U.S.-Swiss Cooperative Research in Index Theory and InfiniteDimensional Analysis (Mathematical Physics)

该奖项支持哈佛大学的Arthur Jaffe等人对瑞士联邦理工学院（ETH）的几次短期研究访问，与Konrad Osterwalder等人在数学物理研究方面进行合作。这两个小组共同感兴趣的一般主题是超对称量子场模型的几何。这些问题的数学框架是对无限维空间的分析。更具体地说，一个适当的希尔伯特空间配备了一个分级，产生一个适当的分解哈密顿算子H的系统作为一个自伴随算子q的平方。这些算子产生一个指标，被证明具有良好的性质，如同伦不变性，并涉及到一个有限维流形上的狄拉克算子。研究人员现在打算研究他们以前工作中产生的几个数学结构，打算发展一个适用于更普遍现象的指标理论。最终，这些结构应该给弦理论以及其他场论如怀特曼场论提供进一步的见解。提出了一组具有无限个非平凡耦合自由度的狄拉克算子的存在性定理和指标结果。哈佛大学和ETH之间的非正式合作有着悠久的历史，在这一领域产生了许多重要的见解。资助定期互访将加强他们的合作，提高他们的研究势头和生产力。

This award supports several short term research visits by Arthur Jaffe and others of Harvard University to the Swiss Federal Institute of Technology (ETH) to collaborate in mathematical physics research with Konrad Osterwalder and others. The general topic of the two groups' mutual interest is the geometry of supersymmetric quantum field models. The mathematical framework for these problems is analysis on an infinite dimensional space. More specifically, a suitable Hilbert space equipped with a grading gives rise to an appropriate decomposition of the Hamiltonian operator H for the system as the square of a self adjoint operator Q. These operators give rise to an index, which is shown to have good properties such as homotopy invariance, and relate to a Dirac operator on a finite dimensional manifold. The researchers now propose to examine several mathematical structures arising from their previous work, intending to develop an index theory applicable to more generalized phenomena. Ultimately, these structures should give further insight into string theory as well as into other field theories such as the Wightman field theories. The mathematical research proposed is the first set of existence theorems and index results for a Dirac operator which couples an infinite number of degrees of freedom in a non-trivial way. There has been a long history of informal collaboration between the Harvard and ETH groups which has produced a number of significant insights in this area. Funding for a regular schedule of exchange visits will intensify their collaboration and increase the momentum and productivity of their research.