2004 Fiscal Year Final Research Report Summary
Studies on topological field theory and hyperbolic structures
| Project/Area Number |
13640062
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Chiba University |
|---|
Principal Investigator |
KUGA Ken'ichi Chiba University, Faculty of Science, Professor, 理学部, 教授 (30186374)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
INABA Takashi Chiba University, Graduate School of Natural Sciences, Professor, 大学院・自然科学研究科, 教授 (40125901)
SUGIYAMA Ken-ichi Chiba University, Faculty of Science, Associate Professor, 理学部, 助教授 (90206441)
|
|---|
| Project Period (FY) |
2001 – 2004
|
| Keywords | 3-manifolds / hyperbolic structure / Kashaev invariant / colored Jones invariant / volume conjecture / L^2-torsion / Chern-Simons invariant / topological field theory |
|---|
| Research Abstract |
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.
|
