Abstract

According to SYZ philosophy the same tropical object should admit two ways to be lifted classically: as a complex object, and as a (T-dual) symplectic object. While the tropical-to-complex correspondence is relatively well-studied, tropical-to-symplectic correspondence remains significantly less well-studied up to date.

In the talk we’ll look at some first instances of tropical- to-symplectic correspondence. As an application of such correspondence in the case of planar tropical curves we’ll reprove a theorem of Givental (from about 30 years ago) on Lagrangian embeddings of connected sums of Klein bottles to C^2. For tropical curves in toric 3-folds the resulting Lagrangians turn out to be Waldhausen graph-manifolds. For this case we’ll relate the enumerative multiplicity of tropical rational curves to the torsion in the first homology group of the corresponding Lagrangian submanifolds (in full compliance with Mirror Symmetry predictions).