Tropical Fukaya Algebras

Abstract

We introduce a tropical version of the Fukaya algebra of a Lagrangian submanifold. Tropical graphs arise as large-scale behavior of pseudoholomorphic disks under a multiple cut operation on a symplectic manifold that produces a collection of cut spaces each containing relative normal crossing divisors, following works of Ionel and Brett Parker. Given a Lagrangian submanifold in the complement of the relative divisors in one of the cut spaces, the structure maps of the broken Fukaya algebra count broken disks associated to rigid tropical graphs. We apply the results to give various computations of potentials, such as those of Lagrangians in cubic surfaces and flag varieties.