W-algebras and higher analogues of the Knizhnik-Zamolodchikov equations

Abstract

The key role in the derivation of the Knizhnik-Zamolodchikov equations in the Wess-Zumino-Witten model is played by the energy-momentum tensor, which is constructed from a second-order Casimir element in the universal enveloping algebra of the corresponding Lie algebra. We investigate the possibility of constructing analogues of Knizhnik-Zamolodchikov equations using higher-order central elements. We consider the Casimir element of the third order for the Lie algebra \(\mathfrak{s}\mathfrak{l}_N\) and of the fourth order for \(\mathfrak{o}_N\). The construction is impossible in the first case, but we succeed in obtaining the sought equation in the second case.