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

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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.