Studies on topological field theory and hyperbolic structures

拓扑场论和双曲结构研究

基本信息

批准号：
13640062
负责人：
KUGA Ken'ichi
金额：
$ 2.18万
依托单位：
Chiba University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

In this project we first tried to solve the volume conjecture on the relation between the volume of the complement of a hyperbolic knot in the 3-sphere and and the Kashaev or the colored Jones invariant of that knot. Our motivation was that this relation was strongly suggested by the Chern-Simons topological field theory based on path-integral argument of Witten. Along the course of study, we began to realize that the difficulty existed in the very difference of the perturbative aspect of the volume and the non-perturbative character of the arguments using R-matices. We then decided to take two approaches : the first was to try to directly solve the volume conjecture based on manipulations on the presentation of the fundamental group, and the second approach was to study more broad aspect of the R-matrices and hyperbolic knots. For the first approach, we found that using a good choice of generators and relators of the fundamental group of the knot complement, L^2-torsion (from which we find volume) has a strong resemblance between the limit of the Kashaev invariant expressed by dilog functions. Unfortunately this approach was not completed for general hyperbolic knots due to the word problem necessary in the computation of the L^2-invariant. For the second approach, a twisted version of Drinfel'd quantum double construction of R-matrices is obtained by the head investigator and graduate student D. Fukuda. Also, topological L-function was defined and studied by the investigator K.Sugiyama.

在这个项目中，我们首先试图解决体积的猜想，这是在3球中的双曲线结的音量与kashaev或该结的彩色琼斯之间的关系之间的关系之间的猜想。我们的动机是，基于Witten的路径综合论证，Chern-Simons拓扑领域理论强烈提出了这种关系。在整个研究过程中，我们开始意识到，使用R-Matices的参数的扰动方面和参数的非扰动特征的差异存在。然后，我们决定采用两种方法：第一种方法是尝试基于对基本组的呈现的操纵来直接解决体积猜想，第二种方法是研究R-矩阵和双曲线结的更广泛方面。对于第一种方法，我们发现，使用了结的基本组的发电机和关系，l^2-torsion（从中找到体积）在dilog函数表达的kashaev不变性的极限之间具有很强的相似度。不幸的是，由于计算l^2-Invariant所需的单词问题，因此未完成这种方法的一般双曲结。对于第二种方法，首席调查员和研究生D. Fukuda获得了Drinfel'D量子双重构造的扭曲版本。此外，研究人员K.Sugiyama还定义了拓扑L功能。