Theory of braids, hyperplane arrangements and applications to conformal field theory

辫子理论、超平面排列及其在共形场理论中的应用

• 批准号: 16340014
• 负责人: KOHNO Toshitake
• 金额: $ 9.51万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16340014/
• 关键词: configuration space; loop space; iterated integral; hypergeometric function; KZ equation; mapping class group; セルバーグ積分; ジョーンズ多項式; 周期積分; 組みひも群; ランキングパターン; 共形場理論; 有限型位相不変量; シュレフリ関数; 超平面配置; 多変数超幾何関数; 共形場の理論; 局所系

1. Loop spaces of orbit configuration spaces}In collaboration with F.Cohen and M.Xicont\'encatl, I developed research on the algebraic structure of the homology of loop spaces of configuration spaces. We described the homology of loop spaces of orbit configuration spaces associated with actions of Fuchsian groups on the upper half plane by means of Lie algebras and established a relation to the algebra of chid diagrams on surfaces.We studied the de Rham cohomobgy of the loop spaces of the configuration spaces based on Chen's iterated integrals. As an application I developed a systematic approach to construct link homotopy invariants based on cohomology classes of the loop spaces of configuration spaces.2. Iterated integrals and hyperbolic volumes}It is known by K. Aomoto that volumes of spherical or hyperbilic simplices are expressed by iterated integrals of logarithmic forms based on Schl\'afli's equality. Using this method, we described the analytic continuation of the volume functions from spherical to hyperbolic geometry and an integrable connection of nilpotent type such that the volume functions appear as horizontal sections. By means of the singularities of such connections I investigatgd the asymptotic behavior of the hyperbolic volumes on the boundary of moduli spaces.

1. 我与F.Cohen和M.Xicont\'encatl合作，开展了构型空间环空间同调的代数结构研究。利用李代数描述了上半平面上与Fuchsian群作用相关的轨道位形空间的环空间的同调性，并建立了与曲面上子图代数的关系。基于Chen的迭代积分，研究了组态空间环空间的de Rham上同调。作为一个应用，我开发了一个系统的方法来构造链同伦不变量的基础上的环空间的上同调类。迭代积分和双曲体积}由K. Aomoto已知，球面或双曲简单体的体积是由基于Schl\'afli等式的对数形式的迭代积分表示的。利用这种方法，我们描述了体积函数从球面几何到双曲几何的解析延拓，以及使体积函数表现为水平截面的幂零型可积连接。利用这类连接的奇异性，研究了双曲体在模空间边界上的渐近性。

项目成果

反復積分の幾何学

重复积分的几何

• 发表时间: 2006
• 作者: Yuzuru Miyata;Hiroyuki Shibusawa;Y.Hirobata;A.Ogai;坂元章;Y.Maeda et al.;河野 俊丈
• 通讯作者: 河野 俊丈

The Hodge filtration and the contact-order filtration of derivations of Coxeter arrangements.

Hodge 过滤和 Coxeter 排列的推导的接触顺序过滤。

• 发表时间: 2005
• 期刊: Marnuscripta Math. 118
• 作者: D.Kotschick;S.Morita;H.Terao
• 通讯作者: H.Terao

Arrangements and ranking patterns.

安排和排名模式。

• 发表时间: 2006
• 期刊: Ann. Comb. 10
• 作者: H;Terao
• 通讯作者: Terao

Homological Representations of the Iwahori-Hecke algebra associated with a Selberg-type integral

与 Selberg 型积分相关的 Iwahori-Hecke 代数的同调表示

• 发表时间: 2005
• 期刊: Intern. Math. Res. Notices 33
• 作者: N.Kurokawa;H.Ochiai;M.Wakayama;K.Mimachi
• 通讯作者: K.Mimachi

Cohomological structure of the mapping class group and beyond. Problems on mapping class groups and related topics

映射类群及其他的上同调结构。

• 发表时间: 2006
• 期刊: Proc. Sympos. Pure Math. 74
• 作者: Morita;Shigeyuki
• 通讯作者: Shigeyuki