Mohammed Abouzaid
We construct the Fukaya category of a closed surface equipped with an area form using only elementary (essentially combinatorial) methods. We also compute the Grothendieck group of its derived category.
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We prove that a positive allowable Lefschetz fibration, PALF in short, admits a structure of exact Lefschetz fibration in the sense of Seidel \cite{Se08}. If the two-fold first Chern class of the total space is zero, we obtain the Fukaya-Seidel category. We prove that the derived Fukaya-Seidel category of PALF is independent of the choice of the symplectic structure. At the end of this paper, we study examples and show that derived Fukaya-Seidel categories have more information than the Milnor lattices of PALFs.
We introduce a formalism which allows us to formulate a version of Fukaya category in presence of curves of higher genus.
We outline an interpretation of Heegaard-Floer homology of 3-manifolds (closed or with boundary) in terms of the symplectic topology of symmetric products of Riemann surfaces, as suggested by recent work of Tim Perutz and Yanki Lekili. In particular we discuss the connection between the Fukaya category of the symmetric product and the bordered algebra introduced by Robert Lipshitz, Peter Ozsvath and Dylan Thurston, and recast bordered Heegaard-Floer homology in this language.
The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of Fukaya categories of symmetric products and Lagrangian correspondences. More specifically, we give a description of the algebra A(F) which appears in the work of Lipshitz, Ozsvath and Thurston in terms of (partially wrapped) Floer homology for product Lagrangians in the symmetric product, and outline how bordered Heegaard-Floer homology itself can conjecturally be understood in this language.
This paper is a companion to the authors' forthcoming work extending Heegaard Floer theory from closed 3-manifolds to compact 3-manifolds with two boundary components via quilted Floer cohomology. We describe the first interesting case of this theory: the invariants of 3-manifolds bounding S^2 union T^2, regarded as modules over the Fukaya category of the punctured 2-torus. We extract a short proof of exactness of the Dehn surgery triangle in Heegaard Floer homology. We show that A-infinity structures on the graded algebra A formed by the cohomology of two basic objects in the Fukaya category …
Given a symplectic manifold M, we consider a category with objects finite ordered families of Lagrangian submanifolds of M (subject to certain additional constraints) and with morphisms Lagrangian cobordisms relating them. We construct a functor that maps this category to a variant of the derived Fukaya category of M in a way that takes into account the triangulated structure of the latter.
Filtered Fukaya categories
本文将Fukaya范畴的自然弱过滤结构升级为真正的过滤结构，主要利用Morse-Bott或“簇”模型及特定摄动数据，并构造了延拓$A_\infty$-函子。
We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A-infinity structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given a pair of smooth manifolds of the same dimension with embeddings of a submanifold B with isomorphic normal bundles, we construct a differential graded category from the singular cochains of these spaces, and prove that it is equivalent to the A-infinity category obtained by considering exact Lagrangian embeddings intersecting cleanly along …
A class of partially wrapped Fukaya categories in $T^* N$ are proven to be well defined and then studied. In the case of $N$ diffeomorphic to $\mathbb{R}^m \times \mathbb{T}^n$, it is shown that these categories provide homological mirrors to equivariant and non-equivariant categories of coherent sheaves on toric varieties. This gives a new Floer theoretic formulation of mirror symmetry in the non-complete case.
Poisson geometry, monoidal Fukaya categories, and commutative Floer cohomology rings
本文建立Poisson几何与Fukaya范畴之间的联系，以辛群胚解释幺半Fukaya结构及其对辛叶子范畴的作用，并据此说明部分Lagrangian Floer上同调环交换的原因。
Given a pointed 4-ended tangle $T \subset D^3$, there are two Khovanov theoretic tangle invariants, $\unicode{1044}_1(T)$ from [arXiv:1910.1458] and $L_T$ from [arXiv:1808.06957], which are twisted complexes over the Fukaya category of the boundary 4-punctured sphere $(S^2,4\text{pt})=\partial (D^3, T)$. We prove that these two invariants are the same.
In this paper we prove a local-to-global principle for the Fukaya category of a closed Riemann surface $Σ$ of genus $g \geq 2$. We show that $\mathrm{Fuk}(Σ)$ can be glued from the Fukaya categories of the pairs-of-pants making up a pants decomposition of $Σ$. This extends our earlier results for the case of punctured Riemann surfaces. Our result has several interesting consequences: we obtain simple proofs of old and new HMS statements for Riemann surfaces, and establish a geometrization theorem for the objects of $\mathrm{Fuk}(Σ)$.
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. This text is based on a series of lectures given…
On the Fukaya categories of projective hypersurfaces of general type
本文利用从仿射超曲面的wrapped Fukaya范畴到其无穷远边界Fukaya范畴的函子，证明了充分高次数射影超曲面的同调镜对称性。
We embed triangulated categories defined by quivers with potential arising from ideal triangulations of marked bordered surfaces into Fukaya categories of quasi-projective 3-folds associated to meromorphic quadratic differentials. Together with previous results, this yields non-trivial computations of spaces of stability conditions on Fukaya categories of symplectic 6-manifolds.
