Noncommutative Geometry Seminar

Classical Riemann-Hilbert correspondence gives an equivalence of the category of bundles with regular singular connections on a projective curve and the category of locally constant sheaves on the curve, with singularities of the connections removed. I am going to recall classical story as well as its generalization to the category of holonomic (possibly irregular) D-modules. It admits various interpretations relating the subject to the deformation quantization of Poisson surfaces, Fukaya categories, Legendrian links, etc. Quantum spectral curve (the term is due to physicists) is a holonomic module over the quantum torus. Therefore, we now consider difference equations instead of differential ones. Aim of the talk is to explain how the Riemann-Hilbert correspondence looks in this case. If time permits, I will explain higher-dimensional generalizations of all that as well its relation to holomorphic version of Floer theory, wall-crossing formulas and analytic properties of Feynman integrals.