Explore gauge theory and the analytic approach to geometric Langlands in this comprehensive lecture by Edward Witten, Professor at the School of Natural Sciences, Institute for Advanced Study. Delivered at the Clay Research Conference on September 30, 2021, the talk delves into the recent developments in the "analytic" approach to geometric Langlands correspondence, as proposed by P. Etingof, E. Frenkel, and D. Kazhdan. Examine the shift from categories and functors to quantum states and operators, and discover how gauge theory can be applied to both the "categorical" and "analytic" versions of geometric Langlands. Investigate key concepts such as quantum field theory, line operators, mirror symmetry, deformation polarization, and geometric quantization. Gain insights into duality, physical states, Hamiltonians, and boundary conditions as Witten explains the gauge theory interpretation of the analytic approach, drawing from his work with D. Gaiotto in arXiv:2107.01732.