Journal articles: 'Fukaya category'

1

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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7

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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8

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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9

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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10

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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Kapustin, Anton, and Dmitri Orlov. "Remarks on A-branes, mirror symmetry, and the Fukaya category." Journal of Geometry and Physics 48, no. 1 (October 2003): 84–99. http://dx.doi.org/10.1016/s0393-0440(03)00026-3.

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12

Johns, Joseph. "Complexifications of Morse functions and the directed Donaldson-Fukaya category." Journal of Symplectic Geometry 8, no. 4 (2010): 403–500. http://dx.doi.org/10.4310/jsg.2010.v8.n4.a3.

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13

Ritter, Alexander F., and Ivan Smith. "The monotone wrapped Fukaya category and the open-closed string map." Selecta Mathematica 23, no. 1 (August 9, 2016): 533–642. http://dx.doi.org/10.1007/s00029-016-0255-9.

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14

Sheridan, Nick. "On the Fukaya category of a Fano hypersurface in projective space." Publications mathématiques de l'IHÉS 124, no. 1 (February 15, 2016): 165–317. http://dx.doi.org/10.1007/s10240-016-0082-8.

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15

Dyckerhoff, Tobias. "-homotopy invariants of topological Fukaya categories of surfaces." Compositio Mathematica 153, no. 8 (June 9, 2017): 1673–705. http://dx.doi.org/10.1112/s0010437x17007205.

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We provide an explicit formula for localizing$\mathbb{A}^{1}$-homotopy invariants of topological Fukaya categories of marked surfaces. Following a proposal of Kontsevich, this differential$\mathbb{Z}$-graded category is defined as global sections of a constructible cosheaf of dg categories on any spine of the surface. Our theorem utilizes this sheaf-theoretic description to reduce the calculation of invariants to the local case when the surface is a boundary-marked disk. At the heart of the proof lies a theory of localization for topological Fukaya categories which is a combinatorial analog of Thomason–Trobaugh’s theory of localization in the context of algebraic$K$-theory for schemes.

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16

Ritter, Alexander. "Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties." Geometry & Topology 20, no. 4 (September 15, 2016): 1941–2052. http://dx.doi.org/10.2140/gt.2016.20.1941.

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17

Hedden, Matthew, Christopher M. Herald, Matthew Hogancamp, and Paul Kirk. "The Fukaya category of the pillowcase, traceless character varieties, and Khovanov cohomology." Transactions of the American Mathematical Society 373, no. 12 (October 5, 2020): 8391–437. http://dx.doi.org/10.1090/tran/8116.

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18

Cooper, Benjamin, and Peter Samuelson. "THE HALL ALGEBRAS OF SURFACES I." Journal of the Institute of Mathematics of Jussieu 19, no. 3 (October 22, 2018): 971–1028. http://dx.doi.org/10.1017/s1474748018000324.

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We study the derived Hall algebra of the partially wrapped Fukaya category of a surface. We give an explicit description of the Hall algebra for the disk with $m$ marked intervals and we give a conjectural description of the Hall algebras of all surfaces with enough marked intervals. Then we use a functoriality result to show that a graded version of the HOMFLY-PT skein relation holds among certain arcs in the Hall algebras of general surfaces.

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19

Nadler, David. "Springer theory via the Hitchin fibration." Compositio Mathematica 147, no. 5 (July 29, 2011): 1635–70. http://dx.doi.org/10.1112/s0010437x1100546x.

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AbstractWe develop the Springer theory of Weyl group representations in the language of symplectic topology. Given a semisimple complex group G, we describe a Lagrangian brane in the cotangent bundle of the adjoint quotient 𝔤/G that produces the perverse sheaves of Springer theory. The main technical tool is an analysis of the Fourier transform for constructible sheaves from the perspective of the Fukaya category. Our results can be viewed as a toy model of the quantization of Hitchin fibers in the geometric Langlands program.

