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Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models

S. Kharchev·A. Marshakov·A. Mironov·A. Morozov
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Abstract

We represent the partition function of the Generalized Kontsevich Model (GKM) in the form of a Toda lattice $τ$-function and discuss various implications of non-vanishing "negative"- and "zero"-time variables: the appear to modify the original GKM action by negative-power and logarithmic contributions respectively. It is shown that so deformed $τ$-function satisfies the same string equation as the original one. In the case of quadratic potential GKM turns out to describe {\it forced} Toda chain hierarchy and, thus, corresponds to a {\it discrete} matrix model, with the role of the matrix size played by the zero-time (at integer positive points). This relation allows one to discuss the double-scaling continuum limit entirely in terms of GKM, $i.e.$ essentially in terms of {\it finite}-fold integrals.

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S. Kharchev, A. Marshakov, A. Mironov, A. Morozov. 1992-03-18. Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models. https://doi.org/10.1016/0550-3213(93)90347-r

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