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A Study of Unified Super String Theory Based on Integrable Systems

基于可积系统的统一超弦理论研究

基本信息

批准号：
10640278
负责人：
SAITO Satoru
金额：
$ 1.02万
依托单位：
Tokyo Metropolitan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

项目摘要

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

我们可以将本研究项目的成果总结为三个主要部分。Berezin量子化和弦相关函数在1994年至1996年的研究项目(项目号06835023)中，我们尝试将MoYAL量子化方法推广到超对称场，以将超弦理论分析为一个可积系统。然而，MoYal量子化方法只能处理平坦的相空间。另一方面，Berezin量子化是一种明显的非对易几何，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之间的区别，并证明了弦模型本身可以通过Berezin量子化的函数积分来自然地表示。离散几何的弦模型实现离散几何是通过推广孤子方程和微分Ge…之间的深层联系而建立的一门新的数学离散可积系统的更多几何性质。在这个项目中，我们尝试用离散几何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。刻画离散可积系统的新方法从超弦理论是由可积系统来描述的观点来看，为了理解超弦理论，重要的是刻画非线性系统中的可积系统本身。我们特别研究了离散的Lotka-Volterra方程，以明确当适当的参数连续变化时，不可积系统变成可积系统的机制。结果发现，存在一个描述该系统的二阶代数方程。只有当二次方程的判别式变为变量的多项式的完美平方时，系统才是可积的。较少