Quantum Field Theory, Noncommutative Geometry and Supergeometry

【摘要】本项目的主要目标是应用现代数学的方法来解决理论物理中的问题。许多问题都与m理论有关。这个理论可以看作是弦理论的延伸。人们相信m理论描述了自然界中存在的所有相互作用。最近有人推测M理论可以表述为一个矩阵模型，称为M（矩阵）理论。在这种方法中很明显，物理空间的几何在某种意义上是非交换的，因此应用非交换几何方法是很自然的。这个想法被用于一些项目的制定，以及我们最近关于M（矩阵）理论的环面紧化的工作。大多数其他问题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。这项多学科研究由数学科学部、物理部和数学与物理科学主任多学科活动办公室共同资助。

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.