Fukaya Category Research Articles

Iterations of symplectomorphisms and $p$-adic analytic actions on the Fukaya category

Abstract Inspired by the work of Bell on the dynamical Mordell-Lang conjecture, and by family Floer cohomology, we construct $p$ p -adic analytic families of bimodules on the Fukaya category of a monotone or negatively monotone symplectic manifold, interpolating the bimodules corresponding to iterates of a symplectomorphism $\phi $ ϕ isotopic to the identity. This family can be thought of as a $p$ p -adic analytic action on the Fukaya category. Using this, we deduce that the ranks of the Floer cohomology groups $HF(\phi ^{k}(L),L';\Lambda )$ H F ( ϕ k ( L ) , L ′ ; Λ ) are constant in $k\in {\mathbb{Z}}$ k ∈ Z , with finitely many possible exceptions. We also prove an analogous result without the monotonicity assumption for generic $\phi $ ϕ isotopic to the identity by showing how to construct a $p$ p -adic analytic action in this case. We give applications to categorical entropy and a conjecture of Seidel.

Spectral Networks and Stability Conditions for Fukaya Categories with Coefficients

Given a holomorphic family of Bridgeland stability conditions over a surface, we define a notion of spectral network which is an object in a Fukaya category of the surface with coefficients in a triangulated DG-category. These spectral networks are analogs of special Lagrangian submanifolds, combining a graph with additional algebraic data, and conjecturally correspond to semistable objects of a suitable stability condition on the Fukaya category with coefficients. They are closely related to the spectral networks of Gaiotto–Moore–Neitzke. One novelty of our approach is that we establish a general uniqueness results for spectral network representatives. We also verify the conjecture in the case when the surface is disk with six marked points on the boundary and the coefficients category is the derived category of representations of an A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_2$$\\end{document} quiver. This example is related, via homological mirror symmetry, to the stacky quotient of an elliptic curve by the cyclic group of order six.

Erratum to “Poisson geometry, monoidal Fukaya categories, and commutative Floer cohomology rings”

This erratum corrects the false submisson date displayed on the last page of [Enseign. Math. 70, 313–381 (2024)].

Poisson geometry, monoidal Fukaya categories, and commutative Floer cohomology rings

We describe connections between concepts arising in Poisson geometry and the theory of Fukaya categories. The key concept is that of a symplectic groupoid, which is an integration of a Poisson manifold. The Fukaya category of a symplectic groupoid is monoidal, and it acts on the Fukaya categories of the symplectic leaves of the Poisson structure. Conversely, we consider a wide range of known monoidal structures on Fukaya categories and observe that they all arise from symplectic groupoids. We also use the picture developed to resolve a conundrum in Floer theory: why are some Lagrangian Floer cohomology rings commutative?

Mini-Workshop: Flavors of Rabinowitz Floer and Tate Homology

Rabinowitz Floer homology originated 15 years ago in symplectic geometry. Recent developments have related it to algebraic topology via string toplogy and Tate homology, and to mirror symmetry via Fukaya categories. This mini-workshop brought together researchers from these different communities, in order to foster exchange and collaborations across research fields.

Spherical adjunctions of stable ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\infty $$\\end{document}-categories and the relative S-construction

We develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\infty }$$\\end{document}-categories. We study the relative Waldhausen S-construction S∙(F)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_\\bullet (F)$$\\end{document} of the spherical functor F and show that it has a natural paracyclic structure (“rotation symmetry”). This fulfills a part of the general program of perverse schobers which are conjectural categorical upgrades of perverse sheaves. If we view a spherical functor as defining a schober on a disk, then each component Sn(F)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_n(F)$$\\end{document} of the S-construction gives a categorification of the cohomology of a perverse sheaf on a disk with support in a union of (n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n+1)$$\\end{document} closed arcs in the boundary. In other words, Sn(F)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_n(F)$$\\end{document} can be interpreted as the Fukaya category of the disk with coefficients in the schober and with support (“stops”) at the boundary arcs. The importance of the paracyclic structure is that it allows us to naturally associate the above data to disks on oriented surfaces. The action of the paracyclic rotation is a categorical analog of the monodromy of a perverse sheaf.

The Quantum A∞-relations on the elliptic curve

We define and prove the existence of the Quantum A∞-relations on the Fukaya category of the elliptic curve, using the notion of the Feynman transform of a modular operad, as defined by Getzler and Kapranov. Following Barannikov, these relations may be viewed as defining a solution to the quantum master equation of Batalin-Vilkovisky geometry.

The wrapped Fukaya category for semi-toric Calabi–Yau

We introduce the wrapped Donaldson–Fukaya category of a (generalized) semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical divisor in a toric Calabi–Yau 3-fold. We compute the wrapped Floer cohomology of a Lagrangian section and find that it is the algebra of functions on the Hori–Vafa mirror. The latter result is the key step in proving homological mirror symmetry for this case. The techniques developed here allow the construction in general of the wrapped Fukaya category on an open Calabi–Yau manifold carrying an SYZ fibration with nice behavior at infinity. We discuss the relation of this to the algebraic vs analytic aspects of mirror symmetry.

Homological mirror symmetry of toric Fano surfaces via Morse homotopy

Strominger–Yau–Zaslow (SYZ) proposed a way of constructing mirror pairs as pairs of torus fibrations. We apply this SYZ construction to toric Fano surfaces as complex manifolds, and discuss the homological mirror symmetry, where we consider Morse homotopy of the moment polytope instead of the Fukaya category.

Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

Exact Calabi–Yau categories and odd-dimensional Lagrangian spheres

An exact Calabi–Yau structure, originally introduced by Keller, is a special kind of smooth Calabi–Yau structure in the sense of Kontsevich–Vlassopoulos (2021). For a Weinstein manifold M , the existence of an exact Calabi–Yau structure on the wrapped Fukaya category \mathcal{W}(M) imposes strong restrictions on its symplectic topology. Under the cyclic open-closed map constructed by Ganatra (2019), an exact Calabi–Yau structure on \mathcal{W}(M) induces a class \tilde{b} in the degree one equivariant symplectic cohomology \mathrm{SH}_{S^1}^{1}(M) . Any Weinstein manifold admitting a quasi-dilation in the sense of Seidel–Solomon [Geom. Funct. Anal. 22 (2012), 443–477] has an exact Calabi–Yau structure on \mathcal{W}(M) . We prove that there are many Weinstein manifolds whose wrapped Fukaya categories are exact Calabi–Yau despite the fact that there is no quasi-dilation in \mathrm{SH}^{1}(M) ; a typical example is given by the affine hypersurface \{x^{3}+y^{3}+z^{3}+w^{3}=1\}\subset\mathbb{C}^{4} . As an application, we prove the homological essentiality of Lagrangian spheres in many odd-dimensional smooth affine varieties with exact Calabi–Yau wrapped Fukaya categories.

Sectorial descent for wrapped Fukaya categories

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 Liouville fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a Künneth 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.

Categorical action filtrations via localization and the growth as a symplectic invariant

Abstract We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A “quantitative” refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories.

Monotone Lagrangians in cotangent bundles of spheres

We study the compact monotone Fukaya category of T⁎Sn, for n≥2, and show that it is split-generated by two classes of objects: the zero-section Sn (equipped with suitable bounding cochains) and a 1-parameter family of monotone Lagrangian tori (S1×Sn−1)τ, with monotonicity constants τ>0 (equipped with rank 1 unitary local systems). As a consequence, any closed orientable spin monotone Lagrangian (possibly equipped with auxiliary data) with non-trivial Floer cohomology is non-displaceable from either Sn or one of the (S1×Sn−1)τ. In the case of T⁎S3, the monotone Lagrangians (S1×S2)τ can be replaced by a family of monotone tori Tτ3.

Homological mirror symmetry for the symmetric squares of punctured spheres

For an appropriate choice of a Z-grading structure, we prove that the wrapped Fukaya category of the symmetric square of a (k+3)-punctured sphere, i.e. the Weinstein manifold given as the complement of (k+3) generic lines in CP2 is quasi-equivalent to the derived category of coherent sheaves on a singular surface Z2,k constructed as the boundary of a toric Landau-Ginzburg model (X2,k,w2,k). We do this by first constructing a quasi-equivalence between certain categorical resolutions of both sides and then localizing. We also provide a general homological mirror symmetry conjecture concerning all the higher symmetric powers of punctured spheres. The corresponding toric LG-models (Xn,k,wn,k) are constructed from the combinatorics of curves on the punctured sphere and are related to small toric resolutions of the singularity x1…xn+1=v1…vk.

Global Fukaya category I

Abstract Let $Ham (M,\omega ) $ denote the Frechet Lie group of Hamiltonian symplectomorphisms of a monotone symplectic manifold $(M, \omega ) $. Let $NFuk (M, \omega )$ be the $A _{\infty } $-nerve of the Fukaya category $Fuk (M, \omega )$, and let $(|\mathbb {S}|, NFuk (M, \omega ))$ denote the $NFuk (M, \omega )$ component of the “space of $\infty $-categories” $|\mathbb {S}| $. Using Floer-Fukaya theory for a monotone $(M, \omega ),$ we construct a natural up to homotopy classifying map $$ \begin{align*}& BHam (M, \omega) \to (|\mathbb{S}|, NFuk (M, \omega)). \end{align*}$$This verifies one sense of a conjecture of Teleman on existence of action of $Ham (M, \omega )$ on the Fukaya category of $(M, \omega ) $. This construction is very closely related to the theory of the Seidel homomorphism and the quantum characteristic classes of the author, and this map is intended to be the deepest expression of their underlying geometric theory. In part II, the above map is shown to be nontrivial by an explicit calculation. In particular, we arrive at a new non-trivial “quantum” invariant of any smooth manifold, which motives the statement of a kind of “quantum” Novikov conjecture.

On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most$3$. As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric$3$-folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.

Disk potential functions for quadrics

We compute the disk potential of Gelfand–Zeitlin monotone torus fiber in a quadric hypersurface by exploiting toric degenerations, Lie theoretical mirror symmetry, and the structural result of the monotone Fukaya category.

Exotic Lagrangian tori in Grassmannians

We describe an iterative construction of Lagrangian tori in the complex Grassmannian \operatorname{Gr}(k,n) , based on the cluster algebra structure of the coordinate ring of a mirror Landau–Ginzburg model proposed by Marsh and Rietsch (2020). Each torus comes with a Laurent polynomial, and local systems controlled by the k -variables Schur polynomials at the n -th roots of unity. We use this data to give examples of monotone Lagrangian tori that are neither displaceable nor Hamiltonian isotopic to each other, and that support nonzero objects in different summands of the spectral decomposition of the Fukaya category over \mathbb{C} .

Wrapped sheaves

We construct a sheaf-theoretic analogue of the wrapped Fukaya category in Lagrangian Floer theory, by localizing a category of sheaves microsupported away from some given Λ⊂S⁎M along continuation maps constructed using the Guillermou-Kashiwara-Schapira sheaf quantization.When Λ is a subanalytic singular isotropic, we also construct a comparison map to the category of compact objects in the category of unbounded sheaves microsupported in Λ, and show that it is an equivalence. The last statement can be seen as a sheaf-theoretical incarnation of the sheaf-Fukaya comparison theorem of Ganatra-Pardon-Shende.