Mohammed Abouzaid
We prove that the algebra of chains on the based loop space recovers the derived (wrapped) Fukaya category of the cotangent bundle of a closed smooth orientable manifold. The main new idea is the proof that a cotangent fibre generates the Fukaya category using a version of the map from symplectic cohomology to the homology of the free loop space introduced by Cieliebak and Latschev.
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Given an exact relatively Pin Lagrangian embedding Q in a symplectic manifold M, we construct an A-infinity restriction functor from the wrapped Fukaya category of M to the category of modules on the differential graded algebra of chains over the based loop space of Q. If M is the cotangent bundle of Q, this functor induces an A-infinity equivalence between the wrapped Floer cohomology of a cotangent fibre and the chains over the based loop space of Q, extending a result proved by Abbondandolo and Schwarz at the level of homology.
We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow (math/0604379, math/0612399) and the authors (math/0701783), before discussing a new approach using family Floer cohomology and the ``wrapped Fukaya category''. The latter, inspired by Viterbo's symplectic homology, emphasises the connection to loop spaces, hence seems particularly suitable when trying to extend the existing theory beyond the simply-connected case.
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…
We study the moduli space of pseudo pointed holomorphic disks with boundaries mapped in the zero section of the cotangent bundle of a manifold. We define perturbations of the equation for which it is possible to describe explicitly all the solutions of the problem in terms of Morse graphs on the manifold. In particular, this proves that the $A_\infty$ structure of the zero section of the cotangent bundle is equivalent to the Morse $A_\infty$ structure of the base manifold.
Density of fibers for the filtered Fukaya category of $T^*N$
本文回答了 Biran 和 Cornea 关于余切丛 Fukaya 范畴中纤维的迭代锥的稠密性问题，证明了取稠密的基点集时，余切纤维的迭代锥在过滤 Fukaya 范畴中是稠密的，并在附录中证明了辛流形中精确拉格朗日子流形关于谱距离永远不会是完全有界的。
Prime-localized Weinstein subdomains
本文提出一种构造方法，针对高维 Weinstein 域和素数集合生成其 wrapped Fukaya category 的素局部化子域，并分类了单连通自旋流形余切丛子域的可能范畴，同时揭示了球面余切丛中同构于真实 Lagrangians 的 twisted complexes。
Let $X$ be a compact real analytic manifold, and let $T^*X$ be its cotangent bundle. Let $Sh(X)$ be the triangulated dg category of bounded, constructible complexes of sheaves on $X$. In this paper, we develop a Fukaya $A_\infty$-category $Fuk(T^*X)$ whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write $Tw Fuk(T^*X)$ for the $A_\infty$-triangulated envelope of $Fuk(T^*X)$ consisting of twisted complexes of Lagrangian branes. Our main result is that $Sh(X)$ quasi-embeds into $Tw Fuk(T^*X)$ as an $A_\infty$-category. Taking cohomology gives an embe…
We (re)consider how the Fukaya category of a Lefschetz fibration is related to that of the fibre. The distinguishing feature of the approach here is a more direct identification of the bimodule homomorphism involved.
Fukaya A_\infty-structures associated to Lefschetz fibrations. VI
本文为单调辛流形上的辛 Lefschetz 纤维束关联了一种非交换几何范畴束结构，揭示其纤维与开纤维的 Fukaya 范畴及新构造的 Fukaya 范畴之间的联系。
We prove that the inclusion of every closed exact Lagrangian with vanishing Maslov class in a cotangent bundle is a homotopy equivalence. We start by adapting an idea of Fukaya-Seidel-Smith to prove that such a Lagrangian is equivalent to the zero section in the Fukaya category with integral coefficients. We then study an extension of the Fukaya category in which Lagrangians equipped with local systems of arbitrary dimension are admitted as objects, and prove that this extension is generated, in the appropriate sense, by local systems over a cotangent fibre. Whenever the cotangent bundle is si…
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。
Spectral equivalence of nearby Lagrangians
该研究构造了稳定极化刘维尔扇区的系数取于交换环谱$R$的包裹Donaldson–Fukaya范畴，证明闭流形余切丛中闭恰当拉格朗日$R$-膜同构于零截面的$R$-膜结构，推广了整数环上的经典结果。
We introduce an $A_\infty$ map from the cubical chain complex of the based loop space of Lagrangian submanifolds with Legendrian boundary in a Liouville Manifold $C_{*}(Ω_{L} \mathcal{L}\mathit{ag})$ to wrapped Floer cohomology of Lagrangian submanifold $\mathcal{CW}^{-*}(L,L)$. In the case of a cotangent bundle and a Lagrangian co-fiber, the composition of our map with a previously constructed map from $\mathcal{CW}^*(L,L) \to C_{*}(Ω_q Q) $ shows that this map is split surjective.
