Research at the Interface of Algebraic Geometry and String Theory

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。对高能物理学、密码学、遗传学、机器人学或控制理论等不同科学学科的基础研究经常揭示出，该学科的关键概念可以用这种几何空间来编码。在某些情况下，这种相互作用提出了代数几何中的新问题，其解决方案是进一步发展所必需的。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，这些问题在其他情况下无法解决。弦理论和量子场论（QFT）在最小的长度尺度上探索物理学，或者相应地在最高的能级上探索物理学。探索这些物理理论与代数几何的相互作用对数学和物理学都是非常有成效的，这种工具和方法的结合的力量似乎只会随着时间的推移而加强。这个项目的目标是探索和推进代数几何与弦理论的接口。这将通过专注于一些特定的研究方向来完成，每个方向代表数学和/或物理学中的一个主要开放问题，其解决方案将对该领域做出重大贡献，并且可能受益于相反学科的技术应用。具体而言，主要研究者提出探索经典曲线及其模理论到超黎曼曲面的扩展，以期建立微扰超弦理论的基础并研究超弦测度;通过非交换Hodge理论证明几何Langlands猜想，并探索其与QFT和镜像对称的关系;将他的实现希钦系统的Calabi-Yau可积系统的构造扩展到亚纯和抛物版本，并探索其物理应用;使用他对6维主极化阿贝尔簇的模空间的新参数化来分析该空间并确定其科代拉维数;进一步探索F理论的各个方面，并试图建立它的数学基础。

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.