Joint IAS/Princeton University Symplectic Geometry Seminar

Given a Legendrian knot, I will construct a category of constructible sheaves, invariant under Legendrian isotopy up to equivalence. The constructble sheaves live on the front plane and are stratified by the front diagram of the Legendrian knot. I will then prove that the "rank-one" subcategory is equivalent to a category of augmentations of the Chekanov-Eliashberg differential graded algebra. The proof boils down to local calculations, which are easy to describe. Next I will apply a similar construction to alternating strand diagrams (such as those generated by bipartite graphs) on surfaces and show how the mooduli spaces of rank-one objects are cluster varieties. The relation to the previous work is that an exact Lagrangian filling defines and augmentation, and the cluster chart near an augmentation is given by the space of line bundles on the filling, recovering the integrable system that Goncharov-Kenyon associate to a bipartite graph. The first part is based on joint works with Vivek Shende and David Treumann (arXiv:1402.0490) and with Lenhard Ng, Dan Rutherford, Vivek Shende, and Steven Sivek (arXiv:1502.04939). The second part is based on work in progress with Vivek Shende, David Treumann and Harold Williams.