Solving Toda field theories and related algebraic and differential properties

TL;DR

This paper develops a method to explicitly solve Toda field theories for classical Lie algebras using a gauge approach, proving algebraic identities and confirming Leznov's solutions through rigorous mathematical proofs.

Contribution

It introduces a gauge-based solving scheme for Toda field theories, establishing algebraic and differential identities, and providing complete proofs for Leznov's solutions.

Findings

Derived algebraic identities for principal minors of special matrices

Proved differential identities for iterated integrals

Validated Leznov's solutions within the new framework

Abstract

Toda field theories are important integrable systems. They can be regarded as constrained WZNW models, and this viewpoint helps to give their explicit general solutions, especially when a Drinfeld-Sokolov gauge is used. The main objective of this paper is to carry out this approach of solving the Toda field theories for the classical Lie algebras. In this process, we discover and prove some algebraic identities for principal minors of special matrices. The known elegant solutions of Leznov fit in our scheme in the sense that they are the general solutions to our conditions discovered in this solving process. To prove this, we find and prove some differential identities for iterated integrals. It can be said that altogether our paper gives complete mathematical proofs for Leznov's solutions.