scholarly journals Fukaya categories of plumbings and multiplicative preprojective algebras

2019 ◽  
Vol 10 (4) ◽  
pp. 777-813
Author(s):  
Tolga Etgü ◽  
Yankı Lekili
2007 ◽  
Vol 313 (1) ◽  
pp. 165-175 ◽  
Author(s):  
Pavel Etingof ◽  
Frédéric Latour ◽  
Eric Rains

1998 ◽  
Vol 205 (2) ◽  
pp. 391-412 ◽  
Author(s):  
Karin Erdmann ◽  
Nicole Snashall

2019 ◽  
Vol 155 (2) ◽  
pp. 372-412 ◽  
Author(s):  
Christopher Brav ◽  
Tobias Dyckerhoff

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.


2020 ◽  
Vol 156 (7) ◽  
pp. 1310-1347
Author(s):  
Yankı Lekili ◽  
Alexander Polishchuk

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.


Author(s):  
Fumihiko Sanda

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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