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We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least two whose Chern character represents a non-zero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank one local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions, and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included.
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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.
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.
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.
Suppose one has found a non-empty sub-category $mathcal{A}$ of the Fukaya category of a compact Calabi-Yau manifold $X$ which is homologically smooth in the sense of non-commutative geometry, a condition intrinsic to $mathcal{A}$. Then, we show $mathcal{A}$ split-generates the Fukaya category and moreoever, that our hypothesis implies (and is therefore equivalent to the assertion that) $mathcal{A}$ satisfies Abouzaids geometric generation criterion [Abo]. An immediate consequence of earlier work [G1, GPS1, GPS2] is that the open-closed and closed-open maps, relating quantum cohomology to the Hochschild invariants of the Fukaya category, are also isomorphisms. Our result continues to hold when $c_1(X) eq 0$ (for instance, when $X$ is monotone Fano), under a further hypothesis: the 0th Hochschild cohomology of $mathcal{A}$ $mathrm{HH}^0(mathcal{A})$ should have sufficiently large rank: $mathrm{rk} mathrm{HH}^0(mathcal{A}) geq mathrm{rk} mathrm{QH}^0(X)$. Our proof depends only on formal properties of Fukaya categories and open-closed maps, the most recent and crucial of which, compatibility of the open-closed map with pairings, was observed independently in ongoing joint work of the author with Perutz and Sheridan [GPS2] and by Abouzaid-Fukaya-Oh-Ohta-Ono [AFO+]; a proof in the simplest settings appears here in an Appendix. Because categories Morita equivalent to categories of coherent sheaves or matrix factorizations are homologically smooth, our result applies to resolve the split-generation question in homological mirror symmetry for compact symplectic manifolds (generalizing a result of Perutz-Sheridan [PS2] proven in the case $c_1(X) = 0$): any embedding of coherent sheaves or matrix factorizations into the split-closed derived Fukaya category is automatically a Morita equivalence when it has large enough $mathrm{HH}^0$ (which it always does if $c_1(X)=0$).
Let $M$ be an exact symplectic manifold with contact type boundary such that $c_1(M)=0$. In this paper we show that the cyclic cohomology of the Fukaya category of $M$ has the structure of an involutive Lie bialgebra. Inspired by a work of Cieliebak-Latschev we show that there is a Lie bialgebra homomorphism from the linearized contact homology of $M$ to the cyclic cohomology of the Fukaya category. Our study is also motivated by string topology and 2-dimensional topological conformal field theory.
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