On sl3 Knizhnik-Zamolodchikov equations and W3 null-vector equations

TL;DR

This work investigates whether the sl3 Knizhnik–Zamolodchikov equations can be mapped to ${\cal W}_3$ null-vector equations via Sklyanin’s separation of variables. While a clean, general equivalence fails for generic values of the level, the authors establish a close structural correspondence that becomes exact in the critical level limit $k\to3$, with a detailed analysis revealing the underlying reasons and the role of a twist function. The study extends the well-known sl2 KZ–BPZ relation to the sl3 setting, highlighting new features from the cubic invariants, the Yangian-to-Gaudin reduction, and the complications from ${\cal W}_3$ degeneracies. The results suggest a quasi-satisfying relation at finite level and invite further exploration at higher rank ${\cal W}$-algebras and their Langlands-type connections. These findings illuminate why the sl3 case aligns only in the critical regime and point to broader structural insights for ${\cal W}_N$ and their relation to affine algebras.

Abstract

Starting from Sklyanin's separation of variables for the sl3 Yangian model, we derive the separation of variables for the quantum sl3 Gaudin model. We use the resulting new variables for rewriting the sl3 Knizhnik-Zamolodchikov equations, and comparing them with certain null-vector equations in conformal field theories with W3-algebra symmetry. The two sets of equations are remarkably similar, but become identical only in the critical level limit. This is in contrast to the sl2 Knizhnik-Zamolodchikov equations, which are known to be equivalent to Belavin-Polyakov-Zamolodchikov equations for all values of the level.