The Chromatic Lagrangian: Wavefunctions and Open Gromov-Witten Conjectures

Inside a symplectic leaf of the cluster Poisson variety of Borel-decorated $PG{L}_2$ local systems on a punctured surface is an isotropic subvariety we will call the chromatic Lagrangian. Local charts for the quantized cluster variety are quantum tori defined by cubic planar graphs, and can be put in standard form after some additional markings giving the notion of a framed seed. The mutation structure is encoded as a groupoid. The local description of the chromatic Lagrangian defines a wavefunction which, we conjecture, encodes open Gromov-Witten invariants of a Lagrangian threefold in threespace defined by the cubic graph and the other data of the framed seed. We also find a relationship we call framing duality: for a family of “canoe” graphs, wavefunctions for different framings encode DT invariants of symmetric quivers.