  We consider the problem of representing the Kac-Moody algebra
$\mathfrak{g}(N)$ specified by an $r\times r$ indecomposable generalised Cartan
matrix $N$ as vector fields on the torus ${{\bb C}^*}^r$. It is shown that, if
the representations are of a certain form, this is possible if and only if
$\mathfrak{g}(N)\cong sl(r+1,{\bb C})$ or $\tilde{sl}(r,{\bb C})$. For
$sl(r+1,{\bb C})$ and $\tilde{sl}(r,{\bb C})$, discrete families of
representations are constructed. These generalise the well-known discrete
families of representations of $sl(2,{\bb C})$ as regular vector fields on
${\bb C}^*$
