Wakimoto construction in the principal gradation

Abstract

It is well known that the bosonized version of the Wakimoto construction allows the explicit realization of any affine algebra $\widehat{g}$, with arbitrary level $k$ in the homogeneous gradation, in terms of $dim(g)$ free bosonic fields. In this paper, we show in the case of the simplest affine algebra $\widehat{sl(2)}$, that the bosonized Wakimoto realization can be extended to the principal gradation only when $k$ is equal to the critical level, i.e., -2. In this case, this construction can be achieved in terms of arbitrary number (larger than 1) of free bosonic fields.