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 …
Ideas of Fukaya and Kontsevich-Soibelman suggest that one can use Strominger-Yau-Zaslow's geometric approach to mirror symmetry as a torus duality to construct the mirror of a symplectic manifold equipped with a Lagrangian torus fibration as a moduli space of simple objects of the Fukaya category supported on the fibres. In the absence of singular fibres, the construction of the mirror is explained in this framework, and, given a Lagrangian submanifold, a (twisted) coherent sheaf on the mirror is constructed.
Fukaya's immersed Lagrangian Floer theory and microlocalization of Fukaya category
本研究利用Fukaya浸入拉格朗日Floer理论的幺半性，证明了余切丛Fukaya范畴的生成结果，并将Nadler的可构造层与Fukaya范畴等价定理推广到Novikov域上。
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。
The wrapped Fukaya category of plumbings
本文基于“局部到全局”的计算方法，给出了沿任意箭图的余切丛管道的包裹 Fukaya 范畴的一般公式，并证明了无势分次箭图的 Ginzburg dg 范畴与之等价。
Consider a Stein manifold M obtained by plumbing cotangent bundles of manifolds of dimension greater than or equal to 3 at points. We prove that the Fukaya category of closed exact Lagrangians with vanishing Maslov class in M is generated by the compact cores of the plumbing. As applications, we classify exact Lagrangian spheres in A2-Milnor fibres of arbitrary dimension, derive constraints on exact Lagrangian fillings of Legendrian unknots in disk cotangent bundles, and prove that the categorical equivalence given by the spherical twist in a homology sphere is typically not realised by any co…
Microlocal Morse theory of wrapped Fukaya categories
本文推广 Nadler–Zaslow 对应，用 Reeb 轨迹与范畴紧的无限秩层刻画无穷远态射，证明停止的部分包裹 Fukaya 范畴等价于微支撑受限层的紧对象范畴，并描述稳定极化 Weinstein 扇形。
Let N be a closed four dimensional manifold which admits a self-indexing Morse function f with only 3 critical values 0,2,4, and a unique maximum and minimum. Let g be a Riemannian metric on N such that (f,g) is Morse-Smale. We construct from (N,f,g) a certain six dimensional exact symplectic manifold M, together with some exact Lagrangian spheres V_4, V_2^j, V_0 in M, j=1,...,k. These spheres correspond to the critical points x_4, x_2^j, x_0 of f, where the subscript indicates the Morse index. (In a companion paper we explain how (M, V_4,{V_2^j},V_0) is a model for the regular fiber and vanis…
Hochschild Cohomology of the Fukaya Category via Floer Cohomology with Coefficients
本文通过Floer上同调系数构造Fukaya范畴的Hochschild上同调，证明了单调极值流形的实部可生成Fukaya范畴，并给出了极值纤维分裂生成Fukaya范畴的新证明。
We study versions of homological mirror symmetry for hypersurface cusp singularities and the three hypersurface simple elliptic singularities. We show that the Milnor fibres of each of these carries a distinguished Lefschetz fibration; its derived directed Fukaya category is equivalent to the derived category of coherent sheaves on a smooth rational surface $Y_{p,q,r}$. By using localization techniques on both sides, we get an isomorphism between the derived wrapped Fukaya category of the Milnor fibre and the derived category of coherent sheaves on a quasi-projective surface given by deleting …
We study symplectic invariants of the open symplectic manifolds $X_Γ$ obtained by plumbing cotangent bundles of 2-spheres according to a plumbing tree $Γ$. For any tree $Γ$, we calculate (DG-)algebra models of the Fukaya category $\mathcal{F}(X_Γ)$ of closed exact Lagrangians in $X_Γ$ and the wrapped Fukaya category $\mathcal{W}(X_Γ)$. When $Γ$ is a Dynkin tree of type $A_n$ or $D_n$ (and conjecturally also for $E_6,E_7,E_8$), we prove that these models for the Fukaya category $\mathcal{F}(X_Γ)$ and $\mathcal{W}(X_Γ)$ are related by (derived) Koszul duality. As an application, we give explicit…
Holomorphic Floer Theory and the Fueter Equation
本研究提出与超Kähler流形相关的2-范畴$\u200b\textrm{Fuet}_M$构建方案，证明Fueter映射的基本分析结果，并在余切丛情形下建立其与复梯度轨迹、Fukaya–Seidel范畴的关联，为全纯Floer理论提供新的范畴化框架。
This is the first of a series of two articles where we construct a version of wrapped Fukaya category $\mathcal W\mathcal F(M\setminus K;H_{g_0})$ of the cotangent bundle $T^*(M \setminus K)$ of the knot complement $M \setminus K$ of a compact 3-manifold $M$, and do some calculation for the case of hyperbolic knots $K \subset M$. For the construction, we use the wrapping induced by the kinetic energy Hamiltonian $H_{g_0}$ associated to the cylindrical adjustment $g_0$ on $M \setminus K$ of a smooth metric $g$ defined on $M$. We then consider the torus $T = \partial N(K)$ as an object in this c…
There is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces. We investigate symplectic aspects of this relationship, with a view to applications in low-dimensional topology. We construct a derived equivalence between the Fukaya category of a curve and the nilpotent summand of the Fukaya category of the associated complete intersection of two quadrics. This essentially determines the instanton Floer homology of a 3-manifold fibred by genus two curves.
