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Moduli of vector bundles, Donaldson-Thomas theory and Gromov-Witten theory

矢量丛模、Donaldson-Thomas 理论和 Gromov-Witten 理论

基本信息

批准号：
0755520
负责人：
Zhenbo Qin
金额：
$ 11.97万
依托单位：
University of Missouri-Columbia
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2008
资助国家：
美国
起止时间：
2008-09-15 至 2013-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0755520&HistoricalAwards=false
关键词：
Moduli vector bundles Donaldson Thomas

项目摘要

The study of vector bundles is a fundamental problem in algebraic geometry. Their connections with physics and representation theory were pioneered in the work of Penrose and Atiyah. They played central roles in Donaldson theory and Seiberg-Witten theory. From 1995, physicists working in string theory have speculated many surprising but deep results concerning vector bundles, Gromov-Witten theory and their interplay with representation theory. In the past few years, elegant relations among moduli spaces of vector bundles (including Hilbert schemes of points and curves), Donaldson-Thomas theory and Gromov-Witten theory have been revealed.In this project, Professor Qin intends to study several problems concerning moduli of vector bundles, Donaldson-Thomas theory and Gromov-Witten theory in the general context of algebraic geometry and its interplay with representation theory and string theory. The main tools are localized virtual fundamental classes, techniques of vertex algebras, quantum cohomology, and stable bundles on surfaces and three folds.Algebraic geometry studies geometric objects described by polynomial equations. It has been at the central stage of recent confluence between mathematics and physics. Many of these interactions have led to profound improvement in the understanding of both mathematics and physics.Professor Qin's research helps to strength these interactions.

向量束的研究是代数几何中的一个基本问题。它们与物理学和表征理论的联系在彭罗斯和阿蒂亚的工作中被开创。它们在Donaldson理论和seberg - witten理论中发挥了核心作用。从1995年开始，从事弦理论研究的物理学家们就向量束、Gromov-Witten理论及其与表示理论的相互作用，推测出了许多令人惊讶但深刻的结果。