Chi Hong Chow, Naichung Conan Leung
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.
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Homological mirror symmetry (HMS) asserts that the Fukaya category of a symplectic manifold is derived equivalent to the category of coherent sheaves on the mirror complex manifold. Without suitable enlargement (split closure) of the Fukaya category, certain objects of it are missing to prevent HMS from being true. One possible solution is to include coisotropic branes into the Fukaya category. This paper gives a construction for linear symplectic tori of a version of Fukaya category including coisotropic branes by using a doubling procedure, and discussing the relation between the Fukaya cate…
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.
Semi-orthogonality in Fukaya-Seidel mirrors to blowups of abelian varieties
本文通过证明阿贝尔簇吹胀的同调镜像猜想，利用半正交性在复侧和辛镜像间的范畴对偶性，展示了1参数家族曲面的镜像结构。
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…
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…
In this paper we establish a version of homological mirror symmetry for punctured Riemann surfaces. Following a proposal of Kontsevich we model A-branes on a punctured surface $Σ$ via the topological Fukaya category. We prove that the topological Fukaya category of $Σ$ is equivalent to the category of matrix factorizations of the mirror LG model $(X,W)$. Along the way we establish new gluing results for the topological Fukaya category of punctured surfaces which might be of independent interest.
We prove a version of homological mirror symmetry statement for toric Calabi-Yau $3$-orbifolds, thus extending arXiv:1604.06448 to the case of orbifolds under the mirror symmetry setting considered in arXiv:1604.07123. The B-model is the matrix factorization category for the toric Calabi-Yau $3$-orbifold with a superpotential; while the A-model is a topologically defined Fukaya-type category on its mirror curve.
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。
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。
On fiber and base decompositions in the Fukaya category of a symplectic Landau-Ginzburg model
本文针对一般辛Landau-Ginzburg模型（奇异辛纤维化），证明权重和Lagrangian分次可分解为基分量和纤维分量，从而推广了精确和Lefschetz纤维化下的权重计算方法。
The derived category of coherent sheaves $\mathcal{T}_B$ associated to a birational cobordism which is either a weighted projective space, a stacky Atiyah flip, or a stacky blow-up of a point has a conjectural mirror Fukaya-Seidel category $\mathcal{T}_A$. The potential $W$ defining $\mathcal{T}_{A}$ has base $\mathbb{C}^*$ and exhibits a great deal of symmetry. This paper investigates the structure of the Fukaya-Seidel category for the mirror potentials. A proof of homological mirror symmetry $\mathcal{T}_A \cong \mathcal{T}_B$ for these birational cobordisms is then given.
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.
Given an algebraic hypersurface $H=f^{-1}(0)$ in $(\mathbb{C}^*)^n$, homological mirror symmetry relates the wrapped Fukaya category of $H$ to the derived category of singularities of the mirror Landau-Ginzburg model. We propose an enriched version of this picture which also features the wrapped Fukaya category of the complement $(\mathbb{C}^*)^n\setminus H$ and the Fukaya-Seidel category of the Landau-Ginzburg model $((\mathbb{C}^*)^n,f)$. We illustrate our speculations on simple examples, and sketch a proof of homological mirror symmetry for higher-dimensional pairs of pants.
Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs $X$ and $Y$ such that the complex geometry on $X$ mirrors the symplectic geometry on $Y$. It allows one to deduce symplectic information about $Y$ from known complex properties of $X$. Strominger-Yau-Zaslow arXiv:hep-th/9606040 described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich arXiv:alg-geom/9411018 conjectured that a complex invariant on $X$ (the bounded derived category of coherent sheaves) sho…
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…
The Hamiltonian reduction of hypertoric mirror symmetry
本研究针对乘法超环面簇上的哈密顿环面作用，证明了福卡亚范畴与约化操作可交换的定理，推进了超环面镜像对称相关研究。
We prove a structural result in mirror symmetry for projective Calabi--Yau (CY) manifolds. Let $X$ be a connected symplectic CY manifold, whose Fukaya category $\mathcal{F}(X)$ is defined over some suitable Novikov field $\mathbb{K}$; its mirror is assumed to be some smooth projective scheme $Y$ over $\mathbb{K}$ with `maximally unipotent monodromy'. Suppose that some split-generating subcategory of (a $\mathsf{dg}$ enhancement of) $D^bCoh( Y)$ embeds into $\mathcal{F}(X)$: we call this hypothesis `core homological mirror symmetry'. We prove that the embedding extends to an equivalence of cate…
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。
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 prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make ex…
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…
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。
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。
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.
