Tropical Fukaya Algebras

In this book, we explore the degeneration of pseudoholomorphic disks bounding a Lagrangian in a symplectic manifold in the large complex structure limit corresponding to a multiple cut. The limit objects, called broken disks, have underlying tropical graphs which in the case of pseudoholomorphic spheres were studied by Brett Parker. In particular, we study the limit of the Fukaya algebra of a Lagrangian submanifold, which is an $A_{\infty }$ algebra whose higher composition maps involve counts of pseudoholomorphic disks.

The goal of the book is to prove an $A_{\infty }$ homotopy equivalence between the ordinary Fukaya algebra of a Lagrangian and a tropical version of the Fukaya algebra defined via counts of broken disks with rigid tropical graphs.

The exposition is self-contained and includes details of the transversality scheme. Various computations of disk potentials of Lagrangian submanifolds, such as those in cubic surfaces and flag varieties, are included.