4-manifolds, quantum field theory and generalized cohomology

在项目的第一部分，主要研究人员将致力于4维流形的拓扑分类，重点是无限基本群和Rob Schneiderman发展的曲面嵌入的障碍理论。该项目的第二部分是与斯蒂芬·斯托尔茨共同研究广义上同调和超对称量子场论之间的关系。对于时空维d=0，1，2，流形X上的(d|1)维欧几里德场论现在有了明确的概念。对于d=0，这些是可证明的闭微分形式，并且对于d=1，人们通过调和类得到了X的K-理论。在Hopkins，Miller和Lurie的普适椭圆上同调理论的“拓扑模形式”中，猜想情况d=2导致了对类的长期期望的几何/物理解释。这一猜想的证据来自于最近证明的事实，即(2|1)维欧氏场论的配分函数是积分模形式。该项目的两个部分都涉及理论物理和数学的方法。在20世纪早期，数学概念(如黎曼几何或泛函分析)被成功地用来解释物理理论(如相对论或量子力学)。在那个世纪的下半叶，角色在某种程度上颠倒了过来，因为现代理论物理学使用量子场论之类的概念做出了令人惊讶的预测，这些预测只有在非常罕见的情况下才能在数学上得到证明。对于数学研究来说，将这些概念结合到已被广泛理解的理论中，从而既提供了数学上的进步，又提供了物理理论基础的精确表述，这是至关重要的。这个项目通过某些量子场理论和数学中一些广为人知的上同调理论之间的特殊关系来为这一目标做出贡献。

In the first part of the project, the principal investigator will work on the topological classification of 4-dimensional manifolds, with an emphasis on infinite fundamental groups and the obstruction theory for embeddings of surfaces developed with Rob Schneiderman. The second part of the project is joint work with Stephan Stolz on the relation between generalized cohomology and super symmetric quantum field theories. For space-time dimensions d=0,1,2 there are now well-defined notions of (d|1)- dimensional Euclidean field theories over a manifold X. For d=0 these are provably closed differential forms and for d=1 one obtains K- theory of X by taking concordance classes. It is conjectured that the case d=2 leads to the long desired geometrical/physical interpretation of classes in the universal elliptic cohomology theory of "topological modular forms" of Hopkins, Miller and Lurie. Evidence for this conjecture comes from the recently proven fact that the partition function of a (2|1)-dimensional Euclidean field theory is an integral modular form.Both parts of the project relate methods from theoretical physics and mathematics. In the early 20th century, mathematical notions (like Riemannian geometry or functional analysis) were successfully used to explain physical theories (like relativity theory or quantum mechanics). In the second part of that century, the roles were somewhat reversed in that surprising predictions, mathematically provable only in very rare instances, were made by modern theoretical physics using notions like that of a quantum field theory. It is of ultimate importance for mathematical research to incorporate such notions into the body of well understood theories, hence providing both, progress in mathematics and a precise formulation of the basics of the physical theories. This project contributes to this goal via a particular relation between certain quantum field theories and some well understood cohomology theories in mathematics.