Examples of tropical-to-Lagrangian correspondence

Monday, 19 February, 2018

Published in: 

arXiv:1802.06473

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.

Author(s): 

Grigory Mikhalkin