Let $X$ be a real analytic manifold, and let $T^*X$ be its cotangent bundle. In a recent paper with E. Zaslow \cite{NZ}, we showed that the dg category $Sh_c(X)$ of constructible sheaves on $X$ quasi-embeds into the triangulated envelope $F(T^*X)$ of the Fukaya category of $T^*X$. We prove here that the quasi-embedding is in fact a quasi-equivalence. When $X$ is complex, one may interpret this as a topological analogue of the identification of Lagrangian branes in $T^*X$ and holonomic $D_X$-modules developed by Kapustin and Kapustin-Witten from a physical perspective. As a concrete application…
Cusped hyperbolic Lagrangians as mirrors to lines in three-space
本文构造了三维环面余切丛中带孤立锥形奇点的尖点双曲拉格朗日子流形及对应浸入拉格朗日，证明其经局部系统修饰后可作为射影三维空间中直线的同调镜像，同时构造了适配浸入对象的包裹Fukaya范畴。
We introduce Morse branes in the Fukaya category of a holomorphic symplectic manifold, with the goal of constructing tilting objects in the category. We give a construction of a class of Morse branes in the cotangent bundles, and apply it to give the holomorphic branes that represent the big tilting sheaves on flag varieties.
We formulate and prove a Riemann-Hilbert correspondence between $\hbar$-differential equations and sheaf quantizations, which can be considered as a correspondence between two kinds of quantizations (deformation and sheaf quantization) of holomorphic cotangent bundles. The latter category is expected to be equivalent to a version of Fukaya category, which is a "quantization" of Lagrangian intersection theory. The ideas of the constructions are based on asymptotic/WKB analysis, which is related to geometric quantization.
This paper explores a refinement of homological mirror symmetry which relates exact symplectic topology to arithmetic algebraic geometry. We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes of coherent sheaves on the Tate curve over the "formal disc" Spec Z[[q]]. It specializes to a derived equivalence, over Z, of the Fukaya category of the punctured torus with perfect complexes on the curve y^2+xy=x^3 over Spec Z, the central fibre of the Tate curve; and, over the "punctured disc" Spec Z((q)), to an integral…
We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We then describe the topology of the regular and singular fibres, in particular calculating their middle Betti numbers. For the example of sl(2,C) we compute the Fukaya--Seidel category of Lagrangian vanishing cycles.
Circle-actions, quantum cohomology, and the Fukaya category of Fano toric varieties
该论文定义容许Fano环面流形，证明其闭弦镜像对称性及缠绕Fukaya范畴的环面生成准则，并推广相关哈密顿类与猜想。
Kodaira fibres and wrapped Floer cohomology
本文证明奇异椭圆纤维邻域为Weinstein域，计算其环绕Floer上同调集中于零次，并建立与显式代数的范畴等价，应用于镜像对称。
We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of…
We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology, over fields of characteristic zero. The key ingredient is the construction of a degree one Hochschild cohomology class on a Floer A-infinity algebra associated to the (k,k)-nilpotent slice Y, obtained by counting holomorphic discs which satisfy a suitable conormal condition at infinity in a partial compactification of Y. The partial compactification is obtained as the Hilbert scheme of a partial compactification of a Milnor fibre. A sequel to this paper will prove forma…
In this expository paper we discuss a project regarding the string topology of a manifold, that was inspired by recent work of Moore-Segal, Costello, and Hopkins and Lurie, on "open-closed topological conformal field theories". Given a closed, oriented manifold M, we describe the "string topology category" S_M, which is enriched over chain complexes over a fixed field k. The objects of S_M are connected, closed, oriented submanifolds N of M, and the complex of morphisms between N_1 and N_2 is a chain complex homotopy equivalent to the singular chains C_*(P_{N_1, N_2}), where C_*(P_{N_1, N_2}) …
Weinstein Handlebodies for the Painlevé Betti Spaces
该研究证明十个Painlevé方程对应的Betti模空间一般纤维等价于沿Stokes Legendrian附着Weinstein柄的结果，给出对应柄体图，并将相关Weinstein流形Gompf范式表示方法推广到带边曲面情形。
Homological mirror symmetry for log Calabi-Yau surfaces
本文针对具有极大边界的对数 Calabi-Yau 曲面构造镜像 Lefschetz 纤维化，建立有向与缠绕 Fukaya 范畴同相干层范畴的显式同构，并与 Gross-Hacking-Keel 的 almost-toric 纤维化及已有镜像势构造相匹配。
This is a survey of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algebraic geometry' of Toen and Lurie. They are a 2-category truncation of the 'derived manifolds' of Spivak (see arXiv:0810.5174, arXiv:1212.1153). The category of manifolds Man embeds in dMan as a full subcategory. We also define 2-categories dMan^b,dMan^c of "d-manifolds with bound…
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