Structure in Topological Field Theory

拓扑场论中的结构

• 批准号: 0709448
• 负责人: Constantin Teleman
• 金额: --
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-15 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0709448&HistoricalAwards=false
• 关键词: Structure; Topological; Field; Theory

This research plan integrates several research projects of the PI, unified by the relation to topological field theories which controls the answer to the various questions. One project builds on the PI's structural classification of 2-dimensional (closed string) semi-simple topological field theories, with implications for several standing conjectures on Gromov-Witten invariants. Beyond a careful treatement of the result and its implications, the aim is to extend it to more general theories; this seems to relate to the factorization properties for GW invariants (Ionel-Parker, Li) and to the structural results in the open/closed string case (Kontsevich, Costello). A second project is the study of gauged Gromov-Witten theory, where twisted K-theories and representations of loop groups appear. Loop groups also feature in the third project, pertaining to the geometric Langlands programme, where the cohomological calculations of the PI (with E. Frenkel) offer a way forward form the results of Beilinson-Drinfeld, toward the 'derived' version of the Langlands correspondence. (Recent work of Gukov-Kapustin-Witten suggest a controlling topological fieldtheory.) Current work by the PI (with C.Woodward) extending to Higgs bundles results of coherent cohomology previously known for principal bundles is an important step. Other, and more speculative projects include a study of non-semisimple 3-dimensional TFT's associated to derived categories.    Topological Field theory is a spectacular and unforeseen application of ideas from modern quantum physics to topology: that is the field of mathematics which, broadly speaking, studies the properties of shapes that are stable under continuous deformations. Previous applications of the foundational problems of quantum physics to mathematics had dominated development in mathematical analysis for decades, but their emergence in topology in the 1980's came as a surprise. Crudely put, the new ideas exploit a topological irreversibility of time flow: a topological change in space-time can usually not be 'undone' in the future. This led to the encoding of information in new kinds of algebraic structures (technically, they are monoids rather than groups).The new methods succeeded in unifying existing invariants of knots and links with those of 3-and 4-dimensional structures (manifolds).New invariants could be defined that bear stunning relations to other fields of mathematics which study finer, but less robust structures (algebraic geometry and complex analysis). The PI's research focuses on instances of these structures where, in addition, a continuous group of symmetries is present in the system, and studies the refined structures that emerge.   

这个研究计划整合了PI的几个研究项目，通过与控制各种问题答案的拓扑场和理论的关系来统一。一个项目建立在PI对二维(闭弦)半简单拓扑场理论的结构和分类的基础上，以及对几个基于Gromov-Witten不变量的现有猜想的影响。除了仔细处理结果及其含义之外，目的是将其扩展到更一般的理论；这似乎与GW不变量(Ionel-Parker，Li)的因式分解性质有关，也与非开/闭弦情况(Kontsevich，Costello)的结构结果有关。第二个项目是对格罗莫夫-维腾理论的研究，在那里出现了扭曲的K-理论和环群的表示。环群也出现在第三个项目中，与几何朗兰兹计划有关，其中PI的上同调计算(与E.Frenkel)提供了一种从Beilinson-Drinfeld的结果向前推进的方法，走向朗兰兹通信的“派生”版本。(Gukov-Kapustin-Witten最近的工作提出了一种控制性的拓扑场理论。)PI(与C.Woodward)目前的工作是将先前已知的主丛的相干上同调的结果推广到Higgs丛，这是重要的一步。其他更具投机性的TFT项目包括对与派生出的TFT类别相关的非半简单3维TFT的研究。拓扑场论是现代量子物理思想在拓扑学中壮观而意想不到的应用：拓扑学是数学领域，广义上研究在连续变形下稳定的形状的基本性质。几十年来，量子物理的基本问题在数学中的应用一直主导着数学分析的发展，但它们在20世纪80年代的拓扑学中的出现令人惊讶。粗略地说，这些新的想法利用了时间流动的拓扑不可逆性：在未来的未来，时空中的拓扑结构变化通常是不可能的。这导致了信息在新类型的代数结构中的编码(从技术上讲，它们是么半群而不是群)。新的方法成功地将现有的纽结和链环的不变量与三维和四维结构(流形)的不变量统一起来。新的不变量可以定义为与其他数学领域具有惊人关系的不变量，这些领域需要研究更精细但不那么健壮的结构(代数几何和复杂分析)。PI的研究重点放在这些结构的实例上，此外，系统中还存在一组连续的对称，并研究出现的精细结构。