Holomorphic curves on knot conormals

Abstract: We give an overview of results from the last few years. We first describe the "skeins on branes” approach (joint with Shende) to open Gromov-Witten invariants and show how this leads to a direct geometric interpretation of the quantization of the augmentation variety of a Legendrian knot conormal, as a quantum curve. We then describe a partially conjectural quiver picture (joint with Kucharski and Longhi) for the holomorphic curve counts on a Lagrangian knot conormal, where all curves stems from a finite set of basic holomorphic disks. Via more refined disk counts, this quiver picture leads to a description of HOMFLY homology. Finally, we apply similar reasoning to the knot complement Lagrangian we find that a count of holomorphic annuli, after SFT stretching, can be viewed as the semi-classical limit of an instance of Gukov-Pei-Putrov-Vafa Z-hat theory. This leads to a direct geometric interpretation of the Z-hat invariant (joint with Guen, Gukov, Kucharski, Park, and Sulkowski).

algebraic geometrysymplectic geometry

Audience: researchers in the topic

M-seminar