Research Proposal in Algebraic Geometry and String Theory

在过去的几年里，代数几何和可积系统与量子场论和弦理论之间有了一些深刻的联系。这项建议的目的是提出一些新的联系，并探索和扩大现有的联系。第一个提出的项目涉及对傅里叶-穆凯变换或谱结构的猜想扩展到几种新的情况，包括没有截面的纤颤、非对易几何和高维复杂或特殊的拉格朗日环面纤维。特别是拉格朗日的特殊情形给出了镜像对称性的Syz方法的新的启示，并表明在Calabi-Yaus的弦模空间上存在一个可积的系统结构。接下来的两个项目涉及量子场论和弦理论这两个核心问题的应用。具体地说，第二个项目试图从特定的M理论真空开始，推导出粒子物理标准模型的一个完全现实的版本。这是基于第一个项目中讨论的亏格1上的向量丛的构造技术。第三个项目研究了另一个主要的猜想弦对偶的几个方面，即杂交弦和F-理论之间的猜想弦对偶。所提出的工作包括在几何边界的所有层上严格建立完全一般性的同构，并在一个重要的情况下将其扩展到内部，在这种情况下，也可能建立两侧的超势相等。弦理论通常被认为是解释我们所知道的关于物理世界的一切的统一物理理论的主要候选者，从最小的亚粒子尺度到整个宇宙。我们对弦理论描述现实世界的能力的信心在很大程度上依赖于最近对弦对偶的发现以及通过几何工具对它们的理解。这项提议旨在探索和扩展这些代数几何在弦理论，特别是对偶理论中的应用。它还包括一系列教育活动，包括制定课程、编写教科书以及与本科生和研究生开展广泛的工作，旨在传播有关数学和高能物理相互作用的新知识。

Some profound connections have been developed, over the past few years, between algebraic geometry and integrable systems, on the one hand, and quantum field theory and string theory on the other. The aim of this proposal is to suggest some new links, and to explore and expand the existing ones. The first proposed project involves a conjectural extension of the Fourier-Mukai transform, or the spectral construction, to several new situations, including fibrations without sections, noncommutative geometries, and higher dimensional complex or special Lagrangian torus fibers. The special Lagrangian case in particular casts new light on the SYZ approach to mirror symmetry and suggests the existence of an integrable system structure on the stringy moduli space of Calabi-Yaus. The next two projects involve applications of the first to two central issues of quantum field theory and string theory. Specifically, the second project seeks to derive a fully realistic version of the Standard Model of particle physics, starting from particular M-theory vacua. This is based on the construction techniques for vector bundles on genus-1 fibrations discussed in the first project. The third project studies several aspects of the other major conjectural string duality, the one between the heterotic string and F-theory. Proposed work includes the rigorous establishment of the isomorphism on all strata of the geometric boundary in complete generality, and its extension to the interior in one important case where it may also be possible to establish the equality of the superpotentials on both sides.String theory is generally considered to be the leading candidate for a unified physical theory which explains everything we know about the physical world, from the smallest sub-particle scales to the entire universe. Our confidence in the power of string theory to describe the real world relies to a great extent on the recent discovery of string dualities and their understanding via geometric tools. This proposal aims to explore and expand these recent applications of algebraic geometry to string theory and especially to dualities. It also includes a range of educational activities, including curriculum development, the writing of a textbook, and extensive work with undergraduate and graduate students, aimed at the dissemination of new knowledge concerning the interactions of mathematics and high energy physics.