Geometrical origin of integrability for Liouville and Toda theory

TL;DR

This paper explores the geometric foundations of integrability in Liouville and Toda theories by extending Lax pairs and Bäcklund transformations to arbitrary Riemann surfaces, revealing the geometric structure behind their integrability.

Contribution

It generalizes key integrability tools for Liouville and Toda theories to arbitrary Riemann surfaces, linking their integrability to geometric structures.

Findings

Lax pairs and Bäcklund transformations are extended to arbitrary Riemann surfaces.

The geometric origin of integrability in Toda theory is clarified.

Connections between W-geometry and Riemannian geometry are established.

Abstract

We generalize the Lax pair and B\"acklund transformations for Liouville and Toda field theories as well as their supersymmetric generalizations, to the case of arbitrary Riemann surfaces. We make use of the fact that Toda field theory arises naturally and geometrically in a restriction of so called $W$--geometry to ordinary Riemannian geometry. This derivation sheds light on the geometrical structure underlying complete integrability of these systems. (Invited talk presented at the 877th meeting of the American Mathematical Society, USC, November 1992 and at the YITP workshop ``Directions on Quantum Gravity", Kyoto, November 1992.)