Moduli space and infinite dimensional geometry

Fukaya, Ono， Ohta和Oh共同构造了拉格朗日交花同源性的一个良好定义的阻碍理论。在此基础上，构造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，我们完成了描述他们的书的初版（350页）。在此基础上对同伦代数部分作了进一步的研究，阐明了同伦类型与经典同伦类型的关系等。所以我们现在增加了更多的材料（大约100页）。拉格朗日交Floer理论是由Fukaya-Ono等数学家完成的Gromov-Witten不变量理论的开弦版本。在我们的开弦版本中，我们需要更仔细地处理同调代数部分，例如，我们需要为此目的发展同调代数本身。模空间的方向也是一个微妙的问题，因为伪全纯d的模空间不携带几乎复杂的结构。此外，我们需要额外的论据来制定分析细节，主要是因为我们需要在链层面上工作。在研究拉格朗日交花理论的细节时，我们对量子场论与20世纪后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，我们观察到费曼图、同局部代数和局部变形理论之间的密切关系。Nakajima致力于在模空间上构造有趣的代数结构。在他的研究中，他详细明确地研究了在理论中起中心作用的各种例子，并构造了各种有趣的新代数结构。Furuta等人进一步研究了四维规范理论中模空间与稳定同伦理论之间的关系。这一观点已经出现在古田对4流形交点形式的10/8定理的证明中。通过近年来的发展，一些有趣的应用，如同调3球的新不变量的构造和曲面在两个K3曲面连通和中的嵌入研究。得到了。f在Less中嵌入曲面

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