Non Abelian Toda Theory : A Completely Integrable Model for Strings on a Black Hole Background

TL;DR

This paper presents a completely integrable conformal field theory model describing string propagation on a 2D black hole background, providing explicit solutions and analyzing its conserved quantities and algebraic structure.

Contribution

It introduces the non-abelian Toda theory as a new integrable model for strings on black hole backgrounds, with explicit classical solutions and a novel algebraic structure of conserved quantities.

Findings

Explicit classical solutions for the model

Identification of conserved quantities and their algebraic structure

Demonstration of the model's integrability and conformal invariance

Abstract

The present paper studies a completely integrable conformally invariant model in 1+1 dimensions that corresponds to string propagation on the two-dimensional black hole background (semi-ininite cigar). Besides the two space-time string fields there is a third (internal) field with a very specific Liouville-type interaction leading to the complete integrability. This system is known as non-abelian Toda theory. I give the general explicit classical solution. It realizes a rather involved transformation expressing the interacting string fields in terms of (three) functions $\varphi_j(u)$ and $\bar\varphi_j(v)$ of one light-cone variable only. The latter are shown to lead to standard harmonic oscillator (free field) Poisson brackets thus paving the way towards quantization. There are three left-moving and three right-moving conserved quantities. The right (left)-moving conserved quantities form a new closed non-linear, non-local Poisson bracket algebra. This algebra is a Virasoro algebra extended by two conformal dimension-two primaries.