Some problems in algebraic geometry and string theory

代数几何和弦论中的一些问题

• 批准号: 1201089
• 负责人: Sheldon Katz
• 金额: $ 35.49万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-05-15 至 2016-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201089&HistoricalAwards=false
• 关键词: Some; problems; algebraic; geometry; string

Problems to be investigated in the proposed research include the determination and properties of correlation functions and quantum sheaf cohomology arising from string theories and gauge theories associated to a smooth projective variety and a vector bundle satisfying the anomaly cancellation conditions. Particular attention will be paid to toric complete intersections and the mathematical formulation of the nonlinear sigma model version of quantum sheaf cohomology. A related problem is the determination of the quantum product in the ordinary quantum cohomology of a toric variety. Another theme is the study of algebro-geometric invariants related to topological string amplitudes: Gromov-Witten, Gopakumar-Vafa, and stable pair invariants, including motivic stable pair invariants. These techniques will be applied to put the definition of Gopakumar-Vafa invariants from string theory on a more firm mathematical foundation. Techniques will be developed for the computation of stable pair invariants in the non-toric setting, and will be applied to verify predictions of mirror symmetry. Problems in the area of F-theory and gauge/gravity duality in string theory will be investigated as well.The research funded by this grant has two modes: bringing current developments in algebraic geometry to bear on fundamental problems of particle physics as modeled by string theory, and bringing physical ideas and intuition to bear on current problems in geometry. The questions to be investigated are of pressing current interest in mathematics and physics. The mathematical techniques under development could make it possible to exactly compute the predicted masses of particles in increasingly realistic string models, and better understand how the observed forces and interactions in nature can arise from grand unified theories which include quantum gravity. Ideas of string theory provide key insight which are expected to directly lead to the solution of several unsolved problems in three-dimensional algebraic geometry. Graduate students will be trained in geometry and physics, and their interaction.This award is jointly funded by the Algebra and Number Theory and the Geometric Analysis programs.

在拟议的研究中需要研究的问题包括与光滑的投射品种相关的弦理论和量学理论以及满足异常取消条件的矢量束相关的量子造函数的确定和特性。 将特别注意曲折的完整交叉点以及量子脱落协同学的非线性Sigma模型版本的数学公式。 一个相关的问题是确定倍态品种的普通量子共同体中的量子产物。另一个主题是研究与拓扑弦振幅相关的代数几何不变的：gromov-witten，gopakumar-vafa和稳定的成对配对，包括动机稳定的稳定对不变式。这些技术将应用于将弦理论中的gopakumar-vafa不变性的定义放在更坚定的数学基础上。 将开发用于计算非态设置中稳定对不变性的技术，并将应用于验证镜像对称性的预测。 还将研究弦理论中F理论和量规/重力二元性领域的问题。该赠款资助的研究具有两种模式：将代数几何形状的当前发展带来粒子物理学的基本问题，以弦乐理论建模，并将物理思想和直觉带来当前的几何学问题。 要研究的问题是迫切需要对数学和物理学的兴趣。 正在开发的数学技术可以准确地计算越来越现实的弦模型中预测的粒子质量，并更好地理解自然界中观察到的力和相互作用如何来自包括量子重力的大统一理论。弦理论的思想提供了关键的见解，这些洞察力有望直接导致解决三维代数几何形状中几个未解决问题的解决方案。 研究生将接受几何和物理学的培训，并将其相互作用。该奖项由代数和数字理论和几何分析计划共同资助。