Algebraic Geometry of Difference Operators and Real Bundles

差分算子和实丛的代数几何

• 批准号: 0401448
• 负责人: David Ben-Zvi
• 金额: $ 11.38万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401448&HistoricalAwards=false
• 关键词: Algebraic; Geometry; Difference; Operators; Real

DMS-0401448 David Ben-ZviThe geometric Langlands program proposes an extraordinary organizing principle for representation theory over algebraic curves, inspired by the Langlands philosophy which binds harmonic analysis and Galois theory over number fields. Namely, it describes a geometric analog of spectral theory on moduli spaces of bundles. Moreover in the work of Beilinson and Drinfeld this global theory for complex algebraic curves (Riemann surfaces) arises from a local-to-global principle, whose local components are algebraic structures underlying conformal field theory (which itself underlies string theory). My proposal, highlighting joint work with D. Nadler, seeks to develop a real version of the geometric Langlands program, describing a harmonic analysis on moduli spaces of real bundles on real algebraic curves, widening the scope of interactions with both classical representation theory and string theory. A strong impetus for this extension is its potential role as an enhancement of the classical representation theory of real semisimple Lie groups. Specifically, we propose that the Langlands classification of representations appears essentially as the special case of our program when the algebraic curve is the projective line. Another motivation arises from the local features of its description on Riemann surfaces with boundary. Here we find a strong interplay with the rapidly emerging physical theory of D-branes, the boundary conditions of conformal field theory.A fundamental theme of modern mathematics is the exploitation of symmetry as an organizing principle, linking diverse and potentially baffling phenomena in an elegant overarching framework. Perhaps the prime example of this trend is the Langlands program, which identifies a general pattern in the appearance of symmetry in number theory, and counts among its successes the solution of Fermat's Last Theorem. In recent years, the Langlands philosophy has been applied in new and more geometric directions, in particular the geometry of surfaces, where it makes contact with the symmetry principles that underly the exciting developments of string theory in physics. My current research proposal (with D. Nadler) suggests an extension of this geometric Langlands program to organize the symmetries associated to surfaces with boundaries or ends. This extension has two main novel attractions. On the one hand, it seeks to encompass, and thereby shed new geometric light on, a classical topic in the study of symmetries involving real numbers. This topic appears naturally in the case when the surface involved is simply a disc. On the other hand, it suggests an intimate new link with one of the most active areas of interest in string theory, the study of the membranes where strings can attach themselves, or end. Thus this proposal provides a new instance of the versatility and unifying appeal of the Langlands philosophy.

DMS-0401448 David Ben-Zvithe几何Langlands计划在代数曲线上提出了代表理论的非凡组织原则，灵感来自Langlands哲学，该哲学将Harmonic Analysis和Galois理论结合了数量领域。也就是说，它描述了束模量空间上光谱理论的几何类似物。此外，在贝林森（Beilinson）和德林菲尔德（Drinfeld）的工作中，这种复杂代数曲线的全球理论（riemann表面）源自局部到全球原理，其局部组件是结构性场理论的代数结构（本身是基础弦乐理论）。我的建议强调了与D. Nadler的联合合作，试图开发几何兰兰兹计划的真实版本，描述了对真实代数曲线的真实捆绑图的模量空间进行的谐波分析，从而扩大了与经典表示理论和弦理论的相互作用范围。这一扩展的强大动力是它作为对真实半圣母谎言群体的经典表示理论的潜在作用。具体而言，我们建议，当代数曲线是投影线时，兰兰人的表征分类实际上是我们计划的特殊情况。另一个动机来自其描述在带边界的Riemann表面上的当地特征。在这里，我们发现与迅速出现的D-BRANES物理理论，共同场理论的边界条件有很强的相互作用。现代数学的基本主题是将对称性作为组织原则的剥削，将多样的和潜在地与优雅的总体框架联系起来。也许这种趋势的主要例子是兰兰兹计划，该计划在数字理论中识别了对称性的一般模式，并且在其成功中计算了Fermat的最后定理的解决方案。近年来，Langlands哲学已用于新的和更多的几何学方向，尤其是表面的几何形状，它与对称原理接触，这些原理是基本的物理学中弦乐理论令人兴奋的发展。我目前的研究建议（与D. Nadler一起）提出了该几何兰兰兹计划的扩展，以组织与边界或末端相关的表面相关的对称性。该扩展具有两个主要的新颖景点。一方面，它试图包容，从而为涉及实数的对称性研究中的一个古典主题提供了新的几何灯光。在涉及的表面仅仅是光盘的情况下，该主题自然而然地出现。另一方面，它暗示了与弦理论中最活跃的领域之一，即串线可以固定或结束的膜的研究。因此，该提议提供了兰兰兹哲学的多功能性和统一吸引力的新实例。