Integrability and Matrix Models

Abstract

The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Discrete 1-matrix, 2-matrix, ``conformal'' (multicomponent) and Kontsevich models are considered in some detail, together with the Ward identites (``W-constraints''), determinantal formulas and continuum limits, taking one kind of models into another. Subtle points and directions of the future research are also discussed.

146 pages, ITEP-M2/93 & ITFA 93-10, LaTeX, minor misprints are corrected

References in corpus (18)

• Two Dimensional Gauge Theories Revisited
• The $n$-component KP hierarchy and representation theory
• Hamiltonian systems of Calogero type and two dimensional Yang-Mills theory
• Conformal Matrix Models as an Alternative to Conventional Multi-Matrix Models
• Induced QCD at Large N
• Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models
• A Unitary S-matrix for 2D Black Hole Formation and Evaporation
• Exact Solution of Induced Lattice Gauge Theory at Large $N$
• Area Law and Continuum Limit in "Induced QCD"
• Pair Correlator in the Itzykson-Zuber Integral
• Exact solution of d=1 Kazakov-Migdal induced gauge theory
• On Connection between Topological Landau-Ginzburg Gravity and Integrable Systems
• On Exact Evaluation of Path Integrals
• Continuum Limits of ``Induced QCD": Lessons of the Gaussian Model at d=1 and Beyond
• Landau-Ginzburg Topological Theories in the Framework of GKM and Equivalent Hierarchies
• Excitation Scattering in Integrable Models and Hall-Littlewood-Kerov Polynomials
• $c=r_g$ Theories of $W_G$-Gravity: The Set of Observables as a Model of Simply Laced $G$
• Symmetry and Observables in Induced QCD

Cited by in corpus (193)

