Higher dimensional Yang-Mills field with symmetry and quaternionic structure

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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.