Algebraic Geometry of Difference Operators and Real Bundles

DMS-0401448 David Ben-Zv几何朗兰兹程序为代数曲线上的表示理论提出了一种非凡的组织原则，灵感来自于朗兰兹哲学，该哲学将调和分析与数域上的伽罗瓦理论捆绑在一起。也就是说，它描述了丛的模空间上的谱理论的几何模拟。此外，在Beilinson和Drinfeld的工作中，复杂代数曲线(Riemann曲面)的全局理论源于局部到全局原理，该原理的局部分量是支撑保形场论(其本身是弦理论的基础)的代数结构。我的提案强调了与D.Nadler的合作，试图开发几何朗兰兹程序的实版，描述了实代数曲线上实丛的模空间上的调和分析，扩大了与经典表示理论和弦理论的交互范围。这一推广的一个强大推动力是它作为实半单李群的经典表示理论的一个潜在的增强作用。具体地说，我们提出，当代数曲线是投影线时，表示的朗兰兹分类实质上是我们程序的特例。另一个动机来自于它在有边界的黎曼曲面上刻画的局部特征。在这里，我们发现了与快速崛起的D膜物理理论，即共形场理论的边界条件的强烈相互作用。现代数学的一个基本主题是利用对称性作为一种组织原则，在一个优雅的总体框架内将各种和潜在的令人费解的现象联系起来。也许这一趋势的主要例子是朗兰兹计划，该计划确定了数论中对称性出现的一般模式，并将费马最后定理的解计算在其成功之列。近年来，朗兰兹哲学已被应用于新的和更多的几何方向，特别是曲面的几何，它与对称性原理相联系，这些原理是弦理论在物理学中令人兴奋的发展的基础。我目前的研究提案(与D·纳德勒)建议扩展这个几何朗兰兹程序，以组织与具有边界或末端的曲面相关联的对称性。这一延伸有两个主要的新奇吸引力。一方面，它试图涵盖并由此揭示涉及实数的对称性研究中的一个经典主题，从而揭示出新的几何意义。当所涉及的表面仅仅是光盘时，该主题自然出现。另一方面，它提出了一种与弦理论中最活跃的感兴趣的领域之一的亲密新联系，即研究弦可以连接自身或结束的膜。因此，这一提议为朗兰兹哲学的多功能性和统一性提供了一个新的实例。

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.