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

Abstract

The Lax operator of the Gaudin type models is a 1-form on the classical level. In virtue of the quantization scheme proposed in [Talalaev04] (hep-th/0404153) it is natural to treat the quantum Lax operator as a connection; this connection is a particular case of the Knizhnik-Zamolodchikov connection [ChervovTalalaev06] (hep-th/0604128). In this paper we find a gauge transformation which 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 [Talalaev04] of the Lax operator (this operator is called the G-oper or Baxter equation). This observation allows to relate solutions of the KZ and Baxter equations in an obvious way, and to prove that the immanent KZ-equations has only meromorphic solutions. As a corollary we obtain the quantum Cayley-Hamilton identity for the Gaudin-type Lax operators (including the general gl(n)[t] case). The presented construction sheds a new light on a geometric Langlands correspondence. We also discuss the relation with the Harish-Chandra homomorphism.