Collaborative Research: Chromatic homotopy theory and open string theory

摘要奖：dms-0705233,0705381主要研究员：Matthew Ando，Eric R.SharpeAndo和Sharpe将研究由开弦理论物理学提出的椭圆上同调中的几个问题。在物理学中，椭圆代数起源于闭弦理论，但最近Stolz和Teichner以及Borisv和Libgover和Ganter的工作有力地表明了开弦理论在椭圆上同调中的重要作用。Ando和Sharpe建议将开弦理论的最新发展，特别是关于D-膜和K-理论的思想引入到椭圆上同调的研究中。例如，他们希望使用开弦理论来研究椭圆上同调和K理论中的k理论之间的关系，就像Baas、Dundas、Richter和Rognes所研究的那样。他们还希望对由于鲍里索夫和利格伯而产生的椭圆属的麦凯对应以及由于布里奇兰、金和里德而产生的束的麦凯对应给出统一的描述。椭圆上同调源于拓扑学，它标志着物理学，特别是弦理论的物理学与数学的至少两个分支--拓扑学和数论之间的深刻关系。这些东西究竟是如何通过椭圆上同调联系在一起的仍然是个谜：物理学家爱德华·威顿将椭圆上同调称为“一件碰巧落在20世纪的21世纪数学作品”。直到最近，对椭圆上同调物理的研究主要集中在“封闭的”弦理论上，即用弦的环来代替粒子。现在有一种丰富的“开放”弦理论，在这种理论中，粒子也可以是未绕圈的弦。Ando和Sharpe提出，开弦理论的物理学在椭圆上同调中有着深刻的含义。他们提议调查其中的几个隐含意义，希望这将有助于揭示这一主题中的一些重要谜团。

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.