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20

AMORIM, LINO, YONG–GEUN OH, and JOANA OLIVEIRA DOS SANTOS. "Exact Lagrangian submanifolds, Lagrangian spectral invariants and Aubry–Mather theory." Mathematical Proceedings of the Cambridge Philosophical Society 165, no. 3 (August 31, 2017): 411–34. http://dx.doi.org/10.1017/s0305004117000561.

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AbstractWe construct graph selectors for compact exact Lagrangians in the cotangent bundle of an orientable, closed manifold. The construction combines Lagrangian spectral invariants, developed by Oh, and results, by Abouzaid, about the Fukaya category of a cotangent bundle. We also introduce the notion of Lipschitz-exact Lagrangians and prove that these admit an appropriate generalisation of graph selector. We then, following Bernard–Oliveira dos Santos, use these results to give a new characterisation of the Aubry and Mañé sets of a Tonelli Hamiltonian and to generalise a result of Arnaud on Lagrangians invariant under the flow of such Hamiltonians.

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21

Abouzaid, Mohammed. "Homological mirror symmetry without correction." Journal of the American Mathematical Society 34, no. 4 (May 24, 2021): 1059–173. http://dx.doi.org/10.1090/jams/973.

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Let X X be a closed symplectic manifold equipped with a Lagrangian torus fibration over a base Q Q . A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space Y Y , which can be considered as a variant of the T T -dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of tautologically unobstructed graded Lagrangians in X X embeds fully faithfully in the derived category of (twisted) coherent sheaves on Y Y , under the technical assumption that π 2 ( Q ) \pi _2(Q) vanishes (all known examples satisfy this assumption). The main new tool is the construction and computation of Floer cohomology groups of Lagrangian fibres equipped with topological infinite rank local systems that correspond, under mirror symmetry, to the affinoid rings introduced by Tate, equipped with their natural topologies as Banach algebras.

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22

LAUDA, AARON D., and HENDRYK PFEIFFER. "STATE SUM CONSTRUCTION OF TWO-DIMENSIONAL OPEN-CLOSED TOPOLOGICAL QUANTUM FIELD THEORIES." Journal of Knot Theory and Its Ramifications 16, no. 09 (November 2007): 1121–63. http://dx.doi.org/10.1142/s0218216507005725.

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We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a two-dimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an open-closed TQFT with a finite set of D-branes using the example of the groupoid algebra of a finite groupoid.

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23

Trinus, K. F., and K. F. Claussen. "International clinical protocol of vestibular disorders (dizziness)." East European Journal of Neurology, no. 4(4) (September 20, 2015): 4–47. http://dx.doi.org/10.33444/2411-5797.2015.4(4).4-47.

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Vertigo - a common symptom, which traditionally presents the results of vestibular dysfunction and non vestibular dysfunction. Vertigo refers to violations of orientation in space. The main symptoms, which produce the patients in the survey: dizziness, subjective vertigo, dizziness objective, pseudo vertigo, imbalance, kinetosis. In many cases, dizziness is functional, rather than an organic nature. Diagnosis of the causes of vertigo arises from the concept of the structure of the vestibular apparatus, the main idea of which is the formation of vertigo in the vestibular system. Morpho-physiological, vestibular system consists of four main projections: vestibular-cortical (sensory), vestibular-motor, vestibular-autonomic and vestibular-limbic. One of the main methods of research of a condition the vestibular apparatus and projections are vestibular evoked potentials (VEP). Owing to this method, have been established objective limits to the sensitivity of the movement. Research of vestibular-spinal responses are based on Romberg test, Unterberger-Fukuda test and Uemura test. Evaluation and treatment of patients with dizziness will differ significantly once the category of dizziness has been determined

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24

Auroux, Denis, and Ivan Smith. "Fukaya categories of surfaces, spherical objects and mapping class groups." Forum of Mathematics, Sigma 9 (2021). http://dx.doi.org/10.1017/fms.2021.21.

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Abstract We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $2$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $1$ 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 wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included.

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25

Sanda, Fumihiko. "Computation of Quantum Cohomology From Fukaya Categories." International Mathematics Research Notices, May 11, 2020. http://dx.doi.org/10.1093/imrn/rnaa089.

