GT seminar: "The worldsheet skein D-module and basic curves on Lagrangian fillings of the Hopf link conormal"

Abstract: We propose a skein-valued lift of the D-module associated to a link in S^3, which is the quotient of the skein of the cylindrical lift of the unit conormal of the link by an ideal which in the classical limit reduces to the defining ideal of the augmentation variety of the link. The Gromov-Witten partition function of a Lagrangian filling can then be viewed as a morphism from the D-module to the skein of the filling. This imposes constraints on the partition function which for some links can be used to determine the partition function uniquely.
I will then exemplify this general framework in the particular case of the Hopf link. For every point of the augmentation variety, we identify a corresponding Lagrangian filling whose partition function is uniquely determined by the constraints. We obtain explicit expressions for the partition functions in terms of contributions of basic holomorphic curves, which can be viewed as skein-valued versions of the Gopakumar-Vafa formula and the knots-quivers correspondence. This is joint work with T. Ekholm and P. Longhi.

This is a talk in our geometry and topology seminar series.