Advances in Theoretical and Mathematical Physics
Volume 22 (2018)
Cubic planar graphs and Legendrian surface theory
Pages: 1289 – 1345
We study Legendrian surfaces determined by cubic planar graphs. Graphs with distinct chromatic polynomials determine surfaces that are not Legendrian isotopic, thus giving many examples of non-isotopic Legendrian surfaces with the same classical invariants. The Legendrians have no exact Lagrangian fillings, but have many interesting non-exact fillings.
We obtain these results by studying sheaves on a three-ball with microsupport in the surface. The moduli of such sheaves has a concrete description in terms of the graph and a beautiful embedding as a holomorphic Lagrangian submanifold of a symplectic period domain, a Lagrangian that has appeared in the work of Dimofte–Gabella–Goncharov. We exploit this structure to find conjectural open Gromov–Witten invariants for the non-exact filling, following Aganagic–Vafa.
Published 2 May 2019