Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models

TL;DR

By representing the Generalized Kontsevich Model as a Toda-lattice $τ$-function and incorporating negative-time $t_{-p}$ and zero-time $t_0$ variables, the paper unifies discrete and continuous matrix models within a single integrable framework. It shows that these deformations preserve the generalized string equation $L_{-1} au=0$ and, for quadratic potential $V(X)\sim X^2/2$, realize a forced Toda-chain reduction corresponding to a discrete matrix model with matrix size identified with $t_0=n$. The authors derive explicit KP–Toda correspondences via Miwa variables, determinant representations, and orthogonal-polynomial techniques, and they discuss how double-scaling limits can be formulated entirely within GKM language, linking $V(X)=X^2/2-n\log X$ to $V(X)=X^{K+1}/(K+1)$ in the continuum. Overall, the work proposes GKM as a universal framework for matrix-model partition functions, enabling a unified treatment of continuum limits and the interplay between continuous and discrete models.

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.