Examples of tropical-to-Lagrangian correspondence

TL;DR

This paper explores the correspondence between tropical curves and Lagrangian submanifolds in symplectic toric varieties, providing new insights into their geometric and topological properties in low dimensions.

Contribution

It establishes a tropical-to-Lagrangian correspondence framework and rederives key theorems, revealing new connections between tropical geometry and symplectic topology.

Findings

Reproves Givental's theorem on Lagrangian embeddings of surfaces in ^2.

Shows rational tropical curves in ^3 produce 3-manifolds with homology determined by tropical multiplicities.

Identifies Lagrangian submanifolds as graph 3-manifolds in the case n=3.

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.