W--Gravity and Generalized Lax Equations for (super) Toda Theory

TL;DR

This paper extends the Lax pair and Bäcklund transformations for Toda and super Toda equations to arbitrary background geometries, linking them to W-gravity and flatness conditions.

Contribution

It introduces a generalized framework for Toda equations in diverse geometries, connecting integrable systems with W-gravity and flatness conditions.

Findings

Generalized Lax pairs for Toda and super Toda equations.

Established connection between Toda equations and W-gravity.

Applicable to arbitrary worldsheet backgrounds.

Abstract

We generalize the Lax pair and B\"acklund transformations for Toda and N=1 super Toda equations to the case of arbitrary worldsheet background geometry. We use the fact that the Toda equations express constant curvature conditions, which arise naturally from flatness conditions equivalent to the W--gravity equations of motion.