The volume and Chern-Simons invariant in topology and number theory.

拓扑和数论中的体积和陈-西蒙斯不变式。

• 批准号: 1145374
• 负责人: Christian Zickert
• 金额: $ 7.56万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1145374&HistoricalAwards=false
• 关键词: volume; Chern; Simons; invariant; topology

By Mostow rigidity, the volume and Chern-Simons invariant of a hyperbolic 3-manifold are topological invariants. They are often regarded as the real and imaginary part of a complex valued invariant known as the complex volume. The complex volume can be defined more generally as an invariant of a parabolic representation of a 3-manifold group into a simple complex Lie group, e.g. SL(n,C). The PI and his collaborators will study such representations and extend several results previously only known for SL(2,C). In particular, we will give a concrete formula for the complex volume. The new idea is that a parabolic representation has a fundamental class in Neumann's extended Bloch group. By a result of the PI, the extended Bloch group can be defined over a number field, and is isomorphic to an algebraic K-group. This may shed new light on the role of the complex volume in number theory.The volume and Chern-Simons invariant are interesting and important invariants that appear in a wide range of areas of mathematics including mathematical physics, hyperbolic geometry and number theory. The proposal introduces new techniques for computing these invariants via an object known as the extended Bloch group. The extended Bloch group is isomorphic to a group appearing in a different branch of mathematics known as algebraic K-theory. The relationship with K-theory may lead to an improved understanding of the role of the volume and Chern-Simons invariant in number theory, which has so far been completely mysterious.

根据莫斯托刚性，双曲 3 流形的体积和 Chern-Simons 不变量是拓扑不变量。它们通常被视为复数值不变量（称为复体积）的实部和虚部。复数体积可以更一般地定义为 3 流形群的抛物线表示到简单复数李群中的不变量，例如SL(n,C)。 PI 和他的合作者将研究此类表示并扩展以前仅针对 SL(2,C) 已知的几个结果。特别是，我们将给出复体积的具体公式。新的想法是，抛物线表示在诺依曼的扩展布洛赫群中具有一个基本类。根据 PI 的结果，扩展的布洛赫群可以在数域上定义，并且与代数 K 群同构。这可能为复数体积在数论中的作用提供新的线索。体积和陈-西蒙斯不变量是有趣且重要的不变量，它们出现在数学物理、双曲几何和数论等广泛的数学领域中。该提案引入了通过称为扩展布洛赫群的对象计算这些不变量的新技术。扩展的布洛赫群与出现在称为代数 K 理论的不同数学分支中的群同构。与 K 理论的关系可能会导致人们更好地理解体积和陈-西蒙斯不变量在数论中的作用，而迄今为止这仍然是完全神秘的。