Grassmannian Slices in Type A Homological Mirror Symmetry for Affine

Abstract

This is joint work with Mina Aganagic, Ivan Danilenko, Vivek Shende and Peng Zhou. In this talk we discuss a homological mirror symmetry result, where both the symplectic side and the algebraic side are related to affine grassmannian slices in type A. In the pre-talk, we will review some basic examples of homological mirror symmetry. In the second part of the talk, we will begin with examples of type A Affine grassmannian slices, which are useful geometric objects for studying representation theory. The primary examples are A_n surfaces, which are the main players of the pure math seminar this week. They also include all type A Slodowy slices.  By the geometric Satake theorem, their homologies are related to weight spaces in tensor products of fundamental representations. Since Fukaya category can be loosely viewed as a categorification of the middle homology, this result can be viewed as one step towards two different categorifications of the geometric Satake theorem.