Unified Theories of Space and Matter

在这项研究的支持下，我们的成果概括为以下三类：(1)量子引力和弦由于量子引力理论是不可重整化的，所以微扰方法是不允许的。因此，为了理解重力的动力学，人们经常研究反映问题某些方面的各种模型理论。二维共形模型和矩阵模型是最常用的两种方法。在后者中，人们可以总结二维流形的所有拓扑，并找到相关函数的非微扰解。在我们的项目下，许多研究都是沿着这条线进行的，其中我们引用了对二维宇宙中环路相关幅度的研究。单圈相关振幅是霍金无边界宇宙振幅的二维版本。一些人研究了拓扑共形场，给出了流形…上的拓扑和的一种新的几何解释更多的DS。在弦理论中，研究了“普遍弦理论”的存在性。所有的超弦和玻色弦都可以作为宇宙弦理论连续自发对称性破缺的结果来推导。(2)规范对称性的起源在大多数情况下，人们在极高的能区假定高规范对称性，然后引入自发对称性破缺来得到标准的或直肠理论。在这个项目中，我们发现了一种新的机制，即使一个理论在高能下具有低规范对称性，在低能区也会产生一些较高的规范对称性。这发生在空间的卡鲁扎-克莱因紧凑化发生时。弦理论共形场理论、w-代数和Bethe azatz方法中发展起来的数学方法被应用于各种数学问题。其中之一是对XXZ自旋链模型的深入研究。研究了引力作用下的自旋链模型，详细分析了自旋链的排斥力。另一个有趣的例子是w-代数在微生物运动研究中的应用。较少

Our achievements under the support of this Grant are summarized into following three categories.(1)Quantum Gravity and StringsSince the theory of quantum gravity is unrenormalizable the perturbational approach is not permitted. To understand the dynamics of gravity, therefore, one frequently studies various model theories that reflect some aspects of the problem. The two dimensional conformal model and the matrix model are most popular approaches. In the latter one can sum up over all topologies of two dimensional manifolds and find non-perturbative solutions for correlation functions. Many of studies were made along this line under our project, among which here we quote the study of loop correlation amplitudes in a two dimensional universe. The one-loop correlation amplitude is nothing but the two-dimensional version of Hawking's no boundary universe amplitude.Some people studied the topological conformal field and gave a new geometrical interpretation to the topology sum over manifol … More ds. In the string theory existence of "the universal string theory" was studied. All the superstrings and the bosonic strings are able to be deduced as a result of successive spontaneous symmetry breakdown of the universal string theory.(2)Origin of Gauge SymmetryIn most of cases one assume a high gauge symmetry in an extremely high energy region, then introduces a spontaneous symmetry breakdown to get the standard or GUT theory. In this project we found a new mechanism that even if a theory had a low gauge symmetry in high energy, some higher gauge symmetry is generated in the low energy region. This happens when the Kaluza-Klein compactification of space is taken place. The basic mechanism working there is shown to be the Berry phase effect.(3)Applications of Mathematical Techniques developed in String Theory Conformal field theory, w-algebra and Bethe anzatz method were applied to various mathematical problems. One of them is the thorough study of the XXZ-Spin Chain Model. The model had been studied in case of attractive force, now the repulsive spin-chain was analyzed in detail. Another interesting example is an application of w-algebra to the study of the motion of micro-organisms. Less