Famous results of N. Hitchin establish existence of an integrable structure of the Hitchin moduli spaces. The goal of this talk will be to discuss a more explicit approach known in the integrable models literature as separation of variables, how it can be applied to the quantisation of the Hitchin system, and how the result is related to the analytic Langlands correspondence studied by Etingof, Frenkel and Kazhdan.

Partly based on arXiv:1707.07873