Conformal blocks of Wess-Zumino-Witten model from its free-field representation

A powerful approach to the celebrated Wess-Zumino-Witten (WZW) model is provided by its free-field realization. However, explicit calculations of conformal blocks are not described in the literature in full detail. We begin this study with the simplest cases of the $\widehat{sl}(2{)}_k$ and $\widehat{sl}(3{)}_k$ WZW models, with special emphasis on their global $sl(2)$ and $sl(3)$ symmetries of the resulting correlators, which are not explicit in this formalism. Also non-trivial is the verification of the Knizhnik-Zamolodchikov equations in the $\widehat{sl}(3{)}_k$ case, where the answers take the form of double integrals over screening charge positions and do not look like ordinary hypergeometric functions.