• Topological Strings and Integrable Hierarchies
• Liouville Field Theory -- A decade after the revolution
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• Hamiltonian systems of Calogero type and two dimensional Yang-Mills theory
• Integrability and Seiberg-Witten Theory: Curves and Periods
• Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory
• Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model
• Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Conformal blocks as Dotsenko-Fateev Integral Discriminants
• A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model
• Nekrasov Functions from Exact BS Periods: the Case of SU(N)
• W-symmetry, topological vertex and affine Yangian
• RG Equations from Whitham Hierarchy
• Instantons and Merons in Matrix Models
• Exact vs. Semiclassical Target Space of the Minimal String
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• Algebra of differential operators associated with Young diagrams
• On KP-integrable Hurwitz functions
• Ding-Iohara-Miki symmetry of network matrix models
• The Power of Nekrasov Functions
• A direct proof of AGT conjecture at beta = 1
• Explicit examples of DIM constraints for network matrix models
• Matrix model version of AGT conjecture and generalized Selberg integrals
• BGWM as Second Constituent of Complex Matrix Model
• Free fermions and tau-functions
• Late time physics of holographic quantum chaos
• An exact formula for general spectral correlation function of random Hermitian matrices
• On the complete perturbative solution of one-matrix models
• The Dijkgraaf-Vafa prepotential in the context of general Seiberg-Witten theory
• Replica field theories, Painleve transcendents, and exact correlation functions
• Correlators in tensor models from character calculus
• Ward identities and combinatorics of rainbow tensor models
• Integrability of Hurwitz Partition Functions. I. Summary
• On some integrals over the U(N) unitary group and their large N limit
• M-Theory of Matrix Models
• Matrix Models for Random Partitions
• Unified description of correlators in non-Gaussian phases of Hermitean matrix model
• Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy
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• Towards a proof of AGT conjecture by methods of matrix models
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• Virasoro Symmetry of Constrained KP Hierarchies
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• Can Renormalization Group Flow End in a Big Mess?
• Colored knot polynomials. HOMFLY in representation [2,1]
• Superintegrability of Kontsevich matrix model
• Sum rules for characters from character-preservation property of matrix models
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• Interpolating Matrix Models for WLZZ series
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• On matrix models and their $q$-deformations
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• FZZT branes in JT gravity and topological gravity
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• Exact 2-point function in Hermitian matrix model
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• From Simplified BLG Action to the First-Quantized M-Theory
• Bogolubov's Recursion and Integrability of Effective Actions
• The master T-operator for the Gaudin model and the KP hierarchy
• New Insights into Superintegrability from Unitary Matrix Models
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• Orthogonal and symplectic matrix integrals and coupled KP hierarchy
• Exact solution for mean energy of 2d Dyson gas at beta = 1
• On the structure of non-planar strong coupling corrections to correlators of BPS Wilson loops and chiral primary operators
• Matrix models with hard walls: Geometry and solutions
• From Kronecker to tableau pseudo-characters in tensor models
• $q$-Virasoro modular double and 3d partition functions
• Non-Abelian W-representation for GKM
• Statistics of thermal to shot noise crossover in chaotic cavities
• CFT approach to constraint operators for ($\beta$-deformed) hermitian one-matrix models
• Matrix model partition function by a single constraint
• Superintegrability in $\beta$-deformed Gaussian Hermitian matrix model from $W$-operators
• Non-Rational 2D Quantum Gravity II. Target Space CFT
• Around spin Hurwitz numbers
• Complex matrix model duality
• NSR Superstring Measures Revisited
• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• Hopf link invariants and integrable hierarchies
• Open intersection numbers, matrix models and MKP hierarchy
• On the property of Cachazo-Intriligator-Vafa prepotential at the extremum of the superpotential
• A New ``Dual'' Symmetry Structure of the KP Hierarchy
• $(q,t)$-deformed (skew) Hurwitz $\tau$-functions
• Higher order loop equations for A_r and D_r quiver matrix models
• The KP Hierarchy in Miwa Coordinates
• An analogue of Schur functions for the plane partitions
• Superintegrability as the hidden origin of Nekrasov calculus
• Solving q-Virasoro constraints
• Hamiltonian Formalism in the Presence of Higher Derivatives
• Position Space Equations for Banana Feynman Diagrams
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• Notes on the algebraic curves in (p,q) minimal string theory
• String Theory or Field Theory?
• Cauchy formula and the character ring
• Geometrical optics in nonlinear media and integrable equations
• Virasoro type algebraic structure hidden in the constrained discrete KP hierarchy
• Comment on integrability in Dijkgraaf-Vafa beta-ensembles
• Generalized Q-functions for GKM
• Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices
• Annulus Amplitudes in the Minimal Superstring
• On Equivalence of two Hurwitz Matrix Models
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• Challenges of Matrix Models
• Quantization of Harer-Zagier formulas
• Tau Functions and Matrix Integrals
• q-Painleve equation from Virasoro constraints
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
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• Virasoro versus superintegrability. Gaussian Hermitian model
• Bootstrapping the $S^5$ partition function
• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• Nonperturbative Model Of Liouville Gravity
• Matrix Quantum Mechanics on $S^1/{\mathbb{Z}}_2$
• On matrix-model approach to simplified Khovanov-Rozansky calculus
• Exact SUSY Wilson loops on $S^3$ from $q$-Virasoro constraints
• 3-Schurs from explicit representation of Yangian $Y({\widehat{\mathfrak{gl}}}_1)$. Levels 1-5
• From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators
• On the shapes of elementary domains or why Mandelbrot Set is made from almost ideal circles?
• Check-Operators and Quantum Spectral Curves
• Fermionic Quantum Gravity
• Hodge-GUE correspondence and the discrete KdV equation
• On skew tau-functions in higher spin theory
• Infinite dimensional geometry and quantum field theory of strings. III. Infinite dimensional W-geometry of a second quantized free string
• Elliptic $q,t$ matrix models
• Deviation from Alday-Maldacena Duality For Wavy Circle
• Universal Algebras of Hurwitz Numbers
• Natanzon-Orlov model and refined superintegrability
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• Note on a duality of topological branes
• From $r$-Spin Intersection Numbers to Hodge Integrals
• Towards elliptic deformation of $q,t$-matrix models
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• Algebra of quantum $\mathcal{C}$-polynomials
• Grothendieck's Dessins d'Enfants in a Web of Dualities. III
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
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• Towards the theory of Yangians
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• Virasoro and W-constraints for the $q$-KP hierarchy
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• Open/closed duality for FZZT branes in c=1
• The "Null-A" superintegrability for monomial matrix models
• Superintegrability and Kontsevich-Hermitian relation
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• Matrix Integrals, Symmetric Functions theory and matrix integrals
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• Asymptotically Free AdS3/CFT2
• Gauging of Geometric Actions and Integrable Hierarchies of KP Type
• On the relations between correlation functions in SYM/pp-wave correspondence
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• Large N and Bethe ansatz
• Genus expansion of matrix models and $\hslash$ expansion of $B$KP hierarchy
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• Vogel's universality and the classification problem for Jacobi identities
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• An hbar-expansion of the Toda hierarchy: a recursive construction of solutions
• Unitary matrix integrals, symmetric polynomials, and long-range random walks
• String Equations of the q-KP Hierarchy
• Duality and integrability of supermatrix model with external source
• String with noncommutative world-sheet and stringy instantons
• Comments on Slavnov products, Temperley-Lieb open spin chains, and KP tau functions
• Additional symmetry of the modified extended Toda hierarchy
• Tau-functions beyond the group elements
• Are Maxwell knots integrable?
• Symmetric deformed 2D/3D Hurwitz-Kontsevich model and affine Yangian of gl(1)
• Negative Generators of the Virasoro Constraints for the BKP Hierarchy
• CMM formula as superintegrability property of unitary model
• Itzykson-Zuber correlators from character expansion
• On the Integrable Structure of Super Yang-Mills Scattering Amplitudes
• AGT correspondence, (q-)Painlevè equations and matrix models
• Random Matrices, Boundaries and Branes
• The ordered exponential representation of GKM using the $W_{1+\infty }$ operator
• A new kind of anomaly: on W-constraints for GKM
• The Schur Expansion of Characteristic Polynomials and Random Matrices
• Comultiplication in ABCD algebra and scalar products of Bethe wave functions
• From superintegrability to tridiagonal representation of $\beta$-ensembles
• Random matrices with external source and KP $\tau$ functions
• Integrability and Matrix Models
• A basic triad in Macdonald theory
• A note on quantum groups and integrable systems