Research at the Interface of Algebraic Geometry and String Theory

代数几何与弦理论的接口研究

• 批准号: 1603526
• 负责人: Ron Donagi
• 金额: $ 51万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1603526&HistoricalAwards=false
• 关键词: Research; Interface; Algebraic; Geometry; String

The award supports the principal investigator's research at the interface of algebraic geometry and string theory. Algebraic geometry is the mathematical study of spaces described by arbitrary algebraic equations. Fundamental investigation of such diverse scientific disciplines as high energy physics, cryptography, phylogenetics, robotics, or control theory, often reveals that key concepts of the discipline can be encoded in terms of such geometric spaces. In some cases, such interactions suggest deep new problems in algebraic geometry whose solution is necessary for further progress. In other instances, the scientific intuition actually suggests new methods for solving old problems in algebraic geometry that were otherwise inaccessible. String theory and quantum field theory (QFT) explore physics at the smallest length scales, or correspondingly at the highest energy levels. Exploration of the interactions of these physical theories with algebraic geometry has been extremely productive for both math and physics, and the power of this combination of tools and approaches only seems to strengthen with time.The goal of this project is to explore and push forward the interface of algebraic geometry with string theory. This will be done by focusing on a number specific research directions, each representing a major open problem in math and/or in physics, whose solution will make a major contribution to the field, and is likely to benefit from the application of techniques of the opposite discipline. Specifically, the principal investigator proposes to explore the extension of the classical theory of curves and their moduli to super Riemann surfaces, with a view towards establishing the foundations of perturbative superstring theories and studying the superstring measure; to prove the geometric Langlands conjecture via non abelian Hodge theory, and explore its relation to QFT and to mirror symmetry; to extend his construction of Calabi-Yau integrable systems realizing Hitchin's system to meromorphic and parabolic versions, and explore the physical applications; to use his new parametrization of the moduli space of 6 dimensional principally polarized abelian varieties to analyze this space and determine its Kodaira dimension; and to explore further aspects of F theory and attempt to establish its mathematical foundations.

该奖项支持主要研究者在代数几何和弦理论的界面上的研究。代数几何形状是对任意代数方程描述的空间的数学研究。对高能量物理学，密码学，系统发育，机器人或控制理论等多样的科学学科的基本研究通常表明，该学科的关键概念可以根据此类几何空间进行编码。在某些情况下，这种相互作用暗示了代数几何形状的深刻新问题，其解决方案对于进一步的进展是必要的。在其他情况下，科学的直觉实际上提出了解决代数几何形状中旧问题的新方法，而这些方法否则无法访问。字符串理论和量子场理论（QFT）以最小的长度尺度或最高能级探索物理。探索这些物理理论与代数几何形状的相互作用对数学和物理学都非常有效，而这种工具和方法组合的力量似乎只会随着时间的推移而增强。该项目的目的是探索和推动用弦理论来探索和推动代数几何学的界面。这将通过关注一个特定的研究方向来完成，每个研究都代表数学和/或物理学中的主要开放问题，其解决方案将对该领域做出重大贡献，并且很可能会从相反学科的技术的应用中受益。具体而言，首席研究者建议探索曲线及其模量的经典理论扩展到超级黎曼表面，以确立扰动超声理论的基础，并研究超造成措施；通过非Abelian Hodge理论证明了几何Langlands猜想，并探索其与QFT的关系并反映对称性；为了扩展他的Calabi-Yau集成系统的构建，以实现Hitchin的系统来实现Meromorormorphic和抛物线版本，并探索物理应用；使用他对6维空间的新参数化，主要是极化的Abelian品种来分析该空间并确定其Kodaira尺寸；并探索F理论的进一步方面，并试图建立其数学基础。