On the string equation at $c=1$

TL;DR

This paper explores the string equation for the $c=1$ string theory with compactification radius $eta$, using the Toda lattice hierarchy to describe its partition function.

Contribution

It introduces a framework connecting the $c=1$ string string equation with the Toda lattice hierarchy, extending understanding of the model at various radii.

Findings

Formulation of the string equation within the Toda hierarchy

Description of the partition function for $c=1$ string with radius $eta$

Extension of the string equation framework to discrete radii

Abstract

The analogue of the string equation which specifies the partition function of $c=1$ string with a compactification radius $\beta \in \mbox{$\bf{Z}$}_{\geq 1} $ is described in the framework of the Toda lattice hierarchy.