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Abstract Assume the existence of a Fukaya category $\textrm{Fuk}(X)$ of a compact symplectic manifold $X$ with some expected properties. In this paper, we show $\mathscr{A} \subset \textrm{Fuk}(X)$ split generates a summand $\textrm{Fuk}(X)_e \subset \textrm{Fuk}(X)$ 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.

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26

Mak, Cheuk Yu, and Weiwei Wu. "Dehn twists and Lagrangian spherical manifolds." Selecta Mathematica 25, no. 5 (November 1, 2019). http://dx.doi.org/10.1007/s00029-019-0515-6.

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Abstract We study Dehn twists along Lagrangian submanifolds that are finite free quotients of spheres. We describe the induced auto-equivalences to the derived Fukaya category and explain their relations to mirror symmetry.

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27

Haiden, Fabian. "Legendrian skein algebras and Hall algebras." Mathematische Annalen, May 31, 2021. http://dx.doi.org/10.1007/s00208-021-02212-8.

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AbstractWe compare two associative algebras which encode the “quantum topology” of Legendrian curves in contact threefolds of product type $$S\times {\mathbb {R}}$$ S × R . The first is the skein algebra of graded Legendrian links and the second is the Hall algebra of the Fukaya category of S. We construct a natural homomorphism from the former to the latter, which we show is an isomorphism if S is a disk with marked points and injective if S is the annulus.

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28

Kinoshita, T., H. Yuzawa, R. Wada, S. Yao, K. Yano, K. Akitsu, M. Shinohara, et al. "P93 The usefulness of dual cardiac autonomic nervous modulation assessment for prediction of mortality in patients with relatively preserved left ventricular ejection fraction." European Heart Journal 41, Supplement_1 (January 1, 2020). http://dx.doi.org/10.1093/ehjci/ehz872.041.

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Abstract Background Recent guidelines have stated that reduced left ventricular ejection fraction (LVEF) is the gold standard marker for identifying patients at risk for cardiac mortality. Although reduced LVEF identifies patients at an increased risk of cardiac arrest, sudden cardiac deaths (SCDs) occur considerably more often in patients with relatively preserved LVEF. Current guidelines on SCDs risk stratification do not adequately cover this general population pool. Heart rate variability (HRV) and heart rate turbulence (HRT) are non-invasive electrocardiography (ECG)-based techniques capable of providing relevant information on the cardiac autonomic nervous modulation. Although a large body of evidence about autonomic nervous modulation markers has been reported, the usefulness of HRV and HRT parameters for risk stratification in such patients with relatively preserved LVEF has not yet been elucidated. Purpose This study aimed to evaluate HRV and HRT parameters for predicting cardiac mortality in patients with structural heart disease (SHD), including ischemic heart disease, dilated cardiomyopathy and valvular heart disease, who have mid-range left ventricular dysfunction (LVD). Methods We prospectively enrolled 229 patients (187 men, age 63 ± 13 years) with SHD who have mid-range LVD (LVEF > 40%). HRV and HRT parameters based on 24-hour ambulatory ECG recordings (Fukuda Denshi Co., Ltd., Tokyo, Japan) were evaluated as follows; SDNN, triangular index, high and low frequency HRV, turbulence onset and slope. The primary endpoint was all-cause mortality. Univariate and multivariate Cox regression analysis were used to assess the association between these cardiac autonomic nervous modulation and mortality. Results During a mean follow-up of 21 ± 11 months, all-cause mortality was seen in 11 (4.8%) patients. Univariate Cox regression analysis showed that reduced SDNN (<50ms), reduced triangular index (<20ms) and HRT category 2 were significantly associated with the primary endpoint (P < 0.05). When HRT category 2 combined with reduced SDNN, Multivariate Cox regression analysis revealed that this combination more strongly associates with the primary endpoint (hazard ratio =7.91, 95%CI, 1.82-34.2; P = 0.006). Conclusion Dual cardiac autonomic nervous modulation assessment which combined HRT and HRV could be a superior technique to predict mortality in patients with relatively preserved LVEF.

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Journal articles on the topic 'Fukaya category'

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