On some 3-point functions in the $W_4$ CFT and related braiding matrix

TL;DR

This paper constructs specific 3-point functions in $W_4$ conformal field theory, enabling the calculation of braiding matrices for certain 4-point blocks, with implications for models with $sl(4)$ symmetry.

Contribution

It introduces new 3-point constants in $W_4$ CFT, extending previous work, and determines a specific braiding matrix relevant for $sl(4)$-related conformal blocks.

Findings

Derived a $3 imes 3$ braiding matrix consistent with fusion rules.

Verified a braiding relation applicable to $sl(4)$ symmetric models.

Compared 3-point constants across different central charge regions.

Abstract

We construct a class of 3-point constants in the $sl(4)$ Toda conformal theory $W_4$, extending the examples in Fateev and Litvinov. Their knowledge allows to determine the braiding/fusing matrix transforming 4-point conformal blocks of one fundamental, labelled by the 6-dimensional $sl(4)$ representation, and three partially degenerate vertex operators. It is a $3 \times 3$ submatrix of the generic $6 \times 6$ fusing matrix consistent with the fusion rules for the particular class of representations. We check a braiding relation which has wider applications to conformal models with $sl(4)$ symmetry. The 3-point constants in dual regions of central charge are compared in preparation for a BPS like relation in the $\hat{sl}(4)$ WZW model.