Research Proposal in Algebraic Geometry and String Theory

在过去的几年中，代数几何和可积系统与量子场论和弦理论之间已经发展了一些深刻的联系。本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- mukai变换或谱结构的推测扩展，以几种新情况，包括无截面的纤维，非交换几何形状，高维复杂或特殊拉格朗日环面纤维。特别地，拉格朗日的特殊情况为SYZ方法的镜像对称提供了新的思路，并表明了Calabi-Yaus弦模空间上可积系统结构的存在。接下来的两个项目涉及到量子场论和弦理论的两个核心问题的应用。具体来说，第二个项目试图从特定的m理论真空出发，推导出粒子物理学标准模型的完全现实版本。这是基于在第一个项目中讨论的1属纤维的载体束的构建技术。第三个项目研究了另一个主要的推测弦对偶性的几个方面，即异质性弦与f理论之间的对偶性。建议的工作包括在完全一般的几何边界的所有层上严格建立同构，并在一个重要的情况下将其扩展到内部，其中也可能建立两边超势的等式。弦理论通常被认为是统一物理理论的主要候选者，它可以解释我们所知道的关于物理世界的一切，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性的发现以及通过几何工具对它们的理解。本提案旨在探索和扩展代数几何在弦理论，特别是对偶性中的最新应用。它还包括一系列教育活动，包括课程开发，教科书的编写，以及与本科生和研究生的广泛合作，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.