4-manifolds, quantum field theory and generalized cohomology

在该项目的第一部分，主要研究人员将致力于四维流形的拓扑分类，重点是无限基本群和与Rob Schneiderman开发的曲面嵌入的障碍理论。 该项目的第二部分是与Stephan Stolz共同研究广义上同调与超对称量子场论之间的关系。对于时空维度d= 0，1，2，现在有定义明确的（d）概念|流形X上的1）维欧氏场论对于d=0这些是可证明的封闭的微分形式和d=1获得K-理论的X采取和谐类。它是澄清的情况下，d=2导致长期所需的几何/物理解释类的通用椭圆上同调理论的“拓扑模形式”的霍普金斯，米勒和Lurie。这个猜想的证据来自最近证明的事实，即a（2）的配分函数|1）维欧几里德场论是一个积分模形式。该项目的两个部分都涉及理论物理和数学的方法。在世纪早期，数学概念（如黎曼几何或泛函分析）被成功地用于解释物理理论（如相对论或量子力学）。在那个世纪的后半叶，角色有些颠倒，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。这是至关重要的数学研究纳入这些概念的身体很好地理解理论，从而提供了两个，在数学和精确制定的基础物理理论的进展。该项目通过某些量子场论和数学中一些很好理解的上同调理论之间的特定关系来实现这一目标。

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.