Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models

Details Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models

1992-03-18

arxiv.org/abs/hep-th/9203043v1

Nucl.Phys. B397 (1993) 339-378

Physics

High Energy Physics

High Energy Physics - Theory

46 p

Scientific paper

10.1016/0550-3213(93)90347-R

We represent the partition function of the Generalized Kontsevich Model (GKM) in the form of a Toda lattice $\tau$-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 $\tau$-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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