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 $$\mathbb {R}^n$$ 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 (Givental in Funct Anal Appl 20(3):197–203, 1986) on Lagrangian embeddability of non-oriented surfaces to $$\mathbb {C}^2$$ , as well as to the case $$n=3$$ , where we see appearance of the graph 3-manifolds studied by Waldhausen (I Invent Math 3:308–333, 1967a, II Invent Math 4:87–117, 1967b) as Lagrangian submanifolds. In particular, rational tropical curves in $$\mathbb {R}^3$$ 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.