Moduli space and infinite dimensional geometry

福谷，小野，太田和哦一起构建了一个障碍理论的拉格朗日相交Floer同调被很好地定义。在此基础上，建立了拉格朗日子流形的（代数）形变理论和拉格朗日子流形的量子形变理论。我们在2000年12月完成了这本书的初步版本（350页）。之后，我们在同伦代数部分取得了进展，并澄清了与经典同伦类型等的关系，因此我们现在正在添加更多的材料（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Jakaya-Ono在内的几位数学家完成。在我们的开弦版本中，例如，我们需要更仔细地处理同调代数部分，因此我们需要为此目的发展同伦代数本身。模空间的定向也是一个微妙的问题，因为伪全纯d的模空间 关于我们 岛不具有几乎复杂的结构。此外，我们需要额外的参数来计算分析的细节，主要是因为我们需要在链的层次上工作。在计算拉格朗日交Floer理论的细节时，我们对量子场论与世纪后半期发展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，同伦代数和局部变形理论之间的密切关系。Nakajima追求他的研究，以构建有趣的代数结构的基础上模空间。在他的研究中，各种例子应该发挥中心作用的理论进行了详细研究明确和各种有趣的新的代数结构的建设。古田和他的合著者进一步研究了模空间之间的关系在4维规范理论和稳定的同伦理论。这一观点在古田证明四维流形的交形的10/8定理中已经出现。通过近年来的研究，给出了一些有趣的应用，如构造同调3球面的新的不变量，研究曲面在两个K3曲面的连通和中的嵌入等。f在Less中的曲面嵌入

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