Seminar

If K is an m-component link in the 3-sphere then its Legendrian conormal is an m-component link of Legendrian tori in the unit cotangent bundle of the 3-sphere. Knot contact homology is the Chekanov-Eliashberg dg-alegbra CE of this Legendrian link. The augmentation variety of CE is a (one parameter family) of Lagrangian sub-varieties in the complex 2m-dimensional torus that is intimately related to the Gromov-Witten disk potential of Lagrangian fillings in the resolved conifold of the Legendrian torus link. We show how to define Legendrain SFT which is the higher genus counterpart of CE and then how it determines a quantization of the augmentation variety. The relation between Chern-Simons gauge theory and open topological string together with large N transition relates this to the colored HOMFLY polynomial of the link. Here the quantization of the augmentation variety corresponds to the recursion relation of the colored HOMFLY.