We study wrapped Floer theory on product Liouville manifolds and prove that the wrapped Fukaya categories defined with respect to two different kinds of natural Hamiltonians and almost complex structures are equivalent. The implication is we can do quilted version of wrapped Floer theory, based on which we then construct functors between wrapped Fukaya categories of Liouville manifolds from certain classes of Lagrangian correspondences, by enlarging the wrapped Fukaya categories appropriately, allowing exact cylindrical Lagrangian immersions. For applications, we present a general Künneth form…
We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein-Gelfan'd-Gelfan'd category $\mathcal{O}$. As an application, we give a new geometric construction of the spectral sequence from annular to ordinary Khovanov homology. The heart of the paper is the development of a cylindrical model to compute Fukaya categories of (affine open subsets of) Hilb…
We define a new class of symplectic objects called "stops", which roughly speaking are Liouville hypersurfaces in the boundary of a Liouville domain. Locally, these can be viewed as pages of a compatible open book. To a Liouville domain with a collection of disjoint stops, we assign an $A_\infty$-category called its partially wrapped Fukaya category. An exact Landau-Ginzburg model gives rise to a stop, and the corresponding partially wrapped Fukaya category 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 re…
We study two functors between (partially) wrapped Fukaya categories. The first is the Orlov functor from the Fukaya category of a stop to the Fukaya category of the ambient sector. We give a geometric criterion for when this functor is spherical in the sense of Anno-Logvinenko. This criterion is a generalization of the situation where the stop comes from a Landau-Ginzburg model. The second functor is the Viterbo transfer map from a Liouville domain to a subdomain. We show that when the domain and subdomain are independently Weinstein, this functor is a homological epimorphism, which means that…
Calabi-Yau structures on Rabinowitz Fukaya categories
本文在wrapped Fukaya范畴由至多可数个满足有限性条件的Lagrangian子流形生成的假设下，证明了Liouville域的导出Rabinowitz Fukaya范畴具有$(n-1)$-Calabi--Yau结构。
Simple homotopy theory for Fukaya categories
该研究建立了Fukaya范畴中单纯同伦理论的范畴框架，证明Weinstein流形的Fukaya范畴中同构的闭连通拉格朗日量在特定条件下单纯同伦等价。
The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in analogy with Morse homology, the Fukaya category might result from gluing together Fukaya categories of Weinstein cells. This can be formalized by a recollement pattern for Lagrangian branes parallel to that for constructible sheaves. Assuming this structure, we exhibit the Fukaya category as the global sections of a sheaf on the conic topology of the Weinstein manifold. This can be viewed as a symplecti…
Sectorial descent for wrapped Fukaya categories
本研究开发了（部分）卷绕Fukaya范畴的计算工具，证明了下降性质、生成元、停点移除与局部化等价等4项核心结果，这些结果源自Künneth公式、正合三角、完全忠实几何判据三个独立要素，为相关范畴计算提供了新方法。
Rabinowitz Fukaya categories as cluster categories
探讨可逆多项式双悬架的米尔诺纤维的拉比诺维茨富卡亚范畴的同调镜像对称，对非卡拉比 - 丘型布里斯克恩 - 法姆多项式证明该对称，可据此计算米尔诺纤维的拉比诺维茨弗洛尔同调。
Realising perfect derived categories of Auslander algebras of type A as Fukaya-Seidel categories
本文证明了特定类型A的Auslander代数完美导出范畴与某些C²上Lefschetz纤维的Fukaya-Seidel范畴等价，并给出了其Milnor纤维的完整描述。
These are notes from a 2010 talk. They concern possible ways in which Fukaya categories might be considered as "local", which means glued together from simpler pieces in a loosely sheaf-theoretic sense. As the title suggests, this is purely speculative. Much of the motivation comes from tropical geometry and mirror symmetry.
Assume the existence of a Fukaya category $\mathrm{Fuk}(X)$ of a compact symplectic manifold $X$ with some expected properties. In this paper, we show $\mathscr{A} \subset \mathrm{Fuk}(X)$ split generates a summand $\mathrm{Fuk}(X)_e \subset \mathrm{Fuk}(X)$ corresponding to an idempotent $e \in QH^\bullet(X)$ if the Mukai pairing of $\mathscr{A}$ is perfect. Moreover we show $HH^\bullet(\mathscr{A}) \cong QH^\bullet(X) e$. As an application we compute the quantum cohomology and the Fukaya category of a blow-up of $\mathbb{C} P^2$ at four points with a monotone symplectic structure.
