Toda field theories and integral curves of standard differential systems

TL;DR

This paper explores the deep connections between Toda field theories linked to simple Lie algebras and the geometry of integral curves on flag varieties, revealing new structural insights and symmetries.

Contribution

It establishes three key relations connecting Toda field theories with standard differential systems on flag varieties, including an isomorphism and a quotient description.

Findings

Isomorphism between jet space functions and unipotent subgroup functions

Toda field theory reduces to integral curves on flag varieties when one variable is fixed

Toda field theory is a quotient of product systems by a natural group action

Abstract

This paper establishes three relations between the Toda field theory associated to a simple Lie algebra and the integral curves of the standard differential system on the corresponding complete flag variety. The motivation comes from the viewpoint on the Toda field theories as Darboux integrable differential systems as developed in \cite{AFV}. First, we establish an isomorphism concerning regular functions on the jet space and on the unipotent subgroup in the setting of a simple Lie group. Using this result, we then show that in the sense of differential systems, after restricting one independent variable to a constant the Toda field theory becomes the system for integral curves of the standard differential system on a complete flag variety. Finally, we establish that, in terms of differential invariants, the Toda field theory is the quotient of the product of two such systems by a natural group action.