Gauge Theory and the Analytic Approach to Geometric Langlands

Recently P. Etingof, E. Frenkel, and D. Kazhdan, following earlier contributions by R. Langlands and J. Teschner, described an “analytic” approach to the geometric Langlands correspondence, in which the main ingredients are quantum states and operators acting on them rather than categories and functors. In this talk, I will review the gauge theory approach to the “categorical” version of geometric Langlands, and then, following the paper arXiv:2107.01732 with D. Gaiotto, I will explain how the same ingredients can be arranged differently to give a gauge theory interpretation of the “analytic” version of geometric Langlands.