Some problems in algebraic geometry and string theory

所要研究的问题包括与光滑射影簇和满足反常抵消条件的向量丛相关的弦理论和规范理论所产生的关联函数和量子层上同调的确定和性质。我们将特别关注环形完全交集和量子层上同调的非线性西格玛模型版本的数学表述。一个相关的问题是确定环簇的普通量子上同调中的量子积。另一个主题是研究与拓扑弦振幅有关的代数几何不变量：Gromov-Witten，Gopakumar-Vafa，以及稳定的对不变量，包括Motivic稳定的对不变量。这些技术将被应用于将弦理论中的Gopakumar-Vafa不变量的定义建立在更坚实的数学基础上。我们将开发技术来计算非圆环境中稳定的对不变量，并将其应用于验证镜像对称性的预测。此外，还将研究弦理论中的F理论和规范/引力对偶问题。由这笔拨款资助的研究有两种模式：将代数几何的当前发展与弦理论所模拟的粒子物理的基本问题相关联，以及将物理思想和直觉与当前几何中的问题相关联。将要研究的问题是当前数学和物理领域的紧迫兴趣。正在开发中的数学技术可以在越来越逼真的弦模型中准确计算粒子的预测质量，并更好地理解自然界中观察到的力和相互作用是如何从包括量子引力的大统一理论中产生的。弦理论的思想提供了关键的见解，有望直接导致三维代数几何中几个悬而未决的问题的解决。研究生将接受几何和物理及其相互作用方面的培训。该奖项由代数和数论以及几何分析课程共同资助。

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.