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.
