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.

本研究要研究的问题包括关联函数和量子束上同调的确定和性质，这些函数和量子束上同调是由弦理论和规范理论产生的，与平滑射影簇和满足异常消除条件的向量丛相关。 将特别关注环面完全交集和量子束上同调的非线性西格玛模型版本的数学公式。 一个相关的问题是复曲面簇的普通量子上同调中量子积的确定。另一个主题是研究与拓扑弦振幅相关的代数几何不变量：Gromov-Witten、Gopakumar-Vafa 和稳定对不变量，包括动机稳定对不变量。这些技术将用于将弦理论中的 Gopakumar-Vafa 不变量的定义置于更坚实的数学基础上。 将开发用于计算非复曲面设置中稳定对不变量的技术，并将应用于验证镜像对称的预测。 弦理论中的 F 理论和规范/引力对偶性领域的问题也将得到研究。该资助的研究有两种模式：将代数几何的当前发展应用于以弦为模型的粒子物理的基本问题理论，并将物理思想和直觉应用于当前的几何问题。 要研究的问题是当前数学和物理领域迫切感兴趣的问题。 正在开发的数学技术可以使在日益现实的弦模型中精确计算粒子的预测质量成为可能，并更好地理解自然界中观察到的力和相互作用如何从包括量子引力在内的大统一理论中产生。弦理论的思想提供了关键的见解，有望直接解决三维代数几何中几个未解决的问题。 研究生将接受几何和物理及其相互作用的培训。该奖项由代数和数论以及几何分析项目共同资助。