Collaborative Research: Chromatic homotopy theory and open string theory

合作研究：色同伦理论和开弦理论

• 批准号: 0705381
• 负责人: Eric Sharpe
• 金额: $ 8.11万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-15 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0705381&HistoricalAwards=false
• 关键词: Collaborative; Research; Chromatic; homotopy; theory

AbstractAward: DMS-0705233, 0705381Principal Investigator: Matthew Ando, Eric R. SharpeAndo and Sharpe will investigate several questions in ellipticcohomology suggested by the physics of open string theory. Inphysics, elliptic genera arise in closed string theory, but therecent work of Stolz and Teichner and of Borisov and Libgover andGanter strongly suggests that open string theory plays animportant role in elliptic cohomology. Ando and Sharpe propose tointroduce recent developments in open string theory, particularlyideas about D-branes and K-theory, to the study of ellipticcohomology. For example they hope to use open string theory toinvestigate the relationship between elliptic cohomology and theK-theory of K-theory, as studied by Baas, Dundas, Richter, andRognes. They also hope to give a unified account of the McKaycorrespondence for elliptic genera, due to Borisov and Libgober,and for sheaves, due to Bridgeland, King, and Reid.Elliptic cohomology arises from topology, and it signals aprofound relationship between physics, particularly the physicsof string theory, and at least two branches of mathematics,topology and number theory. Precisely how these things arerelated through elliptic cohomology remains mysterious: thephysicist Edward Witten called elliptic cohomology "a piece of21st century mathematics that happened to fall into the 20thcentury." Until very recently, the study of the physics ofelliptic cohomology has focused mainly on "closed" string theory,in which particles are replaced by loops of string. There is nowa rich theory of "open" strings, in which particles may also beunlooped strands of string. Ando and Sharpe suggest that thephysics of open string theory has deep implications in ellipticcohomology. They propose to investigate several of theseimplications, which they hope will shed light on some of theimportant mysteries in the subject.

摘要奖项：DMS-0705233、0705381主要研究员：Matthew Ando、Eric R. SharpeAndo 和 Sharpe 将研究开弦理论物理学提出的椭圆同调中的几个问题。 在物理学中，椭圆属出现在闭弦理论中，但 Stolz 和 Teichner 以及 Borisov 和 Libgover 和 Ganter 最近的工作强烈表明开弦理论在椭圆上同调中发挥着重要作用。 Ando 和 Sharpe 提议将开弦理论的最新进展，特别是有关 D 膜和 K 理论的思想引入椭圆上同调的研究中。 例如，他们希望利用开弦理论来研究椭圆上同调与 K 理论的 K 理论之间的关系，正如 Baas、Dundas、Richter 和 Rognes 所研究的那样。 他们还希望对椭圆属的麦凯对应关系（鲍里索夫和利布戈伯的贡献）以及滑轮的麦凯对应关系（布里奇兰、金和里德的贡献）给出统一的解释。椭圆上同调源于拓扑学，它标志着物理学之间的深刻关系，特别是弦理论物理学，以及至少两个数学分支：拓扑学和数论。 这些事物到底是如何通过椭圆上同调联系起来的仍然是个谜：物理学家爱德华·维滕（Edward Witten）将椭圆上同调称为“碰巧属于 20 世纪的 21 世纪数学的一部分”。 直到最近，椭圆上同调物理学的研究主要集中在“闭合”弦理论，其中粒子被弦环取代。 现在有丰富的“开放”弦理论，其中粒子也可以是解开的弦线。 安藤和夏普认为开弦理论的物理学对椭圆上同调具有深远的影响。 他们提议调查其中的几个含义，希望这将揭示该主题中的一些重要谜团。