Quantum structure of Riemann surfaces and its application to low-climensional topology

黎曼面的量子结构及其在低维拓扑中的应用

基本信息

批准号：
10440007
负责人：
MURUKAMI Jun
金额：
$ 4.67万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B).
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

The aim of this research is to construct a new unifying method to study low dimensional topology from a view point of the quantum structure of Riemann surfaces. To do this, research was done for two areas. One is geometric aspect and the other is algebraic aspect. It is revealed that the volume of the complement of a hyperbolic knot is given by quantum invariants of the knot. Moreover, there is some indication that the volume of a hyperbolic three-manifold is also given by the quantum invariants of the manifold. These facts suggest that the quantum invariants of knots and three-manifolds include various geometric information, and efficiency of the method to study geometric properties from quantum invariants are pointed out.We also study about the finite-type invariants and the web diagrams, which are related to certain expansion of quantum invariants, and get some new properties of them.Quantum invariants are closely related to conformal field theory and theory of q-deformation, and some results for these theories are obtained. In the study of vertex operator algebras, the relation between modular forms and the correlation functions in conformal field theory is given. In the study of the theory of q-deformation, modular representations of finite groups are studied. The heart (quotient of the radical by the socle) of projective indecomposable modules are investigated, and the case that the heart is not indecomposable is determined. Moreover, q-deformation of a Frobenius-Schur character of complex reflection groups is defined and computed actually for the symmetric groups and imprimitive complex reflection groups.

这项研究的目的是构建一种新的统一方法，从黎曼表面的量子结构的角度研究低维拓扑。为此，对两个领域进行了研究。一个是几何方面，另一个是代数方面。据揭示，双曲结的补体的体积由结的量子不变式给出。此外，有一些迹象表明，双曲线三序的体积也由歧管的量子不变剂给出。这些事实表明，打结和三个字体的量子不变符包括各种几何信息，并指出了研究从量子不变的方法研究几何特性的效率。我们还研究了有限型不变性和网络图，这些图表与量子的一定属性相关，并获得了一些新的属性，并获得了一些新的属性。获得Q的理论以及这些理论的一些结果。在对顶点操作员代数的研究中，给出了模块化形式与保形场理论中相关函数之间的关系。在研究Q信息理论的研究中，研究了有限基团的模块化表示。研究了投影性不可分解的模块的心脏（基于Socle的基础），并确定心脏不可分解的情况。此外，对对称组和不良复杂反射组的Frobenius-Schur特征的Q- Q-efformation定义和计算。