Toda Lattice Hierarchy and the Topological Description of the c=1 String Theory

TL;DR

This paper explores the connection between the Toda lattice hierarchy and the topological description of the $c=1$ string theory, showing how specific constraints reproduce known string theory results and relate to topological minimal theories.

Contribution

It demonstrates that imposing special constraints on the Toda hierarchy reproduces key results of the $c=1$ string theory, including $W_{1+ abla}$ relations, linking integrable systems with string theory.

Findings

Reproduction of $W_{1+ abla}$ relations in $c=1$ string theory

Identification of string equations as constraints in Toda hierarchy

Landau-Ginzburg superpotential as a $U(1)$ current operator

Abstract

The Toda lattice hierarchy is discussed in connection with the topological description of the $c=1$ string theory compactified at the self-dual radius. It is shown that when special constraints are imposed on the Toda hierarchy, it reproduces known results of the $c=1$ string theory, in particular the $W_{1+\infty}$ relations among tachyon correlation functions. These constraints are the analogues of string equations in the topological minimal theories. We also point out that at $c=1$ the Landau-Ginzburg superpotential becomes simply a $U(1)$ current operator.