On the Tropical/Special Lagrangian Correspondence

Given a tropical curve in $\mathbb{R}^n$, it is a central question in tropical geometry whether it can be lifted to a holomorphic curve in the corresponding toric variety. Inspired by semi-flat mirror symmetry, Mikhalkin showed that they can always lift to a Lagrangian. In this talk, we will show that every locally planar tropical curve can be lifted to a special Lagrangian in $(\mathbb{C}^*)^2$ based on a gluing construction. Moreover, there exists a $1$-parameter family of special Lagrangians such that Gromov-Hausdorff collapses to the given tropical curve in the adiabatic limit. The feature is different from the lifting of tropical curves to holomorphic curves, which can be obstructed. This is a joint work with S.-K. Chiu and Y. Li.