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的联系已经得到了发展。 本提案的目的是提出一些新的联系，并探讨和扩大现有的联系。 第一个提出的项目涉及傅立叶-向井变换或谱结构的几何扩展，以几种新的情况，包括无截面的纤维化，非对易几何，以及高维复杂或特殊的拉格朗日环面纤维。 特别是拉格朗日的情况下投下了新的光的SYZ方法镜像对称，并建议存在一个可积系统结构的弦模空间的卡拉比-姚。 接下来的两个项目涉及量子场论和弦理论的前两个中心问题的应用。 具体来说，第二个项目旨在从特定的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。 这是基于第一个项目中讨论的亏格1纤维化向量丛的构造技术。 第三个项目研究其他主要的弦对偶性的几个方面，杂合弦和F-理论之间的对偶性。 建议的工作包括严格建立同构的所有层次的几何边界在完整的一般性，并将其扩展到内部的一个重要情况下，它也可能建立平等的超势两侧。弦理论通常被认为是领先的候选人，为统一的物理理论，解释一切，我们知道的物理世界，从最小的亚粒子尺度到整个宇宙。 我们对弦理论描述真实的世界的能力的信心，在很大程度上依赖于最近弦对偶的发现，以及通过几何工具对它们的理解。 这个建议的目的是探索和扩大这些最近的应用代数几何弦理论，特别是对偶。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.