Noncommutative Geometry, Supergeometry, Gauge Theory and M-Theory

非交换几何、超几何、规范理论和 M 理论

• 批准号: 0204927
• 负责人: Albert Schwarz
• 金额: $ 25.8万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204927&HistoricalAwards=false
• 关键词: Noncommutative; Geometry; Supergeometry; Gauge; Theory

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.

摘要- DMS 0204927 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）弦理论的发展导致了这样一种想法，即这个理论应该被嵌入到更一般的理论中，称为M理论。这个假设理论应该存在于11维空间中，相应的低能作用应该由11维超引力给出。目前还不知道M理论的严格定义。尽管如此，物理学家们利用各种启发式工具（首先是对偶性）创造了一个美丽而一致的图景，所有版本的（超）弦理论都可以作为极限情况得到。严格定义M理论的一个有希望的方法是基于所谓的M（矩阵）理论。BFSS-M（atrix）模型的作用泛函可由10维超对称Yang-Mills理论（10 DSYM理论）降维到（0+1）维空间得到。(This意味着我们应该只考虑不依赖于空间变量，但可以依赖于时间的规范场。） 它是由A.Connes，M.道格拉斯和A.施瓦茨（1998）指出，M（矩阵）理论的紧化可以导出非对易空间，特别是非对易环面上的规范理论.本文为Connes的非对易几何在弦/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）出现之后，该论文包含了重要的新结果，Nekrasov-施瓦茨非对易瞬子（1998），A.施瓦茨定义的规范（完全）Morita等价（1998）和Pioline-施瓦茨背景独立性（1999） 从弦理论的观点来看。考虑弦/M理论与非对易空间关系的论文数量呈指数级增长（今天已接近一千篇）;其中许多论文受到A.施瓦茨和他的合作者。A.施瓦茨打算继续他的工作onapplications非交换几何弦/M理论。 他计划彻底分析非对易空间上规范场的一般理论及其与物理学中对偶性的关系。他想申请（与M。Movshev）将这一一般理论应用于非对易弯曲空间上规范理论的构造和分析。PI想追求的另一个方向（也是与M. Movshev）是一个寻找公式M（矩阵）理论（更一般地说，最大超对称规范理论）的明显超对称形式。这个问题的解决本身是有趣的，但M。Movshev和A.施瓦茨计划使用它也作为一个起点，在建设一个最大的超对称Dirac出生Infeldaction的非交换规范领域和非交换空间。A.施瓦茨正计划开发Connes的非对易几何和非对易theta函数理论的复杂版本，考虑到非对易瞬时子和其他BPS场的可能应用。他的目标还包括研究K理论在弦/M理论中的作用。M.莫夫雪夫计划分析开弦场论中的代数结构。现在很清楚，非对易几何在物理学中是很有用的。PI打算发展非交换几何的方法，并将这些方法应用于弦/M理论中出现的各种问题。它表明非对易几何在M理论的严格表述中应起重要作用。PI希望这项工作将有助于寻找适当的数学形式主义。