Geometric and Topolocical Structures Related TO M-branes

本项目的目标是研究m -膜在m理论中产生的新的几何和拓扑结构。PI还计划研究m膜的电荷组。这将涉及到束、广义上同调和模形式的高级概念。这里提出的主题表明理论和数学物理与几何和拓扑学之间的丰富互动，并将提供物理学，微分几何和代数拓扑学的新结构。数学通过表征和消除异常，在决定物理理论的一致性方面起着非常重要的作用。异常消去相当于对场的细化，并提供了一个自然的舞台，可以根据（广义）上同调理论对底层空间施加定向。这些理论提供了描述不变量的基本数学机制。另一方面，m理论，虽然还没有被构建，但已经是一个非常丰富的理论，无论是在物理思想方面还是在数学结构方面。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之前的工作强调了这种方法的力量，并展示了这种方法在获得新结果和揭示物理学中产生的新数学结构方面的力量。此外，PI和该领域的其他人的工作表明，应用几何学和拓扑学的思想和技术也意外地产生了对物理理论的非凡见解。该项目涉及到一些微妙的结构，这些结构非常普遍，足以引起几何学家、拓扑学家和物理学家的真正兴趣。几何/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。由于弦理论和m理论包含了量子场论，因此在结构上比量子场论更丰富，因此预计这将是富有成果的。强大的数学不仅解决了深刻的物理问题，而且我们经常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几何和拓扑结构的技术和应用，而且还将提供新的几何和拓扑结构，这些结构将受到物理学的激励和指导，这将使数学家和物理学家都感兴趣。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鸿沟。为了这个目的，PI在过去三年中一直在美国数学研究所组织关于代数拓扑和物理的年度研究会议，并在2011年和2012年就此类跨学科主题共同组织了另外两次会议。此外，由于所提出的问题涉及丰富的思想和技术，PI希望与马里兰的研究生参与他的研究。他目前正在共同组织一个关于几何和物理的团队研究互动（RIT），其中大部分讲座由研究生甚至本科生提出。他还组织了一个关于超行列式和非线性代数的REU，并与三名本科生，两名大一新生和一名大二学生一起发表了一篇文章。

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.