Loop and Path Space Analysis

本文主要研究黎曼流形路径空间和环路空间上的热核和维纳测度的几何分析。P.I.将继续研究紧致黎曼流形上路径和环路空间上路径积分公式的有限维近似。希望这些近似将导致：1)对Atiyah—Singer指数定理的启发式路径积分证明的严格解释，2)路径和环空间上布朗运动的构造，3)环群上非平凡调和微分形式的构造，以及4)环空间上狄拉克算子的构造。这些目标与主要研究人员正在进行的理解循环空间的索引定理和Hodge - de Rham定理的项目是一致的。并尝试将L. Gross和P.I.的一个结果（与Segal - Bargmann变换有关）从复有限维群推广到复环群。循环空间在曲面和高维流形的数学研究中自然出现。例如，如果表面上有一个环，在不撕裂环的情况下不能变形到一个点，那么表面必须包含一个“孔”。（想想一个甜甜圈的表面和一个球体的表面。）物理学家对自然界基本粒子的弦理论描述也使用了环或“弦”。弦理论的量子力学处理导致了类似于那些由P.I.提出研究的问题。这个建议致力于回答关于有限维曲面和这些曲面上的环上的量子力学的（费曼）路径积分公式的基本数学问题。作为这项研究的副产品，私家侦探希望为所谓的阿蒂亚-辛格指数定理提供严谨的、非常美丽的物理论证。这个指标定理可以说是本世纪最引人注目的数学发展之一，它在数学和物理学中都有深远的影响。

9971036 Driver This research is primarily devoted to the study of the geometric analysis associated to heat kernel and Wiener measures on path and loop spaces of Riemannian manifolds. The P.I. will continue his study of finite dimensional approximations to path integral formulas on path and loop spaces over compact Riemannian manifolds. It is hoped that these approximations will lead to: 1) a rigorous interpretation of the heuristic path integral proofs of the Atiyah--Singer index theorem, 2) construction of Brownian motions on path and loop spaces, 3) construction of non-trivial harmonic differential forms on loop groups, and 4) construction of Dirac operators on loop spaces. These goals are consistent with the principal investigators ongoing project to understand index theorems and the Hodge -- de Rham theorem for loop spaces. It is also proposed to try to generalize a result (related to the Segal Bargmann transform) of L. Gross and the P.I. from complex finite dimensional groups to complex loop groups. Spaces of loops naturally arise in the mathematical study of surfaces and of higher dimensional manifolds. For example, if there is a loop on a surface that can not be deformed to a point without tearing the loop apart, then the surface must contain a "hole." (Think of the surface of a doughnut versus the surface of a sphere.) The physicists' string theory description of the fundamental particles of nature also use loops or "strings." The quantum mechanical treatment of string theory leads to problems similar to those proposed for study by the P.I. This proposal is devoted to answering basic mathematical questions about the (Feynman) path integral formulation of quantum mechanics on both finite dimensional curved surfaces and on the loops on such surfaces. As a by-product of this research, the P.I. hopes to make rigorous the very beautiful physics argument for the so-called Atiyah-Singer Index theorem. This index theorem is arguably one of the most striking mathematical developments of this century and it has had far reaching implications in both mathematics and in physics.