Relevant bibliographies by topics / Fukaya category

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  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters

Academic literature on the topic 'Fukaya category'

Author: Grafiati
Published: 4 June 2021
Last updated: 3 February 2022

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Journal articles on the topic "Fukaya category"

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Castronovo, Marco. "Fukaya category of Grassmannians: Rectangles." Advances in Mathematics 372 (October 2020): 107287. http://dx.doi.org/10.1016/j.aim.2020.107287.

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Pascaleff, James, and Nicolò Sibilla. "Topological Fukaya category and mirror symmetry for punctured surfaces." Compositio Mathematica 155, no. 3 (March 2019): 599–644. http://dx.doi.org/10.1112/s0010437x19007073.

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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$\unicode[STIX]{x1D6F4}$via the topological Fukaya category. We prove that the topological Fukaya category of$\unicode[STIX]{x1D6F4}$is equivalent to the category of matrix factorizations of a certain mirror LG model$(X,W)$. Along the way we establish new gluing results for the topological Fukaya category of punctured surfaces which are of independent interest.
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Nadler, David, and Eric Zaslow. "Constructible sheaves and the Fukaya category." Journal of the American Mathematical Society 22, no. 1 (September 3, 2008): 233–86. http://dx.doi.org/10.1090/s0894-0347-08-00612-7.

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Sheridan, Nick. "Versality of the relative Fukaya category." Geometry & Topology 24, no. 2 (September 23, 2020): 747–884. http://dx.doi.org/10.2140/gt.2020.24.747.

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Haydys, Andriy. "Fukaya-Seidel category and gauge theory." Journal of Symplectic Geometry 13, no. 1 (2015): 151–207. http://dx.doi.org/10.4310/jsg.2015.v13.n1.a5.

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Lekili, Yankı, and Alexander Polishchuk. "Homological mirror symmetry for higher-dimensional pairs of pants." Compositio Mathematica 156, no. 7 (June 18, 2020): 1310–47. http://dx.doi.org/10.1112/s0010437x20007150.

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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 $k\geqslant n$, 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 the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$. By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$-dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$. We also prove similar equivalences for finite abelian covers of the $n$-dimensional pair of pants.
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Abouzaid, Mohammed. "A cotangent fibre generates the Fukaya category." Advances in Mathematics 228, no. 2 (October 2011): 894–939. http://dx.doi.org/10.1016/j.aim.2011.06.007.

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Solomon, Jake P., and Misha Verbitsky. "Locality in the Fukaya category of a hyperkähler manifold." Compositio Mathematica 155, no. 10 (September 6, 2019): 1924–58. http://dx.doi.org/10.1112/s0010437x1900753x.

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Let $(M,I,J,K,g)$ be a hyperkähler manifold. Then the complex manifold $(M,I)$ is holomorphic symplectic. We prove that for all real $x,y$, with $x^{2}+y^{2}=1$ except countably many, any finite-energy $(xJ+yK)$-holomorphic curve with boundary in a collection of $I$-holomorphic Lagrangians must be constant. By an argument based on the Łojasiewicz inequality, this result holds no matter how the Lagrangians intersect each other. It follows that one can choose perturbations such that the holomorphic polygons of the associated Fukaya category lie in an arbitrarily small neighborhood of the Lagrangians. That is, the Fukaya category is local. We show that holomorphic Lagrangians are tautologically unobstructed. Moreover, the Fukaya $A_{\infty }$ algebra of a holomorphic Lagrangian is formal. Our result also explains why the special Lagrangian condition holds without instanton corrections for holomorphic Lagrangians.
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Abouzaid, Mohammed. "On the wrapped Fukaya category and based loops." Journal of Symplectic Geometry 10, no. 1 (2012): 27–79. http://dx.doi.org/10.4310/jsg.2012.v10.n1.a3.

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Abouzaid, Mohammed. "A geometric criterion for generating the Fukaya category." Publications mathématiques de l'IHÉS 112, no. 1 (October 14, 2010): 191–240. http://dx.doi.org/10.1007/s10240-010-0028-5.

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Dissertations / Theses on the topic "Fukaya category"

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Harris, Richard. "The Fukaya category, exotic forms and exotic autoequivalences." Thesis, University of Cambridge, 2012. https://www.repository.cam.ac.uk/handle/1810/242376.

