Examples of tropical-to-Lagrangian correspondence

Abstract

The paper associates Lagrangian submanifolds in symplectic toric varieties to certain tropical curves inside the convex polyhedral domains of $\R^n$ that appear as the images of the moment map of the toric varieties. We pay a particular attention to the case $n=2$, where we reprove Givental's theorem on Lagrangian embeddability of non-oriented surfaces to $\C^2$, as well as to the case $n=3$, where we see appearance of the graph 3-manifolds studied by Waldhausen as Lagrangian submanifolds. In particular, rational tropical curves in $\R^3$ produce 3-dimensional rational homology spheres. The order of their first homology groups is determined by the multiplicity of tropical curves in the corresponding enumerative problems.