Research of relation among quantum invariant and number theoretic invariants and modular forms

量子不变量与数论不变量及模形式关系的研究

• 批准号: 17540067
• 负责人: TAKATA Toshie
• 金额: $ 1.54万
• 依托单位: Niigata University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540067/
• 关键词: 3-manifold; knot; quantum invariant; modular form; theta function

Applying to our formula of the colored Jones polynomial for 2-bridge knots Lawrence and Ron's way to compute SU(2)Witten--Reshetikhin--Turaev (WRT) invariant of homology 3-sphere obtained by surgery of a knot from the colored Jones polynomial of the knot, we conjectured some number theoretical properties of the Ohtsuki invariants. Later on, using some results about the colored Jones polynomial given by K.Habiro, we showed the conjecture. Furthermore, using the values of the Ohtsuki invariants for Seifert 3-manifolds given by Hikami, we identified the set of the LMO invariant up to degree 6 for integral homology 3-spheres, and characterized the set of the Ohtsuki invariants up to degree 6.Lawrence and Zagier pointed out that WRT invariant of the Poincare homology sphere is related to the vector modular form with half-integral weight. Namely the WRT invariant coincides with a certain limit of the Eichler integral of the modular forms. As a generalization of this result, we have computed the WRT invariant for the spherical Seifert manifold with three exceptional fibers, and we have found that it coincides with a limiting value of Eicher integral of vector modular form with half-integral weight. By use of the modular property, we have obtained an exact asymptotic expansion of the WRT invariant, and have given an interpretation of topological invariants such as the Ohtsuki series and the Chern--Simons invariant from the viewpoint of modular forms. Furthermore, we have pointed out that the vector modular form is related to the fundamental group of manifolds.

适用于我们的有色琼斯多项式的公式，以适用于劳伦斯（Lawrence）和罗恩（Ron）的方式计算su（2）witten-witten-reshetikhin--turaev（wrt）同源性的3-球手术，通过从彩色的琼斯（the neymial of the Nenomial of the nymial of）属于某些属性中，通过有色人种的多项属性来获得。后来，使用有关K.Habiro给出的彩色琼斯多项式的一些结果，我们显示了猜想。 Furthermore, using the values of the Ohtsuki invariants for Seifert 3-manifolds given by Hikami, we identified the set of the LMO invariant up to degree 6 for integral homology 3-spheres, and characterized the set of the Ohtsuki invariants up to degree 6.Lawrence and Zagier pointed out that WRT invariant of the Poincare homology sphere is related to the vector modular形成半融合的体重。即WRT不变性与模块化形式的Eichler积分的一定极限相吻合。作为对此结果的概括，我们已经计算了带有三个特殊纤维的球形塞菲尔特歧管的WRT不变性，并且我们发现它与矢量模块化形式的Eicher积分的限制值和半构成重量的限制值相吻合。通过使用模块化特性，我们获得了WRT不变性的精确渐近扩展，并从模块化形式的角度对拓扑不变性（例如Ohtsuki系列和Chern）进行了解释。此外，我们指出的是，向量模块化形式与基本歧管群有关。

项目成果

Mock (false) theta functions as quantum invariants

• DOI: 10.1070/rd2005v010n04abeh000328
• 发表时间: 2005-06
• 期刊: Regular & Chaotic Dynamics
• 影响因子: 1.4
• 作者: K. Hikami

Transformation Formula for the "2nd" Order Mock Theta Function

“二阶”模拟 Theta 函数的变换公式

• 发表时间: 2006
• 期刊: Letters in Mathematical Physics 75
• 作者: K. Mine;K. Sakai and M. Yaguchi;樋上 和弘;樋上 和弘
• 通讯作者: 樋上 和弘

On the Topological Structure of Fractal Tilings Generated by Quadratic Number Systems

二次数系统生成的分形瓦片的拓扑结构

• 发表时间: 2005
• 期刊: Computers and Mathematics with Applications 49
• 作者: K. Mine;K. Sakai and M. Yaguchi;樋上 和弘;樋上 和弘;樋上 和弘;K.Hikami;K.Hikami;K.Hikami;樋上 和弘;樋上 和弘;秋山 茂樹
• 通讯作者: 秋山 茂樹

Quantum Invariant, Modular Form, and Lattice Points

量子不变量、模形式和格点

• 发表时间: 2006
• 期刊: Journal of Mathematical Physics 47
• 作者: K. Mine;K. Sakai and M. Yaguchi;樋上 和弘;樋上 和弘;樋上 和弘
• 通讯作者: 樋上 和弘

On the Quantum Invariant for the Spherical Seifert Manifold

关于球形Seifert流形的量子不变量

• 发表时间: 2006
• 期刊: Communication in Mathematical Physics 268
• 作者: K. Mine;K. Sakai and M. Yaguchi;樋上 和弘
• 通讯作者: 樋上 和弘