Noncommutative Geometry, Supergeometry, Gauge Theory and M-Theory

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数学问题。(超级)弦理论的发展导致了这样的想法，即这个理论应该嵌入到更一般的理论中，称为M理论。这一假设理论应该存在于11维空间中，相应的低能量作用应该由11维超引力给出。目前，M-理论的严格定义尚不为人所知。然而，物理学家使用各种启发式工具(首先是对偶性)创建了一幅美丽而一致的图景，其中所有版本的(超弦)理论都可以作为极限情况得到。一种有希望的严格定义M理论的方法是基于所谓的M(Atrix)理论。BFSS M(ATRIX)模型的作用泛函可以从10维超对称Yang-Mills理论(10维SYM理论)通过降维到(0+1)维空间得到(这意味着我们应该只考虑不依赖于空间变量而可以依赖于时间的规范场)。A.Connes，M.Douglas和A.Schwarz(1998)证明了M(Atrix)理论的紧化可以得到非对易空间上的度量理论，特别是非对易拓扑论。本文为Connes非对易几何在弦/M-理论中的应用开辟了道路。它变得非常流行，特别是在有影响力的Seiberg-Witten论文(1999)的出现之后，这篇论文包含了重要的新结果，从弦理论的角度对Nekrasov-Schwarz非对易瞬子(1998)、A.Schwarz定义的规范(完全)Morita等价和Pioline-Schwarz背景独立性(1999)的理解。研究弦/M-理论与非对易空间的关系的论文数量呈指数级增长(今天接近1000篇)；许多研究受到A.Schwarz及其合作者的论文的影响。施瓦茨打算继续他在非对易几何在弦/M理论中的应用方面的工作。他计划深入分析非对易空间上规范场的一般理论，以及它与物理学中的对偶性的关系。他希望(与M.Movshev合作)将这一一般理论应用于非对易曲线空间上规范理论的构造和分析。PI想要追求的另一个方向(也是与M.Movshev一起)是以明显的超对称形式搜索公式M(Atrix)理论(更广泛地说，最大超对称规范理论)。这个问题的解本身很有趣，但M.Movshev和A.Schwarz正计划将其用作构造非阿贝尔规范场和非对易空间的极大超对称Dirac-Born-Infield作用量的起点。A.Schwarz正计划发展一个复杂版本的Connes非对易几何和非对易theta函数理论，考虑到可能应用于非对易瞬子和其他BPS领域。他的目的也是研究K-理论在弦/M-理论中的作用。莫夫谢夫正计划分析在开放弦场理论中出现的经典结构。现在很明显，非对易几何在物理学中非常有用。PI打算发展非对易几何的方法，并将这些方法应用于弦/M理论中出现的各种问题。看来，非对易几何在M-理论的严格表述中应该发挥重要作用。PI希望这项工作将有助于寻找适当的数学形式主义。

Abstract - DMS 0204927 The main goal of the proposal is to apply methods of noncommutative geometry and supergeometry to the mathematical problems arising in theoretical physics. The development of (super)string theory led to the idea that this theory should by embedded into more general theorycalled M-theory. This hypothetical theory should live in11-dimensional space and the corresponding low energy actionshould be given by 11-dimensional supergravity. At thismoment a rigorous definition of M-theory is not known. Nevertheless, using various heuristic tools (first of all,dualities) physicists have created a beautiful andconsistent picture, where all versions of (super)stringtheory can be obtained as limiting cases.A promising approach to a rigorous definition of M-theory is based on socalled M(atrix) theory. The action functional of BFSS M(atrix) model can be obtained from ten-dimensional supersymmetricYang-Mills theory ( 10 D SYM theory) by means of dimensionalreduction to (0+1)-dimensional space.(This means that we should consider only gauge fields that do not depend on spatialvariables, but can be time-dependent.) It was shown by A.Connes, M. Douglas and A. Schwarz (1998) thatcompactification of M(atrix) theory can leadto gauge theories on noncommutative spaces, in particular, on noncommutative tori.This paper opened the way for application of Connes' noncommutative geometry to string/M-theory . Itbecame very popular, especially after the appearance of theinfluential Seiberg-Witten paper (1999) , that containstogether with important new results, the understanding ofNekrasov-Schwarz noncommutative instantons (1998), gauge(complete) Morita equivalence defined by A.Schwarz (1998),and Pioline-Schwarz background independence (1999) from thestring theory viewpoint. The number of papers considering the relation of string/M-theory to noncommutative spaces grows exponentially (today it is close to a thousand); many of thesearticles were influenced by the papers of A. Schwarz and hiscollaborators. A. Schwarz intends to continue his work onapplications of noncommutative geometry to string/ M-theory. He is planning to analyze thoroughly the general theory of gauge fields on noncommutative spaces and its relation to the duality in physics. He would like to apply (in collaboration with M. Movshev) this general theory to the constructionand analysis of gauge theories on noncommutative curved spaces. One more direction the PI would like to pursue ( alsotogether with M. Movshev) is a search of formulationM(atrix) theory ( and, more generally, maximally supersymmetricgauge theory) in a manifestly supersymmetric form. The solution of this problem is interesting by itself, but M. Movshev and A. Schwarz are planning to use it also as a starting point in the construction of a maximally supersymmetric Dirac-Born-Infeldaction for nonabelian gauge fields and on noncommutative spaces. A. Schwarz is planning to develop a complex version of Connes' noncommutative geometry and the theory ofnoncommutative theta functions having in mind possibleapplications to noncommutative instantons and other BPSfields.He aims also to study the role of K-theory in string/ M-theory. M. Movshev is planning to analyzealgebraic structures arising in open string field theory. It is clear now that noncommutative geometry is quite usefulin physics. The PI intends to develop methods ofnoncommutative geometry and to apply these methods tovarious problems arising in string/ M-theory. It seemsthat noncommutative geometry should play an important rolein rigorous formulation of M-theory. The PI hopes thathis work will contribute to the search of appropriatemathematical formalism.