N-fold supersymmetry and its extension to multi-particle system and field theorie

本文研究了超对称量子力学中N重超对称性的推广。作者从一维量子力学中微扰系数的渐近行为中发现了N重超对称性。由于这种对称性有许多普通超对称性的特征，它对多粒子系统和场论的扩展显然是重要的。本项目就是围绕这一点展开的。在项目期间，在几个方向上进行了许多试验。然而，本项目的主要成果是通过一系列求解超对称代数的长系列计算，发现了三维量子力学中的二重超对称。在该构造中，对超荷的形式和哈密顿量做了一些解释。2重超对称代数导出了一组关于ansatz中函数的非线性偏微分方程组。要得到的函数大约有15个，因此很难求出解。然而，我们已经成功地证明了解决方案的存在，并确定了其中的几个解决方案，从而首次能够构建三维模型。可解模型在所有类别中的重要性是显而易见的。在本项目中，形状发明模型是可解的。因此，我们在上述建设中一直在寻找它们。我们即将得出这样的模式不存在的结论，尽管这在某种程度上是初步的。

In this research, extention of the N-fold supersymmetry in supersymmetric quantum mechanics was investigated. N-fold supersymmetry was found from the asymptotic behaviour of the perturbative coefficients in 1-dimensional quantum mechanics by the present investigator. Since this symmetry shares many features of the ordinary supersymmetry, its extension to the multi-particle system and field theories is apparently important. This project was focused on this point. Many trials were made in several directions during the project period. The main result of this project, however, is the finding of the 2-fold supersymmetry in 3-dimensional quantum mechanics, through a long-series of calculation for finding the solution to the supersymmetry algebra.In the said construction, some ansatz were made for the form of the supercharge and the Hamiltonian. The 2-fold supersymmetry algebra induces a set of non-linear partial differential equations for the functions in the ansatz. There are about 15 functions to be obtained and thus solution is rather difficult to come by. We, however, have managed to show that the solution exists and identified several of them, thereby enabling the construction of the 3-dimensional model for the first time.The importance of the solvable model in all categories is evident. In this project, the shape-invatiant models are solvable. Therefore, we have been searching for them in our construction described above. We are close to concluding that such a model does not exist, although this is somewhat preliminary.