Some problems in Algebraic Geometry and String Theory

代数几何和弦论中的一些问题

• 批准号: 1502170
• 负责人: Sheldon Katz
• 金额: $ 25.2万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-05-15 至 2019-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1502170&HistoricalAwards=false
• 关键词: Some; problems; Algebraic; Geometry; String

The investigations in this project are at the interface of geometry and physics. In one direction, our knowledge of the geometric nature of the fundamental forces and particles of physics will be expanded, much as Einstein explained gravity as a manifestation of the curvature of space and time. In another direction, physical ideas and insights will be applied to advance our knowledge about the geometry of certain curved spaces. For the past thirty years, this general area of mathematical investigation has produced many important applications to other areas of mathematics, as well as sometimes surprising applications to physics, and to science and engineering more generally. Doctoral students will be trained to assist in mathematical aspects of these investigations, adding highly skilled workers to the STEM workforce.Several problems in algebraic geometry inspired by string theory will be investigated, in addition to problems in string theory. One set of problems involves the study of three important algebro-geometric curve-counting invariants motivated by string theory: Gopakumar-Vafa (or BPS) invariants, Donaldson-Thomas invariants, and Pandharipande-Thomas invariants, as well as their motivic and refined extensions. Some of these theories are known to describe equivalent data, while others are only conjecturally equivalent. New techniques and examples will be investigated to extend the reach of these theories. Important goals are to establish some of the conjectures, at least in certain situations, and to put the definition of the refined BPS invariants of string theory on a more firm mathematical foundation by connecting the differing approaches. In another direction, the investigation of the small quantum product of a toric variety will be reopened with the goal of answering a question about its structure that was almost answered in the 1990s. Finally, the geometry-physics dictionary for F-theory and its dual descriptions in terms of M-theory and type II string theory will be investigated with the objectives of making the dictionary more precise and then deducing applications to both algebraic geometry and physics.

该项目的研究处于几何学和物理学的交叉领域。 在一个方向上，我们对物理基本力和粒子的几何性质的认识将会得到扩展，就像爱因斯坦将引力解释为空间和时间曲率的一种表现一样。 在另一个方向上，物理思想和见解将被应用于增进我们对某些弯曲空间的几何形状的了解。 在过去的三十年里，数学研究的这一一般领域已经在数学的其他领域产生了许多重要的应用，有时也产生了对物理学以及更广泛的科学和工程的令人惊讶的应用。 博士生将接受培训，协助这些研究的数学方面，为 STEM 劳动力队伍增添高技能工人。除了弦理论中的问题外，还将研究受弦理论启发的代数几何中的几个问题。一组问题涉及由弦理论驱动的三个重要的代数几何曲线计数不变量的研究：Gopakumar-Vafa（或 BPS）不变量、Donaldson-Thomas 不变量和 Pandharipande-Thomas 不变量，以及它们的动机和改进的扩展。已知其中一些理论描述了等效数据，而其他理论仅在推测上等效。 将研究新技术和实例以扩展这些理论的范围。 重要的目标是至少在某些情况下建立一些猜想，并通过连接不同的方法将弦理论的精炼 BPS 不变量的定义置于更坚实的数学基础上。另一方面，对环面簇的小量子积的研究将重新开始，目的是回答有关其结构的问题，这个问题几乎在 20 世纪 90 年代就得到了解答。最后，将研究F理论的几何物理词典及其在M理论和II型弦理论方面的双重描述，目的是使词典更加精确，然后推导其在代数几何和物理学中的应用。