A Study of Unified Super String Theory Based on Integrable Systems

我们可以将本研究项目的成果总结为三个主要部分。Berezin量子化与弦相关函数我们在1994 - 1996年的课题（课题编号06835023）中尝试将Moyal量子化方法推广到超对称场，从超弦理论作为可积分系统的角度进行分析。然而，Moyal量化方法只能处理平坦相空间。另一方面，Berezin量子化是一个明显的非交换几何，因此非平坦空间的量子化可以统一许多超弦理论。在这个项目中，我们澄清了这两种量化方法的区别，并表明字符串模型本身可以用Berezin量化的函数积分自然地表示。离散几何的弦模型实现离散几何是通过推广孤子方程与微分方程之间的深度关联而发现的一门新的数学。在这个项目中，我们试图用离散几何的术语来描述超弦相关函数。我们发现离散几何的坐标对应于弦的量子化动量。表征离散可积系统的新方法我们认为超弦理论是用可积系统来描述的，为了理解超弦理论，在非线性系统中表征可积系统本身是很重要的。我们特别研究了离散Lotka-Volterra方程，以说明当适当的参数连续改变时，在何种机制下不可积系统变成可积系统。结果表明，存在表征该系统的二阶代数方程。只有当二次方程的判别式变成变量的多项式的完全平方时，系统才可积。少

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