Tropical Lagrangians and Homological Mirror Symmetry
J. Hicks · arXiv (Cornell University) · 2019
We produce for each tropical hypersurface $V(\\phi)\\subset Q=\\RR^n$ a Lagrangian $L(\\phi)\\subset (\\CC^*)^n$ whose moment map projection is a tropical amoeba of $V(\\phi)$. When these Lagrangians are admissible in the Fukaya-Seidel category, we show that they are mirror to hypersurfaces in a toric mirror. These constructions are extended to tropical varieties given by locally planar intersections, and to symplectic 4-manifolds with almost toric fibrations. We also explore relations to wall-crossing, dimer models, and Lefschetz fibrations. A complete example is worked out for the mirror pair $(\\CP^2\\setminus E, W), X_{9111}$.