The goal of these notes is to give a short introduction to Fukaya categories and some of their applications. The first half of the text is devoted to a brief review of Lagrangian Floer (co)homology and product structures. Then we introduce the Fukaya category (informally and without a lot of the necessary technical detail), and briefly discuss algebraic concepts such as exact triangles and generators. Finally, we mention wrapped Fukaya categories and outline a few applications to symplectic topology, mirror symmetry and low-dimensional topology.

A Beginner’s Introduction to Fukaya Categories

This paper is about the Fukaya category of a Fano hypersurface $X \subset \mathbf {CP}^{n}$
 . Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed–open string maps, weak proper Calabi–Yau structure, Abouzaid’s split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface 
 $X$
 : we construct a configuration of monotone Lagrangian spheres in $X$
 , and compute the associated disc potential. The result coincides with the Hori–Vafa superpotential for the mirror of $X$
 (up to a constant shift in the Fano index 1 case). As a consequence, we give a proof of Kontsevich’s homological mirror symmetry conjecture for $X$
 . We also explain how to extract non-trivial information about Gromov–Witten invariants of $X$
 from its Fukaya category.

On the Fukaya category of a Fano hypersurface in projective space

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 K\"unneth 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.

Sectorial descent for wrapped Fukaya categories

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.

Microlocal Morse theory of wrapped Fukaya categories

We define a new class of symplectic spaces called ``pumpkin domains'', which roughly speaking comprise a Liouville domain and a Liouville hypersurface of its boundary. To such an object we assign an $A_\infty$-category called its partially wrapped Fukaya category. An exact Landau-Ginzburg model gives rise to a pumpkin domain, and the partially wrapped Fukaya category of this pumpkin domain is meant to agree with the Fukaya category one is supposed to assign to the Landau-Ginzburg model. As evidence, we prove a formula that relates the partially wrapped Fukaya category of a pumpkin domain to the wrapped Fukaya category of its underlying Liouville domain. This operation is mirror to removing a divisor.

/pdf/on-partially-wrapped-fukaya-categories-4ab8hgclr2.pdf

On partially wrapped Fukaya categories

Consider the wrapped Fukaya category W of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that natural geometric maps from the Hochschild homology of W to symplectic cohomology and from symplectic cohomology to the Hochschild cohomology of W are isomorphisms, in a manner compatible with ring and module structures. This is a consequence of a more general duality for the wrapped Fukaya category, which should be thought of as a non-compact version of a Calabi-Yau structure. The new ingredients are: (1) Fourier-Mukai theory for W via a wrapped version of holomorphic quilts, (2) new geometric operations, coming from discs with two negative punctures and arbitrary many positive punctures, (3) a generalization of the Cardy condition, and (4) the use of homotopy units and A-infinity shuffle products to relate non-degeneracy to a resolution of the diagonal.

/pdf/symplectic-cohomology-and-duality-for-the-wrapped-fukaya-3j1knm1gt4.pdf

Symplectic cohomology and duality for the wrapped Fukaya category

We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the open-closed map for Liouville sectors, and we show that these invariants are covariantly functorial with respect to inclusions of Liouville sectors. From this foundational setup, a local-to-global principle for Abouzaid's generation criterion follows.

Covariantly functorial wrapped Floer theory on Liouville sectors

We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the open-closed map for Liouville sectors, and we show that these invariants are covariantly functorial with respect to inclusions of Liouville sectors. From this foundational setup, a local-to-global principle for Abouzaid’s generation criterion follows.

Sheel Ganatra

Papers

Symplectic cohomology and duality for the wrapped Fukaya category

Sectorial descent for wrapped Fukaya categories

Microlocal Morse theory of wrapped Fukaya categories

Covariantly functorial wrapped Floer theory on Liouville sectors

Covariantly functorial wrapped Floer theory on Liouville sectors