Spectral curves and W-representations of matrix models

Abstract

AbstractWe explain how the spectral curve can be extracted from the$$ \mathcal{W} $$W-representation of a matrix model. It emerges from the part of the$$ \mathcal{W} $$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 parametern, when the model depends on additional couplings (dual times). In this case, the relation between topological and 1/Nexpansions is broken. On the other hand, all the WLZZ partition functions areτ-functions of the Toda lattice hierarchy, and these models also celebrate the superintegrability properties.