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 wrap…
We construct an A-infinity structure of the Fukaya category explicitly for any flat symplectic two-torus. The structure constants of the non-transversal A-infinity products are obtained as derivatives of those of transversal A-infinity products.
A sequel to arXiv:1111.1460, this paper elaborates on some of the themes in the above paper. Connections to Symplectic Field Theory (SFT) and mirror symmetry are explored.
This is an informal (and mostly conjectural) discussion of some aspects of Fukaya categories. We start by looking at exact symplectic manifolds which are obtained from a closed Calabi-Yau by removing a hyperplane section. We look at the possible geometric significance of Hochschild cohomology in this situation, and how one can try to get from the Fukaya category of the exact manifold to that of the closed Calabi-Yau. Also included is a brief discussion of the role of Lefschetz pencils, and a bit of general deformation theory. To appear in the Proceedings of the Beijing ICM.
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.
We construct a new cylinder object for semifree differential graded (dg) categories in the category of dg categories. Using this, we give a practical formula computing homotopy colimits of semifree dg categories. Combining it with the result of Ganatra, Pardon, and Shende, we get a formula computing wrapped Fukaya categories of Weinstein manifolds using their sectorial coverings. This formula has lots of applications including a practical computation of the wrapped Fukaya category of any cotangent bundle or plumbing space. In this paper, we compute wrapped Fukaya categories of cotangent bundle…
Explicitly Computing with Fukaya Categories of Surfaces with Boundary
本文通过组合方法显式计算带边曲面的Fukaya范畴，提供大量示例，旨在帮助初学者掌握独立计算工具。
Let $(X,ω)$ be a compact symplectic manifold whose first Chern class $c_1(X)$ is divisible by a positive integer $n$. We construct a $\mathbb{Z}_{2n}$-action on its Fukaya category $Fuk(X)$ and a $\mathbb{Z}_n$-action on the local models of its moduli of Lagrangian branes. We show that this action is compatible with the gluing functions for different local models.
This paper constructs and studies the Rabinowitz (wrapped) Fukaya category, a categorical invariant of exact cylindrical Lagrangians in a Liouville manifold whose cohomological morphisms, ``Rabinowitz wrapped Floer homology groups" measure the failure of wrapped Floer cohomology to satisfy Poincare duality (and in particular vanish for any pair with at least one compact Lagrangian). Our main result, answering a conjecture of Abouzaid, relates the Rabinowitz and usual wrapped Fukaya category by way of a general construction introduced by Efimov, the categorical formal punctured neighborhood of …
Speculations on higher Fukaya categories
本文推测了与$n$-移位辛叠相关的高阶Fukaya范畴理论，通过两个典型例子（移位余切叠和余伴随叠）联系Teleman工作与三维镜像对称，并给出$1$-移位辛几何的新证据。
Mirror symmetry for a toric variety involves Laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety. We show that there is a monodromy action on the monomially admissible Fukaya-Seidel categories of these Laurent polynomials as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated. In the process, we introduce the monomially admissible Fukaya-Seidel category as a new interpretation of the Fukaya-Seidel category…
Fukaya categories of blowups
该研究通过分析辛吹胀操作下的 Fukaya 分类，展示了小参数下拉格朗日膜的分裂生成特性，并验证了最小模型程序中过渡行为对 Fukaya 分类的影响。
It is shown that there is a useful notion of a relative Bridgeland stability condition on the partially wrapped Fukaya category of a marked surface, relative to some part of the surface's boundary. This construction has nice functorial properties, obeying cutting and gluing relations. This reduces the calculation of stability conditions on the Fukaya category of any fully stopped surface into three types of base cases. Calculations of these cases shows that every Bridgeland stability condition on such categories can be described by flat surfaces. In other words, the map constructed by Haiden-K…
Proper modules over Ginzburg dg algebras and compact Fukaya categories of plumbings
