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.

24 pages. Revised version accepted for publication in Nuclear Physics B

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• On the complete perturbative solution of one-matrix models
• Superintegrability of Kontsevich matrix model
• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• $W_{1+\infty }$ constraints for the hermitian one-matrix model
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• Spectral curves and $W$-representations of matrix models
• 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
• Itzykson-Zuber correlators from character expansion