SUPERSTRING FIELD THEORY AND INTEGRABLE SYSTEMS

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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.