Spectral curves and $W$-representations of matrix models

Abstract

We explain how the spectral curve can be extracted from the $\mathcal{W}$-representation of a matrix model. It emerges from the part of the $\mathcal{W}$-operator, which is linear in time-variables. A possibility of extracting the spectral curve in this way is important because there are models where matrix integrals are not yet available, and still they possess all their important features. We apply this reasoning to the family of WLZZ models and discuss additional peculiarities which appear for the non-negative value of the family parameter $n$, when the model depends on additional couplings (dual times). In this case, the relation between topological and $1/N$ expansions is broken. On the other hand, all the WLZZ partition functions are $\tau$-functions of the Toda lattice hierarchy, and these models also celebrate the superintegrability properties.

arXiv comment25 pages

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Cited by in corpus (11)

• Commutative families in $W_{\infty }$, integrable many-body systems and hypergeometric $\tau$-functions
• On W-operators and superintegrability for dessins d'enfant
• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• Summing up perturbation series around superintegrable point
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
• Generalized $\tilde{W}$ algebras
• Genus expansion of matrix models and $\hslash$ expansion of $B$KP hierarchy
• $W_{1+\infty }$ flows and multi-component hierarchy (KP case)
• Phases in WLZZ Matrix Models
• $\beta$-ensembles and higher genera Catalan numbers