String-related structures in homotopy theory

同伦理论中的弦相关结构

• 批准号: 0503814
• 负责人: Po Hu
• 金额: --
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503814&HistoricalAwards=false
• 关键词: String; related; structures; homotopy; theory

The overall theme of the investigator's research is the application ofhomotopy theory to other areas of mathematics and to physics, inparticular to the understanding of strings and similar structures. Amongthese structures, most directly related to of string theory is conformalfield theory, which also appears to be closely connected to ellipticcohomology. This in turn has implications in the area of automorphicforms, and relations with Borcherds' Moonshine module. There is also aremarkable connection with the absolute Galois group of Q, throughGrothendieck's program of dessins d'enfants, and the action ofGrothendieck-Teichmueller structures on the formalism that make upconformal field theories. By way of analogy, studying the space of loopsin a topological space leads to the string topology, which is closelyrelated to operads and deformation theory in more abstract contexts. Toapproach these questions, the investigator uses categorical Koszulduality. Related to Koszul duality is the notion of Grothendieck/Verdierduality in various contexts, which led back to A^1- and equivariant stablehomotopy theory, including Real-oriented homotopy theory. This is directlyanalogous to the A^1-homotopy theory constructed by Morel and Voevodsky.-------------------------------------The overall theme of the investigator's research is the interactionbetween the area of algebraic topology in mathematics, and string theoryin physics. String theory emerged in physics in the 70's and 80's as atheory providing new hope toward gaining a unified theory of all theforces of nature; its basic idea was that the fundamental units of theuniverse are not point-like particles, but tiny one-dimensional strings,which may form a closed loop, or may be open (with two endpoints). Thetheory since underwent a tumultuous development in which many additionalstructures emerged. Today it is still believed in physics that theunification program may be facilitated by some theory closely related tostructures discovered by string theory. Despite the long history, manyfundamental aspects of string theory are not well mathematicallyunderstood, which, it seems, has somewhat hindered the physical theory.The investigator studies some of these issues both directly, and by way ofanalogy and connection with similar structures elsewhere in mathematics.Topology can be thought of as geometry in its purely qualitative aspects.In particular, algebraic topology has built up many powerful methods forstudying topological spaces, by associating to them certainalgebraic/numerical invariants. In physics, spacetime itself can beconsidered as a topological space. However, to understand strings inspacetime, one needs to study not only the points in topological spaces,but loops in such spaces, which can be thought of as models for strings.The investigator studies the structures arising from the space of loops intopological spaces. This includes conformal field theory and stringtopology, as well as certain structures from the area of algebra, such asoperads and Hochschild cohomology. The investigator also considers thestructures arising from loops in certain purely abstract, general context,for instance that of categorical Koszul duality.

调查员的研究的总体主题是同伦理论应用到数学和物理学的其他领域，特别是对字符串和类似结构的理解。在这些结构中，与弦论最直接相关的是共形场论，它似乎也与椭圆上同调密切相关。 这反过来又在自同构形式领域产生了影响，并与Borcherds的Moonshine模有关系。 通过Grothendieck的dessins d 'enfants计划，以及Grothendieck-Teichmueller结构对构成共形场论的形式主义的作用，Q的绝对伽罗瓦群也有显著的联系。通过类比的方式，研究拓扑空间中的环的空间导致弦拓扑，弦拓扑在更抽象的背景下与操作和变形理论密切相关。为了解决这些问题，研究者使用范畴Koszulduality。 与Koszul对偶相关的是Grothendieck/Verdierduality的概念在不同的背景下，它导致了A^1-和等变稳定同伦理论，包括实定向同伦理论。这直接类似于莫雷尔和沃斯基构造的A^1-同伦理论。调查员研究的总主题是数学中代数拓扑学与物理学中弦理论之间的相互作用。弦理论出现在物理学在70年代和80年代作为一个理论提供了新的希望获得一个统一的理论的所有自然力;它的基本思想是，基本单位的宇宙不是点状粒子，但微小的一维弦，这可能会形成一个封闭的循环，也可能是开放的（有两个端点）。此后，这个理论经历了一个动荡的发展过程，出现了许多额外的结构。今天，物理学中仍然相信，统一计划可能会受到某些与弦论发现的结构密切相关的理论的推动。尽管历史悠久，弦理论的许多基本方面并没有很好地理解，这似乎在某种程度上阻碍了物理理论。研究者研究这些问题的一些直接，并通过类比和联系类似的结构在数学的其他地方。拓扑学可以被认为是几何学在其纯粹的定性方面。特别是，代数拓扑学通过与某些代数/数值不变量相联系，建立了许多研究拓扑空间的强有力的方法。在物理学中，时空本身可以被认为是一个拓扑空间。 然而，要理解弦的超时空，人们不仅需要研究拓扑空间中的点，还需要研究拓扑空间中的环，它们可以被认为是弦的模型。这包括共形场论和弦拓扑，以及代数领域的某些结构，如运算和Hochschild上同调。研究者还认为thestructures所产生的循环在某些纯粹抽象的，一般的情况下，例如，分类Koszul对偶。