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

TL;DR

This paper derives a new separation of variables for the quantum sl3 Gaudin model, compares the resulting equations with W3 null-vector equations, and explores their relation at the critical level, highlighting differences from the sl2 case.

Contribution

It introduces a novel separation of variables for the sl3 Gaudin model and relates the resulting equations to W3 null-vector equations, revealing their equivalence only at the critical level.

Findings

The separation of variables for the sl3 Gaudin model is established.

The sl3 Knizhnik-Zamolodchikov equations are rewritten using new variables.

Equivalence with W3 null-vector equations occurs only at the critical level.

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.