Noncommutative geometry, supergeometry, gauge theory and M-theory

AbstractAward：DMS-9971304首席研究员：Albert Schwarz该提案的主要目标是将非对易几何和超几何的方法应用于理论物理中出现的物理问题。这些问题也有内在的数学价值，他们将被严格对待。许多项目中列出的建议是有关杨米尔斯理论（特别是，超对称杨米尔斯理论）的非交换空间和对偶在这个理论。PI计划利用森田等价和对偶之间的关系，在他最近的论文中发现，目标是理解规范理论，弦理论和M理论中的对偶性，并严格对待这些对偶性。 A. Connes，M.道格拉斯和A.施瓦茨指出，非交换环面自然地出现在M-理论的紧化问题中（在M-理论的矩阵表述中）。 PI打算研究其他Poisson流形的非交换变形和非平坦非交换空间上M-理论的紧化。 其他项目涉及超几何在量子场论的各种问题中的应用，包括规范理论和拓扑量子场论的量子化的一般问题。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，它应该被扩展到一个更一般的理论，不仅包括弦，而且包括它们的多维类似物（膜）。这个一般理论被命名为M-理论;它被证明M-理论可以被公式化为矩阵模型（所谓的M（矩阵）理论）。PI等学者最近指出，非对易几何方法在M理论的分析中，特别是在M（矩阵）理论的表述中是非常有用的。PI正计划在这个方向上进一步工作;特别是，他在非对易几何的框架内研究能谱和对偶。

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.