Free Energy of the Two-Matrix Model/dToda Tau-Function

Abstract

We provide an integral formula for the free energy of the two-matrix model with polynomial potentials of arbitrary degree (or formal power series). This is known to coincide with the tau-function of the dispersionless two--dimensional Toda hierarchy. The formula generalizes the case studied by Kostov, Krichever, Mineev-Weinstein, Wiegmann, Zabrodin and separately Takhtajan in the case of conformal maps of Jordan curves. Finally we generalize the formula found in genus zero to the case of spectral curves of arbitrary genus with certain fixed data.

Ver 2: 18 pages added important formulas for higher genus spectral curves, few typos removed (and few added). Ver 3: 19 pages (minor changes). Typos removed, added appendix and improved exposition Ver 4: 19 pages, minor corrections. Version submitted Ver 4; corrections prompted by referee and accepted in Nuclear Phys. B

References in corpus (4)

• Large N expansion of the 2-matrix model
• Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem
• Mixed Correlation Functions of the Two-Matrix Model
• Integrability in SFT and new representation of KP tau-function

Cited by in corpus (27)

• All genus correlation functions for the hermitian 1-matrix model
• Hermitean matrix model free energy: Feynman graph technique for all genera
• Free energy topological expansion for the 2-matrix model
• Large N expansion for the 2D Dyson gas
• Matrix eigenvalue model: Feynman graph technique for all genera
• Algebraic methods in random matrices and enumerative geometry
• Master loop equations, free energy and correlations for the chain of matrices
• Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains
• CFT and topological recursion
• Geometry of Spectral Curves and All Order Dispersive Integrable System
• Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula
• Multi-instantons in minimal string theory and in matrix integrals
• Genus one contribution to free energy in hermitian two-matrix model
• $1/N^2$ correction to free energy in hermitian two-matrix model
• Loop equations for the semiclassical 2-matrix model with hard edges
• Universality of Nonperturbative Effects in c<1 Noncritical String Theory
• Large N expansion of the 2-matrix model, multicut case
• Conifold geometries, topological strings and multi-matrix models
• Second and Third Order Observables of the Two-Matrix Model
• Topological expansion of the chain of matrices
• Conformal Mappings and Dispersionless Toda hierarchy II: General String Equations
• The complex Liouville string: worldsheet boundaries and non-perturbative effects
• Two-matrix model with semiclassical potentials and extended Whitham hierarchy
• Effective inverse spectral problem for rational Lax matrices and applications
• Another algebraic variational principle for the spectral curve of matrix models
• Fay Identities of Pfaffian Type for Hyperelliptic Curves
• Isomonodromic tau-function of Hurwitz Frobenius manifolds and its applications