Collaborative Research: Chromatic homotopy theory and open string theory

项目编号：DMS-0705233, 0705381首席研究员：Matthew Ando， Eric R. SharpeAndo和Sharpe将研究开放弦理论中椭圆上同调的几个问题。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放弦理论在椭圆上同调中起着重要作用。Ando和Sharpe建议将开放弦理论的最新进展，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，他们希望使用开放弦理论来研究椭圆上同性和k理论的k理论之间的关系，正如Baas， Dundas, Richter， andRognes所研究的那样。他们还希望对椭圆属的麦凯对应给出一个统一的解释，这是鲍里索夫和利戈伯的功劳，而对于麦凯对应是布里奇兰、金和里德的功劳。椭圆上同调源于拓扑学，它标志着物理学，特别是弦理论物理学，和至少两个数学分支，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起来的仍然是个谜：物理学家爱德华·威滕（Edward Witten）称椭圆上同调是“碰巧落入20世纪的21世纪数学的一部分”。直到最近，椭圆上同调的物理研究主要集中在“闭”弦理论上，其中粒子被弦的环所取代。现在有一个丰富的“开放”弦理论，其中的粒子也可能是无环的弦。Ando和Sharpe认为开放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含义，他们希望这将揭示这个主题的一些重要谜团。

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.