研究Ginzburg dg代数的proper模生成及其与紧Fukaya范畴的等价关系，揭示其与微局部sheaves的联系。
Fukaya categories of Coulomb branches as unique deformations
通过移除物质除子并利用Z^2分次缩小形变可能性，求解水平Hilbert方案的相对Fukaya范畴的形变问题，恢复Aganagic等人的结果。
Autoequivalences of Fukaya categories of surfaces and graded gentle algebras
本文研究了Fukaya范畴的自动等价与带状温和代数，通过导出Picard群和形性结果，消除了此前结果中的限制。
Equivariant Fukaya categories at singular values
本文研究 Hamiltonian 环面作用下辛商与原流形 Fukaya 范畴的等价关系，将其推广至矩映射奇异值，并借环面镜对称示例及代数范畴结构加以论证。
In this paper, we define $A_{\infty}$-Koszul duals for directed $A_{\infty}$-categories in terms of twists in their $A_{\infty}$-derived categories. Then, we compute a concrete formula of $A_{\infty}$-Koszul duals for path algebras with directed $A_n$-type Gabriel quivers. To compute an $A_\infty$-Koszul dual of such an algebra $A$, we construct a directed subcategory of a Fukaya category which are $A_\infty$-derived equivalent to the category of $A$-modules and compute Dehn twists as twists. The formula unveils all the ext groups of simple modules of the parh algebras and their higher composi…
It follows from the work of Burban and Drozd arXiv:0905.1231 that for nodal curves $C$, the derived category of modules over the Auslander order $\mathcal{A}_C$ provides a categorical (smooth and proper) resolution of the category of perfect complexes $\mathrm{Perf}(C)$. On the A-side, it follows from the work of Haiden-Katzarkov-Kontsevich arXiv:1409.8611 that for punctured surfaces $X$ with stops $Λ$ at their boundary, the partially wrapped Fukaya category $\mathcal{W}(X,Λ)$ provides a categorical (smooth and proper) resolution of the compact Fukaya category $\mathcal{F}(X)$. Inspired by thi…
Mirror symmetry and Fukaya categories of singular hypersurfaces
本文研究Auroux提出的奇异超曲面Fukaya范畴定义，证明其与高维Landau-Ginzburg模型的Fukaya-Seidel范畴导出等价，并由此导出阿贝尔簇大复结构极限退化的同调镜像对称性。
Equivariant Partially Wrapped Fukaya Categories on Liouville Sectors
本文研究了具有Lie群对称性的Liouville扇形的等变Lagrangian Floer理论，并探讨了其商空间的Lagrangian Floer同调之间的对应关系，证明了等变对应在上同调上的同构性。
Generating the Fukaya categories of compact toric varieties
本文通过构造显式类扭曲复形对象，证明紧致环面簇量子上同调各直和项对应的环面纤维在Abouzaid判据下分裂生成相应Fukaya范畴直和项。
Deformations of partially wrapped Fukaya categories of surfaces
本文研究了部分缠绕Fukaya范畴的A∞变形理论，揭示了其与部分紧化的关系，并引入了弱对偶概念以解决曲率问题。
Calabi-Yau structures on topological Fukaya categories
本研究提出构造范畴可构造层/余层整体截面Calabi-Yau结构的局部到全局形式体系，验证其在局部系统范畴下退化为经典定向概念，并将其应用于A模型微局部方法的树状骨架余层。
Quantum Steenrod operations and Fukaya categories
本文研究了闭单调辛流形的量子上同调与Fukaya范畴，证明了量子Steenrod运算可由范畴的Hochschild同调运算解释，并提供了计算Q的新工具及在算术同调镜像对称中的应用。
In this paper, we present partial results towards a classification of symplectic mapping tori using dynamical properties of wrapped Fukaya categories. More precisely, we construct a symplectic manifold $T_ϕ$ associated to a Weinstein domain $M$, and an exact, compactly supported symplectomorphism $ϕ$. $T_ϕ$ is another Weinstein domain and its contact boundary is independent of $ϕ$. In this paper, we distinguish $ϕ$ from $T_{1_M}$, under certain assumptions (Theorem 1.1). As an application, we obtain pairs of diffeomorphic Weinstein domains with the same contact boundary and whose symplectic co…
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.
Continuous and coherent actions on wrapped Fukaya categories
本文建立了包裹Fukaya范畴关于Liouville自同构的连续函子性，证明了Teleman猜想，并在余切丛情形下将作用与微分同胚作用交织，给出非平凡的有理同伦群映射。
On Fukaya categories and prequantization bundles
本文提出了一种通过 sheaf 理论计算非精确有理拉格朗日子流形 Floer 同调的方法，并展示了其在 Weinsten 流形中的应用。
Fukaya categories of orbifold surfaces in representation theory
本文通过轨道面的Fukaya范畴研究，揭示了轨道表面与斜gentle代数之间的导出等价关系，并利用变形理论探讨其联系。
Homological mirror symmetry for functors between Fukaya categories of very affine hypersurfaces
本文证明非常仿射超曲面的同调镜像对称性与部分缠绕Fukaya范畴间的辛函子相容，发现补集具有两个由Knörrer周期性关联的自然镜像，并据此修正Auroux猜想，同时提出Liouville流形粘合新方法。
Hypertoric Fukaya categories and categories O
本文证明超环面超 Kähler 流形的范畴 O 等价于由 J-全纯矩映射纤维停止的 Liouville 流形的部分包裹 Fukaya 范畴，并由此推出形式性、Koszul 对偶性及相对 Calabi–Yau 结构。
Fukaya categories of hyperplane arrangements
本文研究了超平面排列的Fukaya范畴，证明其部分包裹Fukaya范畴由特定拉格朗日子流形生成，并将其与超锥卷积代数关联，验证了相关猜想。
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