KZ~equation, $G$-opers, quantum Drinfeld--Sokolov reduction, and quantum Cayley--Hamilton identity

D. Talalaev ; A. Chervov

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamics systems, combinatorial methods. Part XVI, Tome 360 (2008), pp. 246-259

Résumé

The Lax operator of the Gaudin-type models is a 1-form at the classical level. In virtue of the quantization scheme proposed by D. Talalaev, it is natural to treat the quantum Lax operator as a connection; this connection is a particular case of the Knizhnik–Zamolodchikov connection. In this paper we find a gauge transformation that produces the “second normal form” or the “Drinfeld–Sokolov” form. Moreover, the differential operator naturally corresponding to this form is given precisely by the quantum characteristic polynomial of the Lax operator (this operator is called the $G$-oper or Baxter operator). This observation allows us to relate solutions of the KZ and Baxter equations in an obvious way and to prove that the immanent KZ-equation has only meromorphic solutions. As a corollary, we obtain the quantum Cayley–Hamilton identity for the Gaudin-type Lax operators (including the general $\mathfrak{gl}_n[t]$ case). The presented construction sheds a new light on the geometric Langlands correspondence. We also discuss the relation with the Harish-Chandra homomorphism. Bibl. – 19 titles.