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

TL;DR

This paper provides detailed calculations of conformal blocks in Wess-Zumino-Witten models using free-field realization, focusing on $ ext{sl}(2)$ and $ ext{sl}(3)$ cases and verifying associated equations.

Contribution

It offers explicit computations of conformal blocks in WZW models from free-field representations, including verification of Knizhnik-Zamolodchikov equations for $ ext{sl}(3)$.

Findings

Explicit conformal blocks for $ ext{sl}(2)$ and $ ext{sl}(3)$ WZW models derived.

Verification of Knizhnik-Zamolodchikov equations in complex cases.

Identification of integral representations involving screening charges.

Abstract

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 $\hat{sl}(2)_k$ and $\hat{sl}(3)_k$ 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 $\hat{sl}(3)_k$ case, where the answers take the form of double integrals over screening charge positions and do not look like ordinary hypergeometric functions.