Differential geometry of infinite dimensional manifolds is basic to the study of quantum field theory. The principal investigator will extend his results on iterated integrals and the Witten current for the Dirac operator developed under the previous grant to the setting of supersymmetric conformal field theory. He will continue his development of analytic tools for loop spaces. These include logarithmic Sobolev inequalities, singular integral estimates, deRham cohomology and the Feynman- Kac formula. Lastly, he will investigate applications of Connes' theory of non-commutative differential geometry including links to quantum field theory. Classical field theory involves assigning to each point of four-dimensional space-time a vector or scalar quantity such as the electrical field or temperature. In quantum field theory usually some sort of operator or function is assigned to each point in space-time. Using inner products, from these operators an observable quantity may result which then can be checked by repeated experimentation. In this project subtle properties of quantum fields will be investigated using infinite dimensional hypersurfaces and a mathematician's tool known as loop spaces.
