Toda Lattice Hierarchy and Generalized String Equations

TL;DR

This paper explores the Toda lattice hierarchy's role in string equations of generalized Kontsevich models and $c eq 1$ string theories, clarifying their relationships and constructing broader solution families.

Contribution

It demonstrates that the $p=-1$ generalized Kontsevich model does not match the $c=1$ string theory at self-dual radius and constructs a broader family of solutions satisfying generalized string equations.

Findings

The $p=-1$ model differs from $c=1$ string theory at self-dual radius.

A broader family of solutions to the Toda hierarchy is constructed.

The framework clarifies the status of various $c \,\leq\, 1$ string models.

Abstract

String equations of the $p$-th generalized Kontsevich model and the compactified $c = 1$ string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model at $p = -1$ does not coincide with the $c = 1$ string theory at self-dual radius. A broader family of solutions of the Toda lattice hierarchy including these models are constructed, and shown to satisfy generalized string equations. The status of a variety of $c \le 1$ string models is discussed in this new framework.