Mathematical Sciences: Gauge Theory of 3-Manifolds: Spectral Flow & Chern-Simons Invariants

数学科学：三流形规范理论：谱流

• 批准号: 9401196
• 负责人: Paul Kirk
• 金额: $ 4.5万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-01-01 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401196&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Gauge; Theory; Manifolds

9401196 Kirk Paul Kirk and Eric Klassen intend to develop powerful tools to compute and analyze gauge-theoretic invariants of 3-dimensional manifolds, particularly Chern-Simons invariants and the spectral flow of the signature operator, with an emphasis on manifolds with boundary. The techniques to be used include the theory of boundary-value problems for Dirac operators, Massey products in differential graded Lie algebras, and the representation theory of fundamental groups of 3-manifolds. Over the last decade, gauge theoretic topology has grown into one of the most important current fields of mathematical research. It involves an interaction of ideas from topology, analysis, differential and algebraic geometry, and theoretical physics. The exciting techniques resulting from this interaction have been used to answer fundamental questions about 3- and 4-dimensional manifolds which had been completely resistant to pre-gauge theory methods. The research to be undertaken forms one of the basic ingredients of both the 3- and 4-dimensional theories; it will provide information important for cut-and-paste methods in the 4-dimensional Donaldson theory, as well as providing computations for 3-dimensional topological quantum field theory. ***

小行星9401196 Paul Kirk和Eric Klassen打算开发强大的工具来计算和分析三维流形的规范理论不变量，特别是Chern-Simons不变量和签名算子的谱流，重点是有边界的流形。 将使用的技术包括理论的边值问题的狄拉克运营商，梅西产品在微分分次李代数，和代表理论的基本群体的3-流形。 在过去的十年中，规范理论拓扑已经发展成为当前数学研究中最重要的领域之一。 它涉及到拓扑学、分析学、微分几何和代数几何以及理论物理学的思想的相互作用。 从这种相互作用产生的令人兴奋的技术已被用来回答有关3-和4-维流形的基本问题，这些流形已经完全抵抗了前规范理论方法。 进行的研究形成了3维和4维理论的基本成分之一;它将为4维唐纳森理论中的剪切粘贴方法提供重要信息，并为3维拓扑量子场论提供计算。 ***