Mathematical Sciences: Cohomological and Homotopical Methodsin Mathematical Physics

上同调物理是指物理学的一部分，主要是规范理论和场论，在这一部分中，各种上同调技术的应用越来越广泛。即时项目是关于在研究者以前的分类空间和理性同伦理论作为代数拓扑学的研究中开发的技术的应用。它特别针对四类问题：(1)同伦结合微分梯度代数和李类似物，特别是当它们出现在弦场理论和自旋n-代数中；(2)拟hopf代数与双代数的变形；(3) Zamolodcdhikov四面体方程与广义分类空间；(4)经典和量子约束哈密顿系统约化的同调方面，体现在BRST形式主义和Batalia-Fradkin-Vilkovisky复合体及其推广中。（一）与（二）非常相似；这项研究的一个主要目的是了解这种相似性的潜在原因。(3)涉及“高维代数”，它与(1)和(2)中的代数类似，但从一维图不足以描述的结构开始。虽然在更大和更抽象的一般定义，这样的结构发生在数学物理是本工作的重点。在另一种情况下，一位著名的数学物理学家惊叹于“数学的不可思议的有效性”。本课题致力于该观察的一个特例，即代数拓扑的不合理有效性。它开始于一个发现，大约三十年前，作为一个代数拓扑学家，研究者发表的一篇论文被认为在共形弦理论和相关问题上起着关键作用。他在那里提出的h空间的“高阶结合律”在麦克莱恩的指导下经历了一系列的发展；布鲁斯坦、内曼和斯滕伯格；乔亚尔和斯特里特；最后是德林菲尔德，他在1989年对物理学和低维拓扑学都做出了深刻而巧妙的应用。Stasheff已经成为一个热情的参与者，随之而来的是对这个和相关领域的兴趣爆炸。

Cohomological physics refers to that part of physics, primarily gauge and field theories, in which a variety of cohomological techniques are seeing increasing application. The instant project is concerned with application of techniques developed in the investigator's previous study of classifying spaces and rational homotopy theory as an algebraic topologist. It is particularly directed to four classes of problems: (1) homotopy associative differential graded algebras and Lie analogs, especially as they occur in string field theories and spin n- algebras; (2) quasi-Hopf algebras and deformations of bialgebras; (3) Zamolodcdhikov's tetrahedral equation and generalized classifying spaces; (4) the homological aspects of reduction of constrained Hamiltonian systems, both classical and quantum, as embodied in the BRST formalism and the Batalia-Fradkin-Vilkovisky complex and its generalizations. (1) and (2) bear a very strong resemblance; one major thrust of this research being to understand the underlying reason for this resemblance. (3) involves "higher dimensional algebra," which is the analog of the algebra in (1) and (2) but beginning with structures for which one-dimensional diagrams are inadequate. Although defined in greater and more abstract generality, such structures as occur in mathematical physics are the focus of this work. In another context, a noted mathematical physicist has marveled at the "unreasonable effectiveness of mathematics." This project is devoted to a special case of that observation, namely, the unreasonable effectiveness of algebraic topology. It begins with the discovery that a paper the investigator published almost thirty year ago as an algebraic topologist was slated for a key role in conformal string theory and related matters. The "higher order associativity" of H-spaces that he developed there has gone through a sequence of developments at the hands of MacLane; Brustein, Ne'eman, and Sternberg; Joyal and Street; and finally Drinfeld, who, in 1989, gave deep and ingenious applications to both physics and low dimensional topology. Stasheff has become an enthusiastic participant in the ensuing explosion of interest in this and related areas.