Some problems in algebraic geometry and string theory

拟研究的问题包括关联函数和量子层上同调的确定及其性质，这些关联函数和量子层上同调来自弦理论和与光滑射影簇和满足反常消去条件的向量丛相关的规范理论。 特别注意将支付给复曲面完整的交叉和量子层上同调的非线性西格玛模型版本的数学公式。 一个相关的问题是确定的量子产品在普通的量子上同调的环面品种。另一个主题是与拓扑弦振幅相关的代数几何不变量的研究：Gromov-Witten，Gopakumar-Vafa和稳定对不变量，包括motivic稳定对不变量。这些技术将应用于把Gopakumar-Vafa不变量的定义从弦理论上一个更坚实的数学基础。 技术将被开发用于计算稳定对不变量在非复曲面设置，并将被应用于验证镜像对称的预测。 该基金的研究模式有两种：一种是将代数几何的最新发展应用于弦论所模拟的粒子物理的基本问题，另一种是将物理学的思想和直觉应用于几何学的最新问题。 所要研究的问题是当前数学和物理学中的迫切兴趣。 正在开发的数学技术可以精确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洞察力，有望直接导致三维代数几何中几个未解决的问题的解决。 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。

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.