Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models

TL;DR

This paper links the Generalized Kontsevich Model to Toda hierarchy and discrete matrix models, showing how deformations affect the model and enabling continuum limits to be studied via finite integrals.

Contribution

It demonstrates that the GKM partition function can be expressed as a Toda lattice tau-function and explores the effects of negative and zero-time deformations on the model.

Findings

Deformed tau-function satisfies the same string equation as the original.

GKM with quadratic potential describes a forced Toda chain hierarchy.

The relation enables studying the double-scaling limit through finite integrals.

Abstract

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.