Moduli space and infinite dimensional geometry

模空间和无限维几何

基本信息

批准号：
09304008
负责人：
FUKAYA Kenji
金额：
$ 17.92万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A).
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 2000
项目状态：
已结题

项目摘要

Fukaya, Ono, Ohta together with Oh constructed an obstruction theory for Lagrangian intersection Floer homology to be well defined. Based on it, an (algebraic) deformation theory of Lagrangian submanifold and quantum deformation of Lagrangian submanifold is constructed. We finished a preliminary version of the book (of 350 pages) describing them in Dec. 2000. After than we made a progress on homotopical algebra part and clarify the relation to classical homotopy type etc. So we are now adding more material (approximately 100 pages).Our Lagrangian intersection Floer theory is an open string version of the theory of Gromov-Witten invariant which are completed by several mathematicians including Fukaya-Ono.In our case of open string version, we need more careful treatment on homological algebra part for example so that we need to develop homotopical algebra itself for this purpose.The orientation of the moduli space is also a delicate question since the moduli space of pseudoholomorphic d … More isks do not carry an almost complex structure. Moreover we need additional argument to work out analytic detail mainly because we need to work in the chain level.While working out the detail of the Lagrangian intersection Floer theory, we have a better understanding of the relation between quantum field theory and various notions developed in the late half of the 20 th century. For example we observed a close relation between Feynman diagram, homotopical algebra and of local deformation theory.Nakajima pursuit his study to construct interesting algebraic structures based on moduli spaces. In his study, various example which are supposed to play the central role in the theory is studied in detail explicitly and various interesting new algebraic structures are constructed.Furuta and his coauthors further studied a relation between moduli space in 4 dimensional gauge theory and stable homotopy theory. This point of view is already appeared in Furuta's proof of 10/8 theorem of intersection form of 4 manifolds. By recent development, several interesting applications such as construction of new invariant of homology 3 spheres and study of embedding of surface in the connected sum of two K3 surfaces. are obtained.f embedding of surface in the Less

Fukaya、Ono、Ohta 和 Oh 一起构建了拉格朗日交集弗洛尔同调的阻碍理论，在此基础上，构建了拉格朗日子流形的（代数）变形理论和拉格朗日子流形的量子变形理论。 2000 年 12 月描述它们的书（共 350 页）。之后我们在同伦代数部分取得了进展，并澄清了因此，我们现在正在添加更多材料（大约 100 页）。我们的拉格朗日交集 Floer 理论是 Gromov-Witten 不变量理论的开弦版本，由包括 Fukaya-Ono 在内的几位数学家完成。在我们的开弦版本的情况下，我们需要对同调代数部分进行更仔细的处理，因此我们需要为此目的开发同伦代数本身。模空间也是一个微妙的问题，因为伪全纯的模空间不具有几乎复杂的结构，而且我们需要额外的参数来计算出分析细节，主要是因为我们需要在链级别上工作。通过拉格朗日交弗洛尔理论的细节，我们对量子场论和20世纪下半叶发展起来的各种概念之间的关系有了更好的理解，例如我们观察到了费曼图、同伦图之间的密切关系。中岛致力于基于模空间构造有趣的代数结构。在他的研究中，详细研究了各种被认为在理论中发挥核心作用的例子，并提出了各种有趣的新代数结构。 Furuta 和他的合著者进一步研究了 4 维规范理论中的模空间与稳定同伦理论之间的关系，这一观点已经出现在 Furuta 的 10/8 证明中。通过最近的发展，得到了一些有趣的应用，例如同调3球的新不变量的构造以及两个K3曲面的连通和中的曲面嵌入的研究。f 曲面在Less中的嵌入。