Cohomological and Homotopical Methods in Mathematical Physics

9803435Stasheff这项研究涉及到stasheff早期在分类空间和理性同伦理论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个相互关联的问题类别，旨在阐明它们的内在结构以及重要应用的计算：(I)同伦结合微分梯度代数和李交换类似物，特别是当它们出现在各种物理场理论中；（II）拉格朗日和更一般的外部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）变形理论，产生更高的同伦代数并应用于物理系统。具体应用于高自旋粒子和混合开闭弦场论的问题。上同调物理是指数学物理的一部分，主要是规范理论和其他场论，各种上同调技术在其中得到越来越多的应用。最近，这些技术在物理背景下的进一步发展已经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，而一维图是不够的。尽管这些结构的定义更抽象、更笼统，但它们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（特别是高自旋粒子和混合开闭弦场理论）所必需的数学结构，以及物理和数学概念之间的相互关系。结果还应该具有独立的数学重要性

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.***