Affine Algebras, Langlands Duality and Bethe Ansatz

By Langlands duality one usually understands a correspondence between automorphic representations of a reductive group G over the ring of adels of a field F , and homomorphisms from the Galois group Gal(F/F ) to the Langlands dual group GL. It was originally introduced in the case when F is a number field or the field of rational functions on a curve over a finite field [1]. Recently A. Beilinson and V. Drinfeld [2] proposed a version of Langlands correspondence in the case when F is the field of rational functions on a curve X over C. This geometric Langlands correspondence relates certain D–modules on the moduli stack MG(X) of principal G–bundles on X, and G L–local systems on X (i.e. homomorphisms π1(X) → G L). A. Beilinson and V. Drinfeld construct this correspondence by applying a localization functor to representations of the affine Kac-Moody algebra ĝ of critical level k = −h, where h is the dual Coxeter number. The localization functor assigns a twisted D–module on MG(X) to an arbitrary ĝ– module from a category O0. The fibers of this D–module are analogous to spaces of conformal blocks from conformal field theory. In fact, the D–module, which corresponds to the vacuum irreducible ĝ–module of level k ∈ Z+, is the sheaf of sections of a vector bundle (with projectively flat connection), whose fiber is dual to the space of conformal blocks of the Wess-Zumino-Witten model. The analogy between conformal field theory and the theory of automorphic representations was underscored by E. Witten in [3]. It is at the critical level where this analogy can be made even more precise due to the richness of representation theory of ĝ. The peculiarity of the critical level is that a completion of the universal enveloping algebra of ĝ at this level, U−h∨(ĝ) = U(ĝ)/(K+h ), contains a large center Z(ĝ). This center is isomorphic to the classical W–algebra W(gL) associated to the Lie algebra gL, which is Langlands dual to g [4]. Recall that W(gL) consists of functionals on a certain Poisson manifold, which is obtained by the Drinfeld-Sokolov reduction from a hyperplane in the dual space to ĝL [5]. Elements of this Poisson manifold can be considered as connections of special kind on a GL–bundle over a punctured disc called gL–opers in [6]. For example, sl2–opers are the same as projective connections. Thus,