Structure in Topological Field Theory

本研究计划整合了PI的几个研究项目，通过与拓扑场理论的关系统一起来，控制着各种问题的答案。一个项目建立在PI的二维（闭弦）半简单拓扑场论的结构分类上，包含了关于Gromov-Witten不变量的几个长期猜想。除了对结果及其含义进行仔细处理之外，其目的是将其扩展到更一般的理论；这似乎与GW不变量的因式分解性质（Ionel-Parker, Li）和开/闭弦情况下的结构结果（Kontsevich, Costello）有关。第二个项目是研究测量的Gromov-Witten理论，其中出现了扭曲的k理论和环路群的表示。环路群也出现在第三个项目中，与几何朗兰兹方案有关，其中PI的上同调计算（与E. Frenkel）提供了一种从贝林森-德林菲尔德的结果向朗兰兹对应的“派生”版本前进的方法。（Gukov-Kapustin-Witten最近的研究提出了一种控制拓扑场论。）目前PI（与C.Woodward）将先前已知的主束相干上同的结果推广到希格斯束的工作是重要的一步。其他更具推测性的项目包括与派生类别相关的非半简单三维TFT的研究。拓扑场论是现代量子物理学思想在拓扑学中的一个引人注目和不可预见的应用：拓扑学是数学领域，从广义上讲，研究在连续变形下稳定的形状的性质。量子物理学基础问题在数学上的应用已经主导了数学分析的发展几十年，但它们在20世纪80年代在拓扑学上的出现令人惊讶。粗略地说，这些新想法利用了时间流的拓扑不可逆性：时空的拓扑变化通常不会在未来“撤销”。这导致了在新的代数结构中对信息进行编码（从技术上讲，它们是单群而不是群）。新方法成功地将现有的结点和连杆不变量与三维和四维结构（流形）的不变量统一起来。可以定义新的不变量，它们与研究更精细但不那么健壮的结构（代数几何和复杂分析）的其他数学领域有着惊人的关系。PI的研究重点是这些结构的实例，此外，在系统中存在连续的对称组，并研究出现的精细结构。

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.