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.

该项目的调查是在几何和物理学的界面上。 在一个方向上，我们对物理基本力和物理颗粒的几何特性的了解将被扩展，就像爱因斯坦将重力解释为时空曲率的表现一样。 在另一个方向上，将采用物理思想和见解来促进我们对某些弯曲空间几何形状的了解。 在过去的三十年中，这一数学研究的一般领域为其他数学领域提供了许多重要的应用，有时还为物理学以及对科学和工程的应用更为令人惊讶。 博士生将接受培训，以协助这些调查的数学方面，在STEM劳动力中增加高技能的工人。除弦理论中的问题外，还将研究受字符串理论启发的代数几何问题。一组问题涉及研究由弦理论激发的三个重要代数几何曲线计数的不变性：gopakumar-vafa（或BPS）不变性，唐纳森 - 托马斯 - 托马斯不变性和pandharipande-thomas-thomas novariants，以及他们的动机和精神分子和精神分裂。这些理论中的一些描述了等效数据，而另一些则只是猜想。 将研究新技术和示例，以扩大这些理论的影响。 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来将弦理论的精致BP不变性的定义放在更坚定的数学基础上。在另一个方向上，将重新开放对复曲面品种的小量子产品的研究，目的是回答有关其结构的问题，几乎在1990年代得到了回答。最后，将对F理论的几何形态词典及其双重描述在M理论和II型字符串理论方面进行研究，以使词典更精确，然后将应用程序推论到代数几何和物理学的目标。