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A symplectic manifold is a smooth manifold M together with a choice of aclosed non-degenerate two-form. Recent years have seen the importance of associatingan A∞-category to M, called its Fukaya category, in helping to understandsymplectic properties of M and its Lagrangian submanifolds. One of the principlesof this construction is that automorphisms of the symplectic manifold shouldinduce autoequivalences of the derived Fukaya category, although precisely whatautoequivalences are thus obtained has been established in very few cases. Given a Lagrangian V ≅ CPn in a symplectic manifold (M,ω), there is anassociated symplectomorphism ∅v of M. In Part I, we defi ne the notion of aCPn-object in an A∞-category A, and use this to construct algebraically an A∞-functor Φv , which we prove induces an autoequivalence of the derived categoryDA. We conjecture that Φv corresponds to the action of ∅v and prove this inthe lowest dimension n = 1. We also give examples of symplectic manifolds forwhich this twist can be defi ned algebraically, but corresponds to no geometricautomorphism of the manifold itself: we call such autoequivalences exotic. Computations in Fukaya categories have also been useful in distinguishing certainsymplectic forms on exact symplectic manifolds from the 'standard' forms. In Part II, we investigate the uniqueness of so-called exotic structures on certainexact symplectic manifolds by looking at how their symplectic properties changeunder small nonexact deformations of the symplectic form. This allows us to distinguishbetween two exotic symplectic forms on T*S3∪2-handle, even though thestandard symplectic invariants such as their Fukaya category and their symplecticcohomology vanish. We also exhibit, for any n, an exact symplectic manifoldwith n distinct, exotic symplectic structures, which again cannot be distinguishedby symplectic cohomology or by the Fukaya category.
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Ganatra, Sheel (Sheel Chandrakant). "Symplectic cohomology and duality for the wrapped Fukaya Category." Thesis, Massachusetts Institute of Technology, 2012. http://hdl.handle.net/1721.1/73362.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 313-315).
Consider the wrapped Fukaya category W of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that natural geometric maps from the Hochschild homology of W to symplectic cohomology and from symplectic cohomology to the Hochschild cohomology of W are isomorphisms, in a manner compatible with ring and module structures. This is a consequence of a more general duality for the wrapped Fukaya category, which should be thought of as a non-compact version of a Calabi-Yau structure. The new ingredients are: (1) Fourier-Mukai theory for W via a wrapped version of holomorphic quilts, (2) new geometric operations, coming from discs with two negative punctures and arbitrary many positive punctures, (3) a generalization of the Cardy condition, and (4) the use of homotopy units and A-infinity shuffle products to relate non-degeneracy to a resolution of the diagonal.
by Sheel Ganatra.
Ph.D.
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3

Peiffer-Smadja, Amiel. "Homologies lagrangiennes, symplectiques et attachement d'anse." Thesis, Sorbonne université, 2018. http://www.theses.fr/2018SORUS370.

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Dans cette thèse, je présente une nouvelle construction du complexe de Fukaya enroulé d’une lagrangienne ainsi que de l’algèbre de Chekanov d’une legendrienne en utilisant des techniques développées par Cieliebak, Ekholm et Oancea. Ces constructions vérifient des propriétés de fonctorialité par rapport aux cobordismes et sont donc adaptées pour étudier un attachement d’anse symplectique. Ainsi, je démontre que le complexe de Fukaya enroulé de la coâme est isomorphe à l’algèbre de Chekanov de la sphère d’attachement d’anse et je montre que cet isomorphisme se factorise par l’application « Open-Closed » de Abouzaid. Je présente ensuite une stratégie pour déduire de ces résultats deux théorèmes importants annoncés par Bourgeois, Ekholm et Eliashberg concernant le comportement de l’homologie symplectique par attachement d’anse et la génération de la catégorie de Fukaya enroulée. Dans le dernier chapitre, je définis en suivant une idée de A’Campo un flot géodésique sur le squelette des variétés de Brieskorn-Pham et je relie ce dernier au flot de Reeb sur l’entrelacs de contact de la singularité dans l’optique de généraliser le théorème de Viterbo qui relie homologie symplectique du cotangent et homologie d’un espace de lacets
In this PhD thesis, I present a new construction of the wrapped Fukaya complex of a Lagrangian and of the Chekanov algebra of a Legendrian using techniques developed by Cieliebak, Ekholm and Oancea. These constructions behave well under cobordisms and thus are fit to study the symplectic handle attachment procedure. I prove that the wrapped Fukaya complex of the cocore is isomorphic to the Chekanov algebra of the attachment sphere and show that this isomorphism factors through Abouzaid’s Open-Closed map. I then give a strategy in order to deduce from these results two important theorems announced by Bourgeois, Ekholm and Eliashberg concerning the behaviour of symplectic homology under handle attachment and the generation of the Fukaya category. In the last chapter, I define following an idea of A’Campo a geodesic flow on the skeleton of a Brieskorn manifold and relate this flow to the Reeb flow on the link of the singularity in order to try to generalize Viterbo’s isomorphism between the symplectic homology of a cotangent bundle and the homology of a loop space
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Zhang, Zhongyi. "On Wrapped Fukaya Category and loop space of Lagrangians in a Liouville Manifold." Thesis, 2020. https://doi.org/10.7916/d8-9xbk-hq04.

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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 $C_{*}(\Omega_{L} \Lag)$ to wrapped Floer cohomology of Lagrangian submanifold $\CW^{-*}(L,L)$. In the case of a cotangent bundle and a Lagrangian co-fiber, the composition of our map with the map from $\CW^{-*}(L,L) \to C_{*}(\Omega_q Q) $ as defined in \cite{Ab12} shows that this map is split surjective.
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Book chapters on the topic "Fukaya category"

1

Zhang, Alex Zhongyi. "Introduction to Symplectic Geometry and Fukaya Category." In Springer Proceedings in Mathematics & Statistics, 129–37. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-91626-2_11.

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2

"Fukaya category and Fourier transform." In Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, 261–74. Providence, Rhode Island: American Mathematical Society, 2001. http://dx.doi.org/10.1090/amsip/023/11.

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Journal articles
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