On sℓ 3 Knizhnik-Zamolodchikov equations and W 3 null-vector equations

Abstract

Starting from Sklyanin's separation of variables for the sℓ 3 Yangian model, we derive the separation of variables for the quantum sℓ 3 Gaudin model. We use the resulting new variables for rewriting the sℓ 3 Knizhnik-Zamolodchikov equations, and comparing them with certain null-vector equations in conformal field theories with W 3-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 sℓ 2 Knizhnik-Zamolodchikov equations, which are known to be equivalent to Belavin-Polyakov-Zamolodchikov equations for all values of the level. © 2009 SISSA.