CFT approach to constraint operators for ($\beta$-deformed) hermitian one-matrix models

Abstract

Since the ($\beta$-deformed) hermitian one-matrix models can be represented as the integrated conformal field theory (CFT) expectation values, we construct the operators in terms of the generators of the Heisenberg algebra such that the constraints can be derived by inserting the constructed operators into the integrated expectation values. We also obtain the second order total derivative operators associating with the derived constraint operators and analyze their properties. We explore the intrinsic connection between the derived constraint operators and $W$-representations of some matrix models. For the Gaussian hermitian one-matrix model in the external field and $\beta$-deformed $N\times N$ complex matrix model, we investigate the superintegrability and derive the corresponding character expansions from their $W$-representations. Moreover a conjectured formula for the averages of Jack polynomials in the literature is proved.

arXiv comment24 pages. Revised version accepted for publication in Nuclear Physics B

References in corpus (34)

• Exchange Operator Formalism for Integrable Systems of Particles
• Integrability and Matrix Models
• The Calogero-Sutherland Model and Generalized Classical Polynomials
• Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory
• Conformal Matrix Models as an Alternative to Conventional Multi-Matrix Models
• Character Expansion Methods for Matrix Models of Dually Weighted Graphs
• Five-dimensional AGT Relation and the Deformed beta-ensemble
• On KP-integrable Hurwitz functions
• Algebra of differential operators associated with Young diagrams
• BGWM as Second Constituent of Complex Matrix Model
• On the complete perturbative solution of one-matrix models
• The Calogero-Sutherland Model and Polynomials with Prescribed Symmetry
• Complex Matrix Models and Statistics of Branched Coverings of 2D Surfaces
• Exact Solution of Discrete Two-Dimensional R^2 Gravity
• Sum rules for characters from character-preservation property of matrix models
• Superintegrability of Kontsevich matrix model
• On matrix models and their $q$-deformations
• Almost Flat Planar Diagrams
• Two-Dimensional Chiral Matrix Models and String Theories
• $q$-Virasoro modular double and 3d partition functions
• Matrix model partition function by a single constraint
• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• Super-quantum curves from super-eigenvalue models
• Quantum curves and conformal field theory
• Applications of Laplace-Beltrami operator for Jack polynomials
• From CFT to Ramond super-quantum curves
• Counting planar diagrams with various restrictions
• Virasoro versus superintegrability. Gaussian Hermitian model
• W-representations of the fermionic matrix and Aristotelian tensor models
• Exact correlators in the Gaussian Hermitian matrix model
• $W_{1+\infty }$ constraints for the hermitian one-matrix model
• Hyperdeterminantal computation for the Laughlin wave function
• Superintegrability and Kontsevich-Hermitian relation
• Powers of the Vandermonde determinant, Schur Functions, and recursive formulas

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• On bilinear superintegrability for monomial matrix models in pure phase
• Generalized $\tilde{W}$ algebras
• Supersymmetric polynomials and algebro-combinatorial duality
• Action of $W$-type operators on Schur functions and Schur Q-functions
• $\beta$-WLZZ models from $\beta$-ensemble integrals directly
• Higher order constraints for the ($\beta$-deformed) Hermitian matrix models
• CMM formula as superintegrability property of unitary model
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