Polyfold constructions for Fukaya categories

Fukaya categories capture the chain level information contained in Lagrangian Floer theory and its product structures. There are different constructions (and even more proposals) depending on the properties of the (fixed) ambient symplectic manifold. For example, Seidel [1] considers exact symplectic manifolds and constructs an (uncurved) Fukaya $A_{\infty }$ category whose objects are exact Lagrangians (with a brane structure), whose morphism spaces are Floer complexes (depending on the choice of a Hamiltonian), and whose composition operations are given by counting pseudoholomorphic polygons with boundary on the Lagrangians. (Here the benefit of the exactness assumption is that bubbling is excluded, so that the moduli spaces can be regularized by geometric methods (choices of Hamiltonian perturbations and almost complex structures). On the other hand, exact symplectic manifolds (on which the symplectic 2-form is exact) are necessarily noncompact, so one needs to assume certain boundedness and convexity conditions to ensure that pseudoholomorphic curves do not escape to infinity.) This wiki will focus on the main difficulty that is not addressed in Seidel's book: How to achieve transversality when geometric methods fail (e.g. due to sphere bubbling), and how to capture disk bubbling algebraically. To limit the classical analytic challenges in studying the pseudoholomorphic curves involved, we restrict our constructions to a fixed compact symplectic manifold $(M,\omega )$. Then the Fukaya category of M consists of the following data:

Objects

The Objects of $\operatorname {Fuk}(M)$ are the compact Lagrangian submanifolds $L\subset M$ (i.e.\ $\omega |_{L}=0$ and $\dim L={\frac 12}\dim M$)