Geometry of Quantum Dilogarithm Function

2014 Fiscal Year Final Research Report

• Project/Area Number: 24654041
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Global analysis
• Research Institution: Kyushu University
• Principal Investigator: HIKAMI Kazuhiro 九州大学, 数理(科)学研究科(研究院), 准教授 (60262151)
• Co-Investigator(Kenkyū-buntansha): MURAKAMI Hitoshi 東北大学, 情報科学研究科, 教授 (70192771)
• Co-Investigator(Renkei-kenkyūsha): Rei (INOUE Rei) 千葉大学, 理学研究科, 准教授 (30431901)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 数理物理 / 双曲幾何 / 二重対数関数

Outline of Final Research Achievements

The quantum dilogarithm function was originally introduced by Faddeev to study a discrete analogue of the KdV equation. Sometime ago Kashaev constructed knot invariant by use the quantum dilogarithm function, and he observed that its asymptotic is dominated by the hyperbolic volume of knot complement. Recently it has been realized that the quantum dilogarithm function plays a crucial role in the quantum cluster algebra.
We have used a technique of the quantum cluster algebra to reveal a relationship between the quantum dilogarithm function and the three-dimensional hyperbolic geometry. We constructed the quantum R-operator based on cluster mutation, and derived the Kashaev R-matrix by use of the quantum dilogarithm function.

Free Research Field

数理物理