Description
We will explain how to construct a certain 3 dimensional TFT from the representation theory of quantum groups for arbitrary value of the quantum parameter. It attaches categories to surfaces, which are canonical deformations of the category of sheaves on character varieties, which unifies a variety of well-known constructions in quantum algebra. Those categories carry actions of mapping class groups, and can be used to produce invariants for links in thickened surfaces. To a closed 3 manifold it attaches the corresponding skein module. One obtains this way a link invariant valued in the representation theory of Cherednik's famed spherical double affine Hecke algebra. This is based on joint works with D. Ben-Zvi, D. Jordan, and N. Snyder.
