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.

研究者研究的总体主题是荷兰理论在数学和物理学的其他领域的应用，涉及对字符串和类似结构的理解。在这些结构中，与弦理论的最直接相关是共形菲尔德理论，它似乎与椭圆仪学密切相关。 反过来，这对自动形式的领域具有影响，以及与Borcherds的月光模块的关系。 与Q的绝对Galois组，Deshendieck的Dessins d'Enfants计划以及Grothendieck-teichmueller结构的作用在形式主义上的作用中，也有可见的联系。通过类比，研究循环素的拓扑空间会导致弦拓扑，在更抽象的环境中，该拓扑与策略相关。为了解决这些问题，研究人员使用了分类的koszulduality。 与Koszul二元性相关的是在各种情况下Grothendieck/VerdierDuality的概念，这导致了A^1-和ecorivariant稳定型理论，包括现实的同义理论。 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.弦理论在70年代和80年代的物理学中出现，作为一种理论，为获得所有自然的统一理论提供了新的希望。它的基本思想是，Universe的基本单元不是点状的粒子，而是一个微小的一维字符串，它可能形成一个闭环或可以打开（带有两个端点）。自从经历了动荡的发展以来，出现了许多添加组件。如今，仍然相信物理学，即通过弦理论发现的一些与某些与之紧密相关的结构相关的理论可以促进统一计划。 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空间，通过与他们联系确定性/数值不变。在物理学中，时空本身可以被视为拓扑空间。 但是，要了解弦具有更久的字符串，人们不仅需要研究拓扑空间中的点，而且还需要研究这些空间中的循环，这些空间可以将其视为字符串模型。这包括共形场理论和弦乐学，以及代数区域的某些结构，例如Asoperads和Hochschild共同体。研究者还考虑了在某些纯粹抽象的一般环境中循环引起的造成的，例如分类koszul二元性。