by construction of tau-functions or prepotentials. The complex curves and tau-functions of one- and two- matrix models are discussed in detail.

52 pages, 19 figures, based on several lecture courses and the talks at "Complex geometry and string theory" and the Polivanov memorial seminar; misprints corrected, references added

References in corpus (40)

• Dirichlet-Branes and Ramond-Ramond Charges
• Strings in flat space and pp waves from $\mathcal{N}=4$ Super Yang Mills
• The Bethe-Ansatz for N=4 Super Yang-Mills
• Perturbative Gauge Theory As A String Theory In Twistor Space
• Semiclassical quantization of rotating superstring in AdS_5 x S^5
• Matrix Models, Topological Strings, and Supersymmetric Gauge Theories
• How Algebraic Bethe Ansatz works for integrable model
• Topological Strings and Integrable Hierarchies
• Classical/quantum integrability in AdS/CFT
• A Perturbative Window into Non-Perturbative Physics
• Spin chains and string theory
• Spinning strings in AdS_5 x S^5 and integrable systems
• Integrable Structure of Interface Dynamics
• Large spin limit of AdS_5 x S^5 string theory and low energy expansion of ferromagnetic spin chains
• Higher genus correlators for the hermitian matrix model with multiple cuts
• Breakdown of universality in multi-cut matrix models
• WDVV-like equations in N=2 SUSY Yang-Mills Theory
• Algebraic Curve for the SO(6) sector of AdS/CFT
• Non-Perturbative Effects in Matrix Models and Vacua of Two Dimensional Gravity
• Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models
• Matching quantum strings to quantum spins: one-loop vs. finite-size corrections
• Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions
• The Algebraic Curve of 1-loop Planar N=4 SYM
• Normal random matrix ensemble as a growth problem
• Unification of All String Models with $c<1$
• Matrix models vs. Seiberg-Witten/Whitham theories
• Large N expansion of the 2-matrix model
• Complex Curve of the Two Matrix Model and its Tau-function
• On the Three-point Function in Minimal Liouville Gravity
• DV and WDVV
• Unified description of correlators in non-Gaussian phases of Hermitean matrix model
• Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains
• Holomorphic matrix integrals
• Complex Geometry of Matrix Models
• Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the KdV hierarchy
• String Theory or Field Theory?
• Electric-Magnetic Duality and WDVV Equations
• Matrix Models, Complex Geometry and Integrable Systems. II
• Determinant Formulas for Matrix Model Free Energy
• Matrix Model and Stationary Problem in Toda Chain

Cited by in corpus (7)

• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• On Microscopic Origin of Integrability in Seiberg-Witten Theory
• Reconstructing 1/2 BPS Space-Time Metrics from Matrix Models and Spin Chains
• Notes on the algebraic curves in (p,q) minimal string theory
• Matrix Models, Complex Geometry and Integrable Systems. II
• Comments on Slavnov products, Temperley-Lieb open spin chains, and KP tau functions
• Correlation functions and spectral curves in models of minimal gravity