June 6, 2019
TITLE: Graphs, Lagrangians and Open Gromov-Witten Conjectures
ABSTRACT: This talk picks up on the one of David Treumann, describing our joint work. A trivalent graph on a sphere defines a higher-genus Legendrian surface in complex three-space. This Legendrian serves as a boundary condition for Lagrangian Fukaya objects, equivalently as a singular support condition for constructible sheaves. The moduli space of objects in this category of sheaves itself embeds as a “chromatic Lagrangian” subspace of the space local systems on the surface. Generalizing work of Aganagic-Vafa, we conjecture that the defining function for this chromatic Lagrangian is a generating function for open Gromov-Witten invariants counting holomorphic disks with boundary on the Lagrangian Fukaya objects.
Time permitting, I will discuss joint work with Linhui Shen on how to exploit cluster structures to compute these conjectural open Gromov-Witten invariants at all genus and relate them to cohomological quiver invariants.