Global Aspects of the WZNW Reduction to Toda Theories

It is well-known that the Toda Theories can be obtained by reduction from theWess-Zumino-Novikov-Witten (WZNW) model, but it is less known that this WZNW$\rightarrow$ Toda reduction is \lq incomplete'. The reason for thisincompleteness being that the Gauss decomposition used to define the Todafields from the WZNW field is valid locally but not globally over the WZNWgroup manifold, which implies that actually the reduced system is not just theToda theory but has much richer structures. In this note we furnish a frameworkwhich allows us to study the reduced system globally, and thereby present somepreliminary results on the global aspects. For simplicity, we analyze primarily0 $+$ 1 dimensional toy models for $G = SL(n, {\bf R})$, but we also discussthe 1 $+$ 1 dimensional model for $G = SL(2, {\bf R})$ which corresponds to theWZNW $\rightarrow$ Liouville reduction.Comment: 22 pages, INS-Rep.-104