Mathematical Physics

Geometry of a pair of complex Lagrangian submanifolds of a complex symplectic manifold appears in many areas of mathematics and physics,  including exponential integrals in finite and infinite dimensions,  wall-crossing formulas in 2d and 4d, representation theory, resurgence of WKB series and so  on.

In 2014 we started a joint project with Maxim Kontsevich which we  named "Holomorphic Floer Theory" (HFT for short) in order to study all these (and other) phenomena as a part of a bigger picture.

It is expected that the Betti version of the geometric Langlands program should ultimately be about the equivalence of two 4-dimensional topological field theories. In this talk I will give an overview of ongoing work in categorified sheaf theory and explain how one can use it to describe the categories of boundary conditions arising on the spectral side.

Let G be a complex reductive group, and X be any smooth projective G-variety. In this talk, we will construct an algebra homomorphism from the G-equivariant homology of the affine Grassmannian Gr_G to the G-equivariant quantum cohomology of X. The construction uses shift operators in quantum cohomologies. We will also discuss the possible extension to the loop rotation equivariant setting and the relation with the Peterson isomorphism when X is the flag variety associated with G. This is based on joint work with Alexander Braverman.