Structure in Topological Field Theory

该研究计划整合了PI的几个研究项目，通过与拓扑场论的关系统一起来，控制着各种问题的答案。一个项目建立在PI的2维（闭弦）半简单拓扑场论的结构分类基础上，并对Gromov-Witten不变量的几个常设结构产生影响。除了仔细解释这个结果及其含义之外，我们的目标是将它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。第二个项目是规范Gromov-Witten理论的研究，其中扭曲的K-理论和循环群的表示出现。Loop群也是第三个项目的特色，属于几何Langlands程序，其中PI的上同调计算（E。Frenkel）提供了一种从Beilinson-Drinfeld的结果向Langlands对应的“派生”版本前进的方法。（古科夫-卡普斯京-维滕最近的工作提出了一种控制拓扑场论。）目前的工作由PI（与C.伍德沃德）延伸到希格斯束相干上同调先前已知的主束的结果是一个重要的一步。其他的，和更多的投机项目，包括研究非半简单的三维TFT的相关派生类别。 拓扑场论是一个壮观的和不可预见的应用的想法从现代量子物理学的拓扑：这是数学领域，广义上说，研究性质的形状是稳定的连续变形。量子物理学的基础问题在数学中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年代出现在拓扑学中令人惊讶。粗略地说，新的想法利用了时间流的拓扑不可逆性：时空中的拓扑变化通常在未来无法“撤销”。这导致了新的代数结构中的信息编码（从技术上讲，它们是幺半群而不是群）。新的方法成功地将现有的结和链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有着惊人的关系，这些数学领域研究更精细，但不那么鲁棒的结构（代数几何和复分析）。PI的研究重点是这些结构的实例，此外，系统中存在连续的对称性组，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.