Explicit computation of Drinfeld associator in the case of the fundamental representation of gl(N)

TL;DR

This paper explicitly computes the Drinfeld associator for the fundamental representation of gl(N), revealing its connection to WZW conformal blocks and introducing a simplified symmetrized version characterized by a single symmetric function.

Contribution

It provides an explicit solution for the Drinfeld associator in the fundamental representation of gl(N), linking it to conformal blocks and defining a new symmetrized associator with simplified structure.

Findings

Components match WZW conformal blocks

Symmetrized associator retains knot invariants

Introduces the Drinfeld prepotential

Abstract

We solve the regularized Knizhnik-Zamolodchikov equation and find an explicit expression for the Drinfeld associator. We restrict to the case of the fundamental representation of $gl(N)$. Several tests of the results are presented. It can be explicitly seen that components of this solution for the associator coincide with certain components of WZW conformal block for primary fields. We introduce the symmetrized version of the Drinfeld associator by dropping the odd terms. The symmetrized associator gives the same knot invariants, but has a simpler structure and is fully characterized by one symmetric function which we call the Drinfeld prepotential.