Research Project in Algebraic Geometry and String Theory

代数几何和弦理论研究项目

• 批准号: 0612992
• 负责人: Ron Donagi
• 金额: $ 28.46万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0612992&HistoricalAwards=false
• 关键词: Research; Project; Algebraic; Geometry; String

The recent breakthrough in producing the Standard Model of particlephysics within heterotic string theory is a perfect illustration of thepower of algebraic geometry at the service of physics. Using techniquesfor construction of non simply connected Calabi-Yau threefolds and ofbundles on them satisfying various constraints on their chern classes andcohomology, the PI and a collaborator have recently constructed the onlyknown example of a heterotic string compactification which producesexactly the Minimal Suppersymmetric Standard Model (MSSM) spectrum ofparticles and forces, with no unwanted exotic matter. This opens upnumerous questions related to investigation of the High Country region ofthe Landscape of string vacua. These problems are of great interest inboth algebraic geometry and string phenomenology. Other physics questionsinvestigated in this proposal using techniques of algebraic geometryinclude exploration of the duality between the heterotic string andF-theory, and the Large N Duality. The purely geometric issues includeseveral extensions of the spectral construction and Fourier- Mukaitransforms to gerbes and non-commutative and related situations where suchextensions would have many applications. These yield a proof of Langlandsduality for Hitchin systems with arbitrary structure groups, and includeseveral problems about the moduli of Calabi-Yaus, their degenerations, andthe integrable systems arising from them.String theory is the leading physical candidate for a unified fieldtheory, incorporating both general relativity and quantum field theory.Algebraic geometry is the mathematical investigation of spaces defined byalgebraic equations. It provides an extermely versatile and powerful toolfor applications to string theory and to many other areas. The main thrustof this proposal is the use of techniques from algebraic geometry toderive the Standard Model of particle physics as a concrete instance ofstring theory.

最近在异质弦理论中产生粒子物理学标准模型的突破完美地说明了代数几何为物理学服务的力量。使用构造非简单连接的 Calabi-Yau 三重及其束的技术，满足其陈类和上同调的各种约束，PI 和合作者最近构造了唯一已知的异质弦紧致化示例，该示例精确地产生了最小超对称标准模型 (MSSM)粒子和力的谱，没有不需要的外来物质。这引发了与弦真空景观高地地区调查相关的许多问题。这些问题在代数几何和弦唯象学中都很有趣。本提案中使用代数几何技术研究的其他物理问题包括探索异质弦和 F 理论之间的对偶性以及大 N 对偶性。纯粹的几何问题包括谱构造的几种扩展和傅里叶-穆凯变换到格布以及非交换和相关情况，这些扩展将有许多应用。这些证明了具有任意结构群的希钦系统的朗兰兹对偶性，并包括有关卡拉比-尤斯模数、它们的简并以及由它们产生的可积系统的几个问题。弦理论是统一场论的主要物理候选者，它结合了一般场论相对论和量子场论。代数几何是对代数方程定义的空间的数学研究。它为弦理论和许多其他领域的应用提供了一个极其通用和强大的工具。该提案的主旨是使用代数几何技术来推导粒子物理的标准模型作为弦理论的具体实例。