Global Aspects of the WZNW Reduction to Toda Theories

TL;DR

This paper investigates the global structure of the reduction from WZNW models to Toda theories, revealing that the process is incomplete locally and exploring the richer global structures involved, with analysis on simplified models.

Contribution

It introduces a framework to study the global aspects of WZNW to Toda reduction, extending understanding beyond local Gauss decomposition limitations.

Findings

Reduction is locally valid but not globally over the WZNW group manifold.

The reduced system has richer structures than the standard Toda theory.

Preliminary results on global properties are obtained from simplified models.

Abstract

It is well-known that the Toda Theories can be obtained by reduction from the Wess-Zumino-Novikov-Witten (WZNW) model, but it is less known that this WZNW $\rightarrow$ Toda reduction is \lq incomplete'. The reason for this incompleteness being that the Gauss decomposition used to define the Toda fields from the WZNW field is valid locally but not globally over the WZNW group manifold, which implies that actually the reduced system is not just the Toda theory but has much richer structures. In this note we furnish a framework which allows us to study the reduced system globally, and thereby present some preliminary results on the global aspects. For simplicity, we analyze primarily 0 $+$ 1 dimensional toy models for $G = SL(n, {\bf R})$, but we also discuss the 1 $+$ 1 dimensional model for $G = SL(2, {\bf R})$ which corresponds to the WZNW $\rightarrow$ Liouville reduction.