Cohomological and Homotopical Methods in Mathematical Physics

数学物理中的上同调和同伦方法

• 批准号: 9803435
• 负责人: James Stasheff
• 金额: $ 8.98万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803435&HistoricalAwards=false
• 关键词: Cohomological; Homotopical; Methods; Mathematical; Physics

9803435Stasheff This research is concerned with application of techniques Stasheffdeveloped earlier in his study of classifying spaces and rational homotopytheory and involves his combining of those techniques with developmentsintroduced ad hoc by physicists. The current project is concernedparticularly with three inter-related classes of problems and aims atelucidating their intrinsic structure as well as computation of significantapplications: (I) homotopy associative differential graded algebras and Lie andcommutative analogs, particularly as they occur in various physicalfield theories, (II) the homological aspects of Lagrangian and more general exteriordifferential systems, both classical and quantum, as embodied in theanti-field formalism of Batalin-Vilkovisky and its generalizations, (III) deformation theory as giving rise to higher homotopy algebra andas applied to physical systems. Specific applications are projectedto the problems of higher spin particles and of mixed open-closedstring field theory. Cohomological physics refers to that part of mathematical physics,primarily gauge and other field theories, in which a variety of cohomologicaltechniques are seeing increasing application. Recently, further developmentof these techniques within the physical context has begun to have an effecton more purely mathematical research, for example, providing new applicationsof the existing theory of ``higher dimensional algebra'' for which1-dimensional diagrams are inadequate. Although defined in greater and moreabstract generality, such structures, as they occur in or are inspired bymathematical physics, are the focus of this research. The results shouldaid in deeper understanding of the mathematical structures essential to thephysics (especially of higher spin particles and of mixed open-closed stringfield theory) and of the inter-relation of physical and mathematicalconcepts. The results should also be of independent mathematical importance.***

Stasheff这项研究是关于Stasheff在他早期对分类空间和有理同伦理论的研究中发展起来的技术的应用，并涉及到他将这些技术与物理学家特别介绍的发展相结合。目前的项目特别关注三类相互关联的问题，目的是阐明它们的内在结构和重要应用的计算：(I)同伦结合微分分次代数和Lie和交换类似物，特别是当它们出现在各种物理场理论中时，(Ii)拉格朗日和更广泛的外微分系统的同调方面，包括经典和量子，体现在Batalin-Vilkovisky的反场形式主义及其推广中，(Iii)形变理论引起更高的同伦代数并应用于物理系统。具体的应用被投射到高自旋粒子和混合开闭弦场论的问题。上同调物理指的是数学物理中的一部分，主要是规范和其他领域的理论，其中各种上同调技术正在得到越来越多的应用。最近，这些技术在物理环境中的进一步发展已开始对更纯粹的数学研究产生影响，例如，为现有的“高维代数”理论提供了新的应用，而一维图表对此是不够的。尽管在更大、更抽象的概括性中定义了这种结构，但这些结构在数学物理中出现或受到数学物理的启发，是本研究的重点。这些结果应该有助于更深入地理解物理学所必需的数学结构(特别是高自旋粒子和混合开闭弦场理论)，以及物理和数学概念之间的相互关系。结果也应该具有独立的数学重要性。*