Abstract

Inside a symplectic leaf of the cluster Poisson variety of Borel-decorated $PGL_2$ 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.

This talk is based on joint work with Gus Schrader and Linhui Shen.

About the speaker

Prof. Eric Zaslow is currently a professor  of mathematics at Northwestern University. His research focuses on mathematical questions arising from duality symmetries in theoretical physics such as mirror symmetry. With Andrew Strominger and Shing-Tung Yau, he formulated the famous SYZ conjecture in mirror symmetry. He was named as a fellow of the American Mathematical Society “for contribution to mathematical physics and mirror symmetry”.