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.

该奖项支持首席研究员在代数几何和弦理论的界面上的研究。代数几何是对由任意代数方程描述的空间的数学研究。对高能物理、密码学、系统发育学、机器人学或控制论等不同科学学科的基础研究往往表明，该学科的关键概念可以用这样的几何空间来编码。在某些情况下，这种相互作用暗示了代数几何中深刻的新问题，这些问题的解决对于进一步的发展是必要的。在其他情况下，科学直觉实际上为解决代数几何中的旧问题提供了新的方法，否则这些问题是无法解决的。弦理论和量子场论(QFT)在最小的长度尺度上探索物理，或者相应地在最高能级上探索物理。探索这些物理理论与代数几何的相互作用对数学和物理来说都是非常有成效的，这种工具和方法的组合的力量似乎只会随着时间的推移而增强。这个项目的目标是探索和推动代数几何与弦理论的接口。这将通过集中于一些具体的研究方向来实现，每个方向代表数学和/或物理中的一个重大悬而未决的问题，其解决方案将对该领域做出重大贡献，并可能受益于相反学科的技术应用。具体地说，主要研究人员提出将经典曲线理论及其模推广到超黎曼曲面，以期建立摄动超弦理论的基础并研究超弦测度；通过非阿贝尔Hodge理论证明几何朗兰兹猜想，并探索其与QFT和镜像对称性的关系；将他构造的实现Hitchin系统的Calabi-Yau可积系统推广到亚纯和抛物形式，并探索其物理应用；利用他对6维主极化阿贝尔变种的模空间的新的参数化来分析该空间并确定其Kodaira维度；并进一步探讨F理论，试图建立它的数学基础。