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We introduce a tropical version of the Fukaya algebra of a Lagrangian submanifold and use it to show that tropical Lagrangian tori are weakly unobstructed. 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 introduce a further degeneration of the matching conditions (similar in spirit to Bourgeois version of symplectic field theory) which results in a tropical Fukaya algebra whose structure maps are, in good cases, sums of products over vertices of tropical graphs. We show the tropical Fukaya algebra is homotopy equivalent to the original Fukaya algebra. In the case of toric Lagrangians contained in a toric component of the degeneration, an invariance argument implies the existence of projective Maurer-Cartan solutions.
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We use the technique of stabilizing divisors introduced by Cieliebak-Mohnke to construct finite dimensional, strictly unital Fukaya algebras of compact, oriented, relatively spin Lagrangians in compact symplectic manifolds with rational symplectic classes. The homotopy type of the algebra and the moduli space of solutions to the weak Maurer-Cartan equation are shown to be independent of the choice of perturbation data. The Floer cohomology is the cohomology of a complex of vector bundles over the space of solutions to the weak Maurer-Cartan equation and is shown to be independent of the choice of perturbation data up to gauge equivalence.
We compute the Fukaya category of the symplectic blowup of a compact rational symplectic manifold at a point in the following sense: Suppose a collection of Lagrangian branes satisfy Abouzaids criterion for split-generation of a bulk-deformed Fukaya category of cleanly-intersecting Lagrangian branes. We show that for a small blow-up parameter, their inverse images in the blowup together with a collection of branes near the exceptional locus split-generate the Fukaya category of the blowup. This categorifies a result on quantum cohomology by Bayer and is an example of a more general conjectural description of the behavior of the Fukaya category under transitions occuring in the minimal model program, namely that mmp transitions generate additional summands.
We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a `stop removal equals localization result, and (4) that the Fukaya--Seidel category of a Lefschetz fibration with Weinstein fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a Kunneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful.
The Nadler-Zaslow correspondence famously identifies the finite-dimensional Floer homology groups between Lagrangians in cotangent bundles with the finite-dimensional Hom spaces between corresponding constructible sheaves. We generalize this correspondence to incorporate the infinite-dimensional spaces of morphisms at infinity, given on the Floer side by Reeb trajectories (also known as wrapping) and on the sheaf side by allowing unbounded infinite rank sheaves which are categorically compact. When combined with existing sheaf theoretic computations, our results confirm many new instances of homological mirror symmetry. More precisely, given a real analytic manifold $M$ and a subanalytic isotropic subset $Lambda$ of its co-sphere bundle $S^*M$, we show that the partially wrapped Fukaya category of $T^*M$ stopped at $Lambda$ is equivalent to the category of compact objects in the unbounded derived category of sheaves on $M$ with microsupport inside $Lambda$. By an embedding trick, we also deduce a sheaf theoretic description of the wrapped Fukaya category of any Weinstein sector admitting a stable polarization.
Let $M$ be an exact symplectic manifold with $c_1(M)=0$. Denote by $mathrm{Fuk}(M)$ the Fukaya category of $M$. We show that the dual space of the bar construction of $mathrm{Fuk}(M)$ has a differential graded noncommutative Poisson structure. As a corollary we get a Lie algebra structure on the cyclic cohomology $mathrm{HC}^bullet(mathrm{Fuk}(M))$, which is analogous to the ones discovered by Kontsevich in noncommutative symplectic geometry and by Chas and Sullivan in string topology.
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