Higher dimensional Yang-Mills field with symmetry and quaternionic structure

我们主要学习了三个领域。首先讨论了Yang-Mills场与一元不变微分方程Garnier系统之间的关系。二是对费曼路径积分的重新表述。第三种是基础不完善的sit理论的非标准模型。我们在下面解释这些。(i)①^6作为坐标之一嵌入G_2（①^5）中。G_2（ⅱ^5）是一个四分离子kahler流形，在其上得到了广义反自对偶连接（GASD）的概念。由于四维Jordan群作用于¢^5，它作用于G_2（¢^5）。这些Jordan群对应于young图进行分类。另一方面，对于线性方程d/d3 u = A(3, s, t)u，其中A为ⅱ^2值，我们有一个等单调变形的概念。众所周知，奇点的分类也与约旦群相对应。主要的结果是：(1)我们得到了一系列保持young图的GASD方程，(2)其中两个方程正好等于Garnier系统。（ii）费曼定义了“费曼路径积分”来量子化经典力学和场论。它在某种意义上是一个积分，但是是在无限小的维度空间上。我们定义了双超积、双无穷小晶格和无穷小傅立叶变换，并根据费曼-希布斯的文本重新表述了量子电动力学的费曼路径积分。(3)不具有正则性的集合论称为非成立良好的集合论。例如，Acsel、Scott、Finsler Boffa集合论都是众所周知的。我们构造一个具有任意序长圆形和非圆形类型的路径。此外，对于Boffa集合论，我们研究了一个包含包含映射的原始Boffa集合论的非标准模型。

We have studied mainly three fields. First is about the relationship between Yang-Mills field and Garnier system that is a differential equation of monodromy invariant. The second is a reformulation of Feynman path integral. The third is a nonstandard model of nonwell founded sit theory. We explain these in the following.(i) ¢^6 is embedded in G_2(¢^5) as one of coordinates. G_2(¢^5) is a quater nionic kahler manifold, over it we have a concept of generalized anti-self-dual connection (GASD). Since 4 dimensional Jordan groups act on ¢^5, it act on G_2 (¢^5). These Jordan groups are classified corresponding to young diagrams. On the other hand for linear equation d/d3 u = A(3, s, t)u, where A is ¢^2-valued, we have a concept of isomonodromic deformation. It is known that the singularities are classified corresponding also to Jordan groups. The main results are (1) we write down a list of GASD equations preserving young diagram, (2) two of the equations are just equal to Garnier systems.(ii) Feynman defined "Feynman path integral" to quantise classical mechanics and field theory. It is some sence an integral but on an infinitesimally dimensional space. We define a double ultra product and a double infinitesimal lattice and an infinitesimal Fourier transformation, and reformulate a Feynman path integral of quantum electrodynamics followed Feynman-Hibbs'text.(iii) A set theory without regularity are called non-well-founded set theory. For example, Acsel, Scott, Finsler Boffa set theory are known. We construct a path with any ordinal length of circular and noncircular types. Furthermore, for Boffa set theory, we study a nonstandard model that includes an original Boffa set theory with an inclusion mapping.