Some problems in algebraic geometry with connections to string theory

在弦理论的启发下，提出了一系列代数几何问题。 并提出了弦论中与代数几何有关的一些问题。 其中包括与拓扑弦振幅相关的代数几何不变量的研究：Gromov-Witten，Gopakumar-Vafa和Donaldson-Thomas不变量，它们都是相互关联的。 特别的问题包括Gopakumar-Vafa不变量的严格定义和它们从Calabi-Yau三重到更一般的三重的推广，它们在例子中的计算，以及GW-GV对应的验证，从环面簇和收缩曲线开始。 这些不变量将被推广到非Calabi-Yau的三重不变量。 提出了Gromov-Witten和Donaldson-Thomas不变量的计算方法。 本课程将探讨与物理学的几种联系，包括黑洞熵技术。该项目是物理学中的弦理论和数学中的几何学之间令人兴奋的交叉过程的延续。 在弦理论中，自然界的基本粒子是振动的小弦，物理性质取决于弦传播空间的几何形状，就像鼓的形状影响它发出的声音一样。 物理上相关的性质可以通过纯几何的计算来确定，在这种情况下，计算是数学中的内在兴趣。 当对偶性的物理方法提供了一个以上的数学模型时，其结果是对几何学的一种意想不到的深刻洞察。 在这个项目中，PI将开发工具来调查这些预测的数学有效性，并将几何调查扩展到物理相关的背景之外。 PI也将应用几何技术来推进对弦理论的理解，特别是当它描述黑洞的统计力学时。主要的数学内容是在称为卡-丘三重的弯曲空间中计算曲线，这些曲线形成了弦理论中的小额外维度。

A series of problems are proposed in algebraic geometry, inspired bystring theory. Some problems in string theory relating to algebraicgeometry are also proposed. These include the study ofalgebro-geometric invariants related to topological string amplitudes:Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants, allconjecturally related to each other. Particular problems include arigorous definition of Gopakumar-Vafa invariants and theirgeneralization from Calabi-Yau threefolds to more general threefolds,their computation in examples, and verification of the GW-GVcorrespondence, beginning with toric varieties and contractiblecurves. These invariants will be extended to threefolds which are notCalabi-Yau. Computational methods for Gromov-Witten andDonaldson-Thomas invariants are proposed. Several connections tophysics will be pursued, including black hole entropy techniques.The project is a continuation of the exciting process ofcross-fertilization between string theory in physics and geometry inmathematics. In string theory, the fundamental particles of natureare small vibrating strings, and physical properties depend on thegeometry of the space that strings propagate in, much as the shape ofa drum affects the sound that it makes. Physically relevantproperties can be determined by calculations in pure geometry, in thisinstance calculations of intrinsic interest in mathematics. Whenphysical methods of duality provide more than one mathematical model,a consequence is a deep unexpected insight into geometry. In this project,the PI will develop tools to investigate the mathematical validity ofsome of these predictions and will expand the geometric investigationbeyond the physically relevant context. The PI will apply alsogeometric techniques to advance understanding of string theory,especially as it describes the statistical mechanics of a black hole.The primary mathematical content is the counting of curves in curvedspaces called Calabi-Yau threefolds which form the small extradimensions in string theory.