A Study of Unified Super String Theory Based on Integrable Systems

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We can summarize the results of this research project in three main parts.1. Berezin quantization and string correlation functionsWe have attempted, in the research project from 1994 to 1996 (project number 06835023), to generalize the Moyal quantization method to supersymmetric fields from the view to analyze super string theory as an integrable system. The Moyal quantization method, however, can deal with only flat phase space. On the other hand the Berezin quantization is a manifestly noncommutative geometry, so that a quantization of nonflat space is possible to unify many super string theories. In this project we clarified the difference between these two quantization methods and showed that the string model itself can be represented naturally by functional integration of Berezin quantization.2. String model realization of discrete geometryDiscrete geometry is a new mathematics which is found by a generalization of the deep correlation between soliton equations and differential ge … More ometry to the discrete integrable systems. In this project we attempted to describe the super string correlation functions in terms of the discrete geometry. We found that the coordinates of the discrete geometry correspond to the quantized momenta of strings.3. New method to characterize discrete integrable systemsFrom our point of view that the super string theory is described by integrable systems, it is important to characterize the integrable systems themselves within the nonlinear systems in order to understand the super string theory. We have investigated in particular the discrete Lotka-Volterra equation to make clear under which mechanism a nonintegrable system turns to an integrable one when a suitable parameter is changed continuously. As a result we found that there exists an algebraic equation of the 2nd order which characterizes the system. The system is integrable only if the discriminant of the quadratic equation turns to a perfect square of a polynomial of the variables. Less