Description:

Title:  The Chromatic Lagrangian for N-graphs

Abstract:  

We extend the notion of the chromatic Lagrangian, originally defined for cubic planar graphs on punctured 2-spheres in \cite{SSZ}, to a class of N-graphs related to N-triangulations by edge mutations. An N-triangulation of a punctured surface defines a local chart for the cluster Poisson variety of Borel-decorated PGL_N local systems. Inside a symplectic leaf of unipotent monodromy, we define a Lagrangian subvariety whose underlying monodromy is trivial, known as the chromatic Lagrangian. We prove that upon quantization, the chromatic Lagrangian is defined by an ideal. The system of chromatic ideals is compatible with quantum cluster mutations induced by edge mutations in the N-graphs. Exploiting cluster theory, we derive \textit{wavefunctions} for Lagrangian branes in threespace that fill Legendrian surfaces associated to the N-graphs. From the wavefunctions, we can make explicit predictions about their all-genus open Gromov-Witten invariants.