Free field representation of Toda field theories

TL;DR

This paper demonstrates that classical $sl_n$ Toda field theories on a cylindrical spacetime can be represented using free bosonic oscillators via a Drinfeld--Sokolov construction, establishing a correspondence for hyperbolic solutions.

Contribution

It provides a new free field representation for classical Toda theories, linking solutions to free bosonic oscillators and extending the understanding of their structure.

Findings

One-to-one correspondence between hyperbolic solutions and free bosonic oscillators.

Representation established for real hyperbolic solutions on cylindrical spacetime.

Discussion of solutions with non hyperbolic monodromy included.

Abstract

We study the following problem: can a classical $sl_n$ Toda field theory be represented by means of free bosonic oscillators through a Drinfeld--Sokolov construction? We answer affirmatively in the case of a cylindrical space--time and for real hyperbolic solutions of the Toda field equations. We establish in fact a one--to--one correspondence between such solutions and the space of free left and right bosonic oscillators with coincident zero modes. We discuss the same problem for real singular solutions with non hyperbolic monodromy.