Noncommutative geometry, supergeometry, gauge theory and M-theory

[摘要]获奖：dms -9971304首席研究员：Albert schwarz该提案的主要目标是将非交换几何和超几何的方法应用于理论物理中出现的数学问题。这些问题也具有内在的数学价值；他们将受到严格对待。提案中列出的许多项目都与非交换空间的Yang-Mills理论（特别是超对称Yang-Mills理论）和该理论中的对偶性有关。PI计划利用他在最近的论文中发现的森田等价和对偶之间的关系，目的是理解规范理论、弦理论和m理论中的对偶性，并对这些对偶性进行严格的处理。A. Connes， M. Douglas和A. schwarz在最近的一篇论文中表明，非交换环面在m理论的紧化问题中（在m理论的矩阵表述中）是自然产生的。PI打算研究其他泊松流形的非交换变形和m理论非平坦非交换空间的紧化。其他项目涉及超几何在量子场论各种问题中的应用，包括量子化广义问题和拓扑量子场论。几年来，超弦理论被认为是描述自然界中存在的所有相互作用的理论的候选者。现在我们知道超弦理论是不完整的；它应该扩展到一个更一般的理论，不仅包含弦，而且包含它们的多维类似物（膜）。这个广义理论被命名为m理论；人们推测M理论可以被表述为矩阵模型（即M（矩阵）理论）。近年来，PI和其他学者证明了非交换几何的方法在M理论的分析中是非常有用的，特别是在M（矩阵）的表述中。PI正计划朝这个方向进一步努力；特别地，他在非交换几何的框架下研究了能谱和对偶性。

AbstractAward: DMS-9971304Principal Investigator: Albert SchwarzThe main goal of the proposal is to apply methods ofnoncommutative geometry and supergeometry to the mathematicalproblems arising in theoretical physics. These problems have alsointrinsic mathematical value; they will be treated rigorously.Many of the projects listed in the proposal are related to Yang-Mills theory ( in particular, supersymmetric Yang-Mills theory)on noncommutative spaces and to duality in this theory. The PI isplanning to exploit the relation between Morita equivalence andduality discovered in his recent papers with the goal tounderstand dualities in gauge theories, string theory andM-theory and to give rigorous treatment of these dualities. Itwas shown in recent paper by A. Connes, M. Douglas and A.Schwarzthat noncommutative tori arise naturally in the problem ofcompactification of M-theory ( in Matrix formulation of thistheory). The PI intends to study noncommutative deformations ofother Poisson manifolds and compactifications of M-theory onnon-flat noncommutative spaces. Other projects are related toapplications of supergeometry to various problems of quantumfield theory, including general problem of quantization of gaugetheories and topological quantum field theory.For several years superstring theory was considered as acandidate for a theory describing all interactions that exist inthe Nature. Now we know that superstring theory is not complete;it should be extended to a more general theory, containing notonly strings, but also their multidimensional analogs(branes).This general theory was named M-theory; it wasconjectured that M-theory can be formulated as a matrix model (socalled M(atrix) theory). It was shown recently by PI and by otherscholars that the methods of noncommutative geometry are veryuseful in the analysis of M-theory, especially in M(atrix)formulation of this theory. The PI is planning to work further inthis direction; in partucular, he studies the energy spectrum anddualities in the framework of noncommutative geometry.