论文标题
来自福卡亚类别的开放式Gromov-witten不变性
Open Gromov-Witten invariants from the Fukaya category
论文作者
论文摘要
本文提出了一个框架，以表明Sympletic歧管的福卡亚类别$ x $决定了Lagrangians $ l \ subset x $的开放式gromov-witten不变性。我们将$ a_ \ infty $ - 类别中的对象关联为负循环同源性的扩展，称为\ emph {相对环状同源}。我们将Getzler-Gauss-Manin连接扩展到相对循环同源性。然后，我们（在简化技术假设下）相对环状开放式地图，该地图映射了fukaya $ l $ l $ lagrangian $ l $ fukaya类别的相对环状同源性，符合符号歧管$ x $ to $ s^1 $ equivariant相对量子同源物的相对量子同源物为$（x，l）$。相对量子同源性是所罗门·图金斯基（Solomon-Tukachinsky）构建的相对量子共同体的双重。这是量子共同体的扩展，并配备了扩展量子连接的连接。我们证明，相对开放的地图尊重连接。作为该框架的应用，我们表明，假设在更广泛的技术设置中构造了相对循环开放式地图，则Calabi-yau品种的福卡亚类别决定了开放的Gromov-witten不变性剂，其中一个内部标记了任何无效的Lagrangian Lagrangian Brane。
This paper proposes a framework to show that the Fukaya category of a symplectic manifold $X$ determines the open Gromov-Witten invariants of Lagrangians $L \subset X$. We associate to an object in an $A_\infty$-category an extension of the negative cyclic homology, called \emph{relative cyclic homology}. We extend the Getzler-Gauss-Manin connection to relative cyclic homology. Then, we construct (under simplifying technical assumptions) a relative cyclic open-closed map, which maps the relative cyclic homology of a Lagrangian $L$ in the Fukaya category of a symplectic manifold $X$ to the $S^1$-equivariant relative quantum homology of $(X,L)$. Relative quantum homology is the dual to the relative quantum cohomology constructed by Solomon-Tukachinsky. This is an extension of quantum cohomology, and comes equipped with a connection extending the quantum connection. We prove that the relative open-closed map respects connections. As an application of this framework, we show, assuming a construction of the relative cyclic open-closed map in a broader technical setup, that the Fukaya category of a Calabi-Yau variety determines the open Gromov-Witten invariants with one interior marked point for any null-homologous Lagrangian brane.
