Talalaev's formula, Maslov–Weinstein–Tyurin correspondence and the geometric Langlands correspondence

We give simple explanation and explicit construction of the geometric Langlands correspondence and its generalization to many integrable systems. It appears to be related with the ideas developed recently by A. N. Tyurin and previously by V. P. Maslov, A. Weinstein et al. The key ingredient of the considerations is the ingenious formula of “quantum characteristic polynomial” (or “quantum spectral curve”) found recently by D. Talalaev (arXiv: hep-th/0404153). The constructions shed a new light on the problem of higher-dimensional Langlands correspondence, the related open questions will be discussed.