Quantum Field Theory, Noncommutative Geometry and Supergeometry

施瓦茨这个项目的主要目标是将现代数学的方法应用于理论物理中出现的问题。许多问题都与M-理论有关。这一理论可以看作是弦理论的推广。人们认为M-理论描述了自然界中存在的所有相互作用。最近有人猜测，M-理论可以表示为一个矩阵模型，称为M(Atrix)理论。这种方法清楚地表明，物理空间的几何在某种意义上是非对易的，因此应用非对易几何的方法是很自然的。这一思想被用于一些项目的制定以及我们最近关于M(Atrix)理论的环紧化的工作中。其他问题大多与超几何在量子场论的各种问题中的应用有关，包括规范理论的量子化和拓扑量子场论的一般问题。这项多学科研究是由数学科学系、物理系和数学和物理科学主任多学科活动办公室联合资助的。

AbstractSchwarzThe main goal of this project is to apply methods of modern mathematics to problems arising in theoretical physics. Many of problems are related to M-theory. This theory can be considered as an extension of string theory. It is believed that M-theory describes all interactions existing in nature. It was conjectured recently that M-theory can be formulated as a matrix model, called M(atrix) theory. It is clear in this approach that the geometry of physical space is non-commutative in some sense, and so it is natural to apply methods of non-commutative geometry. This idea was used in the formulation of some projects as well as in our recent work concerning toroidal compactifications of M(atrix) theory. Most of the other problems are related to applications of super-geometry to various problems of quantum field theory, including general problems of quantization of gauge theories and topological quantum field theory.This multidisciplinary research is jointly funded by the Division of Mathematical Sciences, the Division of Physics, and the Office of Multidisciplinary Activity of the Director of Mathematical and Physical Sciences.