scholarly journals Fukaya Categories as Categorical Morse Homology

2014 ◽  
Author(s):  
David Nadler
Keyword(s):  
Fukaya Categories ◽  
Morse Homology

MORSE HOMOLOGY (Progress in Mathematics 111)

1995 ◽  
Vol 27 (5) ◽  
pp. 505-507
Author(s):  
Dietmar Salamon
Keyword(s):  
Morse Homology

scholarly journals Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

2017 ◽  
Vol 263 (11) ◽  
pp. 7162-7186 ◽  
Author(s):  
Marek Izydorek ◽  
Thomas O. Rot ◽  
Maciej Starostka ◽  
Marcin Styborski ◽  
Robert C.A.M. Vandervorst
Keyword(s):  
Hilbert Spaces ◽  
Conley Index ◽  
Homotopy Invariance ◽  
Morse Homology

scholarly journals Relative Calabi–Yau structures

2019 ◽  
Vol 155 (2) ◽  
pp. 372-412 ◽  
Author(s):  
Christopher Brav ◽  
Tobias Dyckerhoff
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Riemann Surfaces ◽  
Fukaya Categories ◽  
Composition Law

We introduce relative noncommutative Calabi–Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi–Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi–Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.

scholarly journals Homological mirror symmetry for higher-dimensional pairs of pants

2020 ◽  
Vol 156 (7) ◽  
pp. 1310-1347
Author(s):  
Yankı Lekili ◽  
Alexander Polishchuk
Keyword(s):  
Mirror Symmetry ◽  
Derived Category ◽  
Affine Variety ◽  
Fukaya Category ◽  
Homological Mirror Symmetry ◽  
Endomorphism Algebra ◽  
Generating Set ◽  
Fukaya Categories ◽  
Higher Dimensional ◽  
Abelian Covers

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.

scholarly journals Computation of Quantum Cohomology From Fukaya Categories

2020 ◽  
Author(s):  
Fumihiko Sanda
Keyword(s):  
Symplectic Manifold ◽  
Symplectic Structure ◽  
Blow Up ◽  
Quantum Cohomology ◽  
Fukaya Category ◽  
Fukaya Categories ◽  
Compact Symplectic Manifold

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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