Geometric and Topolocical Structures Related TO M-branes

与 M 膜相关的几何和拓扑结构

• 批准号: 1102218
• 负责人: Hisham Sati
• 金额: $ 12.6万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1102218&HistoricalAwards=false
• 关键词: Geometric; Topolocical; Structures; Related; branes

The goal of this project is to investigate new geometric and topological structures arising from M-branes in M-theory. The PI also plans to study the group of charges of the M-branes. This will involve higher notions of bundles, generalized cohomology, and modular forms. The subject matter proposed here suggests a rich interaction between theoretical and mathematical physics on one hand and geometry and topology on the other, and will provide new constructions in physics, differential geometry, and algebraic topology. Mathematics plays a very important role in deciding on the consistency of a physical theory by characterizing and canceling anomalies. Anomaly cancellations amount to refinements of the fields and provide a natural arena for imposing orientations on the underlying spaces with respect to (generalized) cohomology theories. These theories provide a fundamental mathematical machinery to describe invariants. On the other hand, M-theory, while not yet constructed, is already a very rich theory, both in terms of physical ideas as well as mathematical structures. Thus, the connection between the two is expected to yield a wealth of new mathematical structures. In fact, there has already been indication of this from the work of many researchers, including that of the PI. Previous work of the PI highlights the power of this method and demonstrates the strength of this approach to obtain new results and uncover new mathematical structures arising from physics. In addition, the work of the PI and others in this area has shown that application of ideas and techniques from geometry and topology unexpectedly yields nontrivial insights into physical theories as well. The project involves subtle constructions that are general enough to be of real interest to geometers, topologists, and physicists. The interaction between geometry/topology and field theory has recently been extended to string theory and M-theory. This is expected to be fruitful since string theory and M-theory subsume, and thus are structurally richer than, quantum field theory. Not only does powerful mathematics solve deep physical problems but many times we see that the physics inspires new directions in mathematics and sheds light on interesting constructions. The intended research will not only use techniques and have applications of constructions from geometry and topology, but will also provide new geometric and topological constructions motivated and guided by physics which will be of interest to both mathematicians and physicists. The research will also expand collaboration and bridge the cultural gap between differential geometry/algebraic topology and theoretical/mathematical physics. The PI has been organizing annual research meetings at the American Institute of Mathematics on Algebraic Topology and Physics in the last three years for that purpose and is co-organizing two other meetings 2011 and 2012 on such interdisciplinary topics. In addition, the PI would like to engage graduate students at Maryland in his research since the questions raised involve a wealth of ideas and techniques. He is currently co-organizing a Research Interactions in Teams (RIT) on Geometry and Physics, in which most of the lectures are presented by graduate and even undergraduate students. He has also organized an REU on Hyperdeterminants and Nonlinear Algebra, which has resulted in a publication with three undergraduates, two freshman and a sophomore.

这个项目的目标是研究新的几何和拓扑结构所产生的M膜在M理论。PI还计划研究M膜的电荷群。这将涉及到更高的概念丛，广义上同调，和模形式。这里提出的主题表明理论和数学物理学与几何学和拓扑学之间存在着丰富的相互作用，并将为物理学、微分几何学和代数拓扑学提供新的结构。数学通过刻画和消除反常，在决定物理理论的一致性方面起着非常重要的作用。反常抵消相当于场的精化，并提供了一个自然的竞技场，用于对（广义）上同调理论的基础空间施加方向。这些理论提供了一个基本的数学机制来描述不变量。另一方面，M理论虽然还没有建立起来，但已经是一个非常丰富的理论，无论是在物理思想还是数学结构方面。因此，两者之间的联系有望产生大量新的数学结构。事实上，包括PI在内的许多研究人员的工作已经表明了这一点。PI以前的工作突出了这种方法的力量，并证明了这种方法的力量，以获得新的结果，并揭示新的数学结构产生的物理。此外，PI和其他人在这一领域的工作表明，应用几何和拓扑学的思想和技术也意外地产生了对物理理论的重要见解。该项目涉及微妙的建设，是一般足以真实的兴趣的几何学家，拓扑学家和物理学家。几何学/拓扑学和场论之间的相互作用最近已经扩展到弦理论和M理论。由于弦论和M理论是量子场论的后继者，因而在结构上比量子场论更丰富，因此这一研究有望取得丰硕成果。强大的数学不仅解决了深刻的物理问题，而且很多时候我们看到物理学激发了数学的新方向，并揭示了有趣的结构。预期的研究将不仅使用技术，并从几何和拓扑结构的应用，但也将提供新的几何和拓扑结构的动机和物理学的指导下，这将是感兴趣的数学家和物理学家。该研究还将扩大合作，弥合微分几何/代数拓扑和理论/数学物理之间的文化差距。在过去三年中，PI一直在为此目的在美国数学研究所组织关于代数拓扑学和物理学的年度研究会议，并在2011年和2012年共同组织关于这些跨学科主题的另外两次会议。此外，PI希望让马里兰州的研究生参与他的研究，因为提出的问题涉及大量的想法和技术。他目前正在共同组织一个研究互动团队（RIT）的几何和物理，其中大部分的讲座是由研究生，甚至本科生。他还组织了一个超行列式和非线性代数的REU，这导致了三个本科生，两个大一和一个大二的出版物。