Research Project in Algebraic Geometry and String Theory

最近在杂化弦理论中产生粒子物理标准模型方面的突破是代数几何为物理服务的力量的完美例证。利用构造满足各种陈类和上同调约束的非单连通的Calabi-Yau三重体及其丛的技术，PI和一位合作者最近构造了唯一已知的异质弦紧凑化的例子，它恰好产生了粒子和力的最小超对称标准模型(MSSM)谱，没有多余的外来物质。这就提出了许多与弦真空地貌的高地调查有关的问题。这些问题在代数几何和弦现象学中都很有意义。在这项建议中使用代数几何技术研究的其他物理问题包括探索杂交弦和f-理论之间的对偶性，以及大N对偶性。纯几何问题包括谱结构和傅立叶-Mukait变换到GERBE的几种推广，以及这种推广将有许多应用的非对易和相关的情况。这些结果给出了具有任意结构群的Hitchin系统的Langland对偶的证明，其中包括关于Calabi-Yaus的模、它们的退化以及由此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，它结合了广义相对论和量子场论。代数几何是对由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非常通用和强大的工具。这一提议的主要推动力是使用代数几何的技术来推导粒子物理的标准模型，作为弦理论的一个具体实例。

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.