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

该奖项支持亚瑟的几次短期研究访问 贾菲和其他哈佛大学到瑞士联邦 理工学院（ETH）在数学物理方面进行合作 康拉德·奥斯特瓦尔德和其他人的研究。 的话题 这两个团体的共同兴趣是超对称的几何 量子场模型 这些问题的数学框架 是对无限维空间的分析 更具体地称为 适当的希尔伯特空间配备了一个分级产生了一个 适当分解的哈密顿算子H的 系统作为自伴算子Q的平方。 这些运营商 产生一个指数，它被证明具有良好的性质， 作为同伦不变性，并涉及到狄拉克算子上的有限 维流形 研究人员现在提出， 从他们以前的工作中产生的几个数学结构， 旨在发展一种适用于更广义的指数理论， 现象。 最终，这些结构应该提供进一步的见解， 弦理论和其他场论， 怀特曼场理论。 提出的数学研究是第一套存在的 定理和指标结果的Dirac算子耦合， 无限多的自由度。 那里 一直是一个长期的非正式合作的历史， 哈佛和ETH集团已经产生了一些重要的 这方面的见解。 定期交流计划的供资 访问将加强他们的合作并增加势头 和他们研究的生产力。

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.