Nonperturbative effects in QFT and new supersymmetry

该研究项目旨在促进对量子理论中非微扰效应的理解，并进一步探索超对称性的新可能性。能够处理非微扰效应的分析方法是虚路径积分形式主义中的谷方法，包括当前项目负责人H。青山利用这种方法对一维量子力学中的非对称双阱势模型进行了研究，发现了N重超对称性，因此打算进一步探索这种新的超对称性，寻找新的模型，这个为期四年的研究项目刚刚完成了这项任务。谷方法适用于各种量子力学模型。从这种方法得到的非微扰效应中，提取了微扰级数的高阶行为。其中一些导致了成功：周期模型被发现具有N重超对称的形式，这比非对称双阱模型更普遍，在某种意义上，它的超荷不仅仅是动量的N次方，而是N阶多项式。受这一发现的启发，另一个模型，一个六重体模型，很快被发现具有N重超对称性，所有这些N重超对称模型都有一个特殊的超荷可因子分解形式。基于此，我们定义了N重超对称的一个子类“A型”，并确定了N重超对称的充要条件。我们完全解决了这些条件，对所有这类模型进行了分类，并写出了它们的势和超荷，有了这些结果，我们现在就有了一个完整的A型理论。探索这个子类之外的模型是未来最重要的任务之一。

This research project was planned for advancement of understanding of the nonperturbative effects in quantum theory and further explore new possibilities of supersymmetry. The analytical method that can handle nonperturbative effects was the valley method in imaginary path-integral formalism, developed a group including the current project leader, H. Aoyama. Using this method for an asymmetric double-well potential model in one-dimensional quantum mechanics, the N-fold supersymmetry had been found. Therefore it was intended to further explore this new supersymmetry, searching for new models.This four-year research project has just accomplished this task. The valley method was applied to various quantum mechanical models. From the nonperturbative effects obtained by this method, the large order behaviour of perturbative series were extracted. Some of them lead to success : a periodic model was found to have a form of N-fold supersymmetry, which were more general than the assymmetric double-well model, in a sense that its supercharge is not just the N-th power of the momentum, but an N-th order polynomial. Inspired by this finding, another model, a sextet model, was soon found to have an N-fold supersymmetry.All these N-fold supersymmetry models shared a special factorizable form of the supercharge. Motivated by this, we defined a subclass "type-A" of N-fold supersymmetry, and identified the necessary and sufficient conditions for N-fold supersymmetry. We solved these conditions completely, classified all the models of this type, and wrote down their potentials and supercharge.With these results, we now have a complete theory of type-A. Exploring models outside this subclass is one of the most important task for future.