Using Conservation Laws to Solve Toda Field Theories

TL;DR

This paper demonstrates how conservation laws in Toda field theories can be used to reduce the problem to solving ordinary differential equations, revealing an intrinsic transformation group related to the Lie algebra structure.

Contribution

It introduces a method to solve Toda field theories using conservation laws in one-sided form and identifies an intrinsic transformation group related to the Lie algebra.

Findings

Reduction of Toda models to ODE systems via conservation laws

Identification of an intrinsic transformation group for each model

Application to $A_1$, $A_2$, and $B_2$ Toda theories

Abstract

We investigate the question of how the knowledge of sufficiently many local conservation laws for a model can be utilized to solve the model. We show that for models where the conservation laws can be written in one-sided forms, like $\barpartial Q_s = 0$, the problem can always be reduced to solving a closed system of ordinary differential equations. We investigate the $A_1$, $A_2$, and $B_2$ Toda field theories in considerable detail from this viewpoint. One of our findings is that there is in each case a transformation group intrinsic to the model. This group is built on a specific real form of the Lie algebra used to label the Toda field theory. It is the group of field transformations which leaves the conserved densities invariant.