Antisymplectic involution and Floer cohomology
Antisymplectic involution and Floer cohomology
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DOI：
10.2140/gt.2017.21.1
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发表时间：
2009-12
影响因子：
2
通讯作者：
K. Fukaya;Y. Oh;H. Ohta;K. Ôno
中科院分区：
文献类型：
--
作者：
K. Fukaya;Y. Oh;H. Ohta;K. Ôno
本文的主要目的是研究边界位于\emph{实}拉格朗日子流形上的伪全纯盘的模空间的取向，即辛流形上的反辛对合$\tau$的不动点集。对于反辛对合$\tau$，我们引入了$\tau$ -相对自旋结构的概念，并研究了对合$\tau$下模空间上的取向行为。我们也将此应用于实拉格朗日子流形的拉格朗日花理论的研究。特别地，我们研究了辛流形$\tau$ -不动点集的不可阻塞性，特别是证明了Calabi-Yau流形的不可阻塞性。并对$\R P^{2n+1}$ / $\Lambda_{0,nov}^{\Z}$的Floer上同构进行了显式计算，给出了一个Floer上同构与其经典上同构不同构的例子。研究了辛流形的平方对角线的Floer上同调，得到了辛流形的量子Massey积的完全广义的严格构造。
The main purpose of the present paper is a study of orientations of the moduli spaces of pseudo-holomorphic discs with boundary lying on a \emph{real} Lagrangian submanifold, i.e., the fixed point set of an anti-symplectic involutions $\tau$ on a symplectic manifold. We introduce the notion of $\tau$-relatively spin structure for an anti-symplectic involution $\tau$, and study how the orientations on the moduli space behave under the involution $\tau$. We also apply this to the study of Lagrangian Floer theory of real Lagrangian submanifolds. In particular, we study unobstructedness of the $\tau$-fixed point set of symplectic manifolds and in particular prove its unobstructedness in the case of Calabi-Yau manifolds. And we also do explicit calculation of Floer cohomology of $\R P^{2n+1}$ over $\Lambda_{0,nov}^{\Z}$ which provides an example whose Floer cohomology is not isomorphic to its classical cohomology. We study Floer cohomology of the diagonal of the square of a symplectic manifold, which leads to a rigorous construction of the quantum Massey product of symplectic manifold in complete generality.