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

该奖项支持哈佛大学Arthur Jaffe和其他人对瑞士联邦理工学院(ETH)的几次短期研究访问，以便与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话题是超对称量子场模型的几何。这些问题的数学框架是无限维空间上的分析。更具体地说，适当的带分次的Hilbert空间导致系统的哈密顿算子H的适当分解为自伴算子Q的平方。这些算子产生一个指数，该指数被证明具有良好的性质，如同伦不变性，并且与有限维流形上的Dirac算子有关。研究人员现在建议检查他们之前工作中产生的几种数学结构，意在开发一种适用于更普遍现象的指数理论。最终，这些结构应该会进一步深入了解弦理论以及其他场论，如怀特曼场论。所提出的数学研究是狄拉克算子的第一组存在定理和指数结果，该算子以非平凡的方式耦合了无限多个自由度。哈佛大学和高等理工学院之间的非正式合作由来已久，在这一领域产生了许多重要的见解。为定期交流访问提供资金将加强他们的合作，并增加他们研究的势头和生产力。

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.