We work in the setting of Calabi-Yau mirror symmetry. We establish conditions under which Kontsevich's homological mirror symmetry (which relates the derived Fukaya category to the derived category of coherent sheaves on the mirror) implies Hodge-theoretic mirror symmetry (which relates genus-zero Gromov-Witten invariants to period integrals on the mirror), following the work of Barannikov, Kontsevich and others. As an application, we explain in detail how to prove the classical mirror symmetry prediction for the number of rational curves in each degree on the quintic threefold, via the third-…
We introduce a new way to encode semicyclic structures using a stack of broken cycles. (We also prove an analogue for paracyclic structures.) This was motivated not only by higher algebra but also by Fukaya-categorical considerations. We also openly speculate about some Fukaya-categorical implications. For example, this stack sees moduli of stopped Liouville disks, and hence yields another platform for gluing together Fukaya categories. We also see that Lagrangian cobordisms with multiple ends may not only serve to detect $K_0$ groups of Fukaya categories, but higher $K$-theory groups as well.…
For a stopped Liouville manifold arising from a Liouville sector, we construct a symplectic analogue of the formal neighborhood of the stop on the level of Fukaya categories. This geometric construction is performed via Floer-theoretic methods by allowing wrappings in the negative direction. On the other hand, inspired by homological mirror symmetry for pairs, where the mirror is the formal neighborhood of a divisor in an ambient projective variety, there is a different approach by taking a `categorical formal completion' introduced by Efimov. Our main results establishes equivalence of these …
We discuss homological mirror symmetry of Fermat polynomials in terms of derived Morita equivalence between derived categories of coherent sheaves and Fukaya-Seidel categories (a.k.a. perfect derived categories of directed Fukaya categories), and some related aspects such as stability conditions, (kinds of) modular forms, and Hochschild homologies.
Here we carefully construct an equivalence between the derived category of coherent sheaves on an elliptic curve and a version of the Fukaya category on its mirror. This is the most accessible case of homological mirror symmetry. We also provide introductory background on the general Calabi-Yau case of The Homological Mirror Symmetry Conjecture.
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…
Open Gromov-Witten invariants from the Fukaya category
本文构造相对循环同调及保持联络的相对循环开-闭映射，建立由 Fukaya 范畴恢复 Lagrange 子流形开 Gromov-Witten 不变量的框架。
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.
Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures
本文构造Fukaya范畴循环同调到$S^1$-等变上同调的循环开-闭映射，用以建立Calabi-Yau结构并证明非交换Hodge-de Rham退化。
This paper gives a new way of constructing Landau-Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger-Yau-Zaslow and Fukaya-Oh-Ohta-Ono. Moreover we construct a canonical functor from the Fukaya category to the mirror category of matrix factorizations. This functor derives homological mirror symmetry under some explicit assumptions. As an application, the construction is applied to spheres with three orbifold points to produce their quantum-corrected mirrors and derive homological mirror symmetry. Furthermore we discover an enumerativ…
We prove a version of equivariant split generation of Fukaya category when a symplectic manifold admits a free action of a finite group $G$. Combining this with some generalizations of Seidel's algebraic frameworks from Seidel's book, we obtain new cases of homological mirror symmetry for some symplectic tori with non-split symplectic forms, which we call \textit{special isogenous tori}. This extends the work of Abouzaid-Smith. We also show that derived Fukaya categories are complete invariants of special isogenous tori.
We show homological mirror symmetry results relating coherent analytic sheaves on some complex elliptic surfaces and objects of certain Fukaya categories. We first define the notion of a non-algebraic Landau-Ginzburg model on $\mathbb{R} \times S^1$ and its associated Fukaya category, and show that non-Kähler sur-faces obtained by performing two logarithmic transformations to the product of the projective plane and an elliptic curve have non-algebraic Landau-Ginzburg models as their mirror spaces; this class of surface includes the classical Hopf surface $S^1\times S^3$ and other elliptic prim…
Mirror symmetry for Berglund-Hübsch Milnor fibers
该研究通过先计算甚仿射Fukaya范畴再进行形变的方法，计算Berglund-Hübsch可逆多项式的Milnor纤维的Fukaya范畴，大体证明了相关同调镜像对称猜想，核心是确定反常schober的典范延拓。
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.
We formulate some conjectures about the K-theory of symplectic manifolds and their Fukaya categories, and prove some of them in very special cases.
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…
We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety. In particular, we realize tropical Lagrangian sections as objects of a partially wrapped category and construct a Lagrangian correspondence mirror to the inclusion of a toric divisor. As a corollary, we prove that tropical sections generate the Fukaya-Seidel category, completing a Floer-theoretic proof of homological mirror symmetry for projective toric varieties. In the course of the proof, we dev…
We establish a $\mathbb{Z}[[t_1,\ldots, t_n]]$-linear derived equivalence between the relative Fukaya category of the 2-torus with $n$ distinct marked points and the derived category of perfect complexes on the $n$-Tate curve. Specialising to $t_1= \ldots =t_n=0$ gives a $\mathbb{Z}$-linear derived equivalence between the Fukaya category of the $n$-punctured torus and the derived category of perfect complexes on the standard (Néron) $n$-gon. We prove that this equivalence extends to a $\mathbb{Z}$-linear derived equivalence between the wrapped Fukaya category of the $n$-punctured torus and the…
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。
Using Auroux's description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^n$, for $k \geq n$, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $(n+2)$-generic hyperplanes in $\mathbb{C}P^n$ ($n$-dimensional pair-of-pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of…
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…
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。
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 …
Towards Homological Mirror Symmetry
本文综述孔采维奇同调镜像对称猜想的A侧，系统阐述A无穷范畴、福山范畴与弗洛尔理论，并联系镜像对称背景推导主要预测。
We prove that the inverse of a mirror map for a toric Calabi-Yau manifold of the form $K_Y$, where $Y$ is a compact toric Fano manifold, can be expressed in terms of generating functions of genus 0 open Gromov-Witten invariants defined by Fukaya-Oh-Ohta-Ono \cite{FOOO10}. Such a relation between mirror maps and disk counting invariants was first conjectured by Gross and Siebert \cite[Conjecture 0.2 and Remark 5.1]{GS11} as part of their program, and was later formulated in terms of Fukaya-Oh-Ohta-Ono's invariants in the toric Calabi-Yau case in \cite[Conjecture 1.1]{CLL12}.
We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus. We show that the derived Fukaya category of this mirror coincides with the derived category of coherent sheaves on the original manifold.
The cyclic open--closed map and variations of Hodge structures
本文在半正情形下为体形变的相对Fukaya范畴构造循环开-闭映射，证明其是半无限霍奇结构极化变分的态射，并给出同构判据，从而在Calabi-Yau镜像对中由同调镜像对称推出枚举镜像对称。
Homological mirror symmetry for weighted projective spaces and Morse Homotopy
本文将环面流形上的加权Morse同伦范畴与凝聚层导出范畴的同调镜像对称研究推广至环面轨形，讨论了加权射影空间的对应版本同调镜像对称。
We prove homological mirror symmetry for Lefschetz fibrations obtained as disconnected sums of polynomials of types A or D. The proof is based on the behavior of the Fukaya category under the addition of a polynomial of type D.
A twin Lagrangian fibration, originally introduced by Yau and the first author, is roughly a geometric structure consisting of two Lagrangian fibrations whose fibers intersect with each other cleanly. In this paper, we show the existence of twin Lagrangian fibrations on certain symplectic manifolds whose mirrors are fibered by rigid analytic cycles. Using family Floer theory in the sense of Fukaya and Abouzaid, these twin Lagrangian fibrations are shown to be induced from fibrations by rigid analytic subvarieties on the mirror. As additional evidences, we discuss two simple applications of our…
$A_\infty$ categories are a mathematical structure that appears in topological field theory, string topology, and symplectic topology. This paper studies the cyclic homology of a Calabi-Yau $A_\infty$ category, and shows that it is naturally an equivariant topological conformal field theory, and in particular, contains an involutive Lie bialgebra. Applications of the theory to string topology and the Fukaya category are given; in particular, it is shown that there is a Lie bialgebra homomorphism from the cyclic cohomology of the Fukaya category of a symplectic manifold with contact type bounda…
Homological mirror symmetry for hypersurfaces in $(\mathbb{C}^*)^n$
该研究证明$(\u2102^*)^n$中极大退化超曲面族的同调镜像对称，通过定义纤维式缠绕Fukaya范畴构造对应拉格朗日子流形，得出凝聚层导出范畴拟嵌入镜像范畴的结论，还提及完全交推广方向。
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 …
Given a smooth 3-fold $Y$, a line bundle $L \to Y$, and a section $s$ of $L$ such that the vanishing locus of $s$ is a normal crossings surface $X$ with graph-like singular locus, we present a way to reconstruct the singularity category of $X$ as a homotopy limit of several copies of the category of matrix factorizations of $xyz : \mathbb{A}^{3} \to \mathbb{A}^{1}$ (the mirror to the Fukaya category of the pair of pants). This extends our previous result for the case where $L$ is trivialized. The key technique is the classification of non-two-periodic autoequivalences of the category of matrix…
Fixing a weakly unobstructed Lagrangian torus in a symplectic manifold X, we define a holomorphic function W known as the Floer potential. We construct a canonical A-infinity functor from the Fukaya category of X to the category of matrix factorizations of W. It provides a unified way to construct matrix factorizations from Lagrangian Floer theory. The technique is applied to toric Fano manifolds to transform Lagrangian branes to matrix factorizations. Using the method, we also obtain an explicit expression of the matrix factorization mirror to the real locus of the complex projective space.
We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many example, the derived category $D^b(coh(X))$ of coherent sheaves on a toric manifold $X$ is compared with the Fukaya-Seidel category of the Milnor fiber of the corresponding Landau-Ginzburg potential. We instead consider the dual torus fibration $π:M \to B$ of the complement of the toric divisors in $X$, where $\bar{B}$ is the dual polytope of the toric manifold $X$. A natural formulation of homological mirror symmetry in this set-up is to define $Fuk(\bar{M}…
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