Lagrangian cobordism and tropical curves

TL;DR

This paper explores the structure of Lagrangian cobordism groups for torus fibers in symplectic manifolds using tropical geometry and Floer theory, revealing diverse group properties and connections to mirror symmetry.

Contribution

It introduces a new framework for analyzing Lagrangian cobordisms via tropical geometry and Floer theory, providing explicit examples and group classifications.

Findings

Some symplectic tori have trivial cobordism relations between fibers.

The degree zero cobordism group can be divisible.

Results relate to mirror symmetry and rational equivalence.

Abstract

We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relation to rational equivalence of 0-cycles on the mirror rigid analytic space.