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

TL;DR

This work develops and explicitly implements free-field (Wakimoto/Feigin–Fuks) representations to construct holomorphic conformal blocks for the WZW models based on $\\hat{sl}(2)_k$ and $\\hat{sl}(3)_k$. By building vertex operators, screening charges, and Dotsenko–Fateev–type integrals, the authors derive concrete integral representations for four-point (and higher) blocks and verify their satisfaction of Knizhnik–Zamolodchikov equations, revealing both hypergeometric and more general multi-contour structures. The analysis uncovers how global and local $SU(2)$ (and $SU(3)$) symmetries, as well as fusion rules, govern non-vanishing correlators and block content, and it extends the framework toward multipoint correlators and connections with higher algebras and representations. The study thus provides a practical, symmetry-aware pathway to explicit WZW conformal blocks and highlights avenues for matrix-model interpretations and generalizations to broader algebraic structures.

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.

Figures (2)

• Figure 1: Weight diagrams of $\mathfrak{sl}_3$ for the fundamental $(1,0)$ and anti-fundamental $(0,1)$ representations.
• Figure 2: Connection between multipoint correlator, higher algebra, and representations