SUPERSTRING FIELD THEORY AND INTEGRABLE SYSTEMS

从三年前开始到现在，粒子物理学的主旋律已经发生了很大的变化。当时超弦模型与完全可积系统之间的密切联系才刚刚被发现，粒子论的基础研究应该沿着哪个方向发展还没有定论。但是，由于量子群理论在数学上的进步，各个领域之间的关系变得清晰起来，现在一个伟大的研究潮流似乎在继续。关于这项研究的主题，即，为了从可积系统的观点来构造超弦场论，现在很自然地将基于量子群理论的理论进行扩展，以便将弦模型推广到量子场论。从这个观点出发，我们首先试图把描述弦模型大对称性的Virasoro代数推广到量子群。这并不完全成功，但通过这项研究，我们可以找到一个重要的结果。也就是说，它是可积系统对独立变量离散化的稳定性，在可积系统保持可积性的变形中起着最重要的作用。为了进行交流和完成研究，我们在东京首都大学组织了这个主题的研讨会。通过大量的讨论，阐明了可积系统与动力系统之间的关系。此外，证明了弦的关联函数所满足的Hirota差分方程可以表示为一个离散动力系统。这些结果在一些会议上提出，其中一些也将在论文中发表。

Comparing the time when this project started three years ago and present, the main theme in the theory of particle, physics has been greatly changed. At that time the intimate connection between the super-string model and the completely integrable systems has just been discovered, there was not known to which direction the fundamental study of particle theory should go along. But, by means of the mathematical progress of the theory of quantum groups, relations among various fields became clear, and now a great stream of study seems to go on.Concerning the theme of this study, i.e., to construct a super-string field theory from the view of integrable systems, it is now natural to extend the theory based on the theory of quantum groups in order to generalize the string model to a quantum field theory. From this point of view we first tried to extend the Virasoro algebra, which characterizes the large symmetry of the string model, to a quantum group. This was not completely successful but through this research we could find an important result. Namely it is the stability of the integrable systems against a discretization of independent variables, that plays the most important role in the deformation of integrable systems preserving the integrability. To get communication and complete our research we organized symposium of this theme at Tokyo Metropolitan University. There through many discussions the relation between integrable systems and dynamical systems has been clarified. In addition it was shown that the Hirota difference equation, which is satisfied by correlation functions of strings, could be formulated as a discrete dynamical systems. These results were presented at some conferences and also some of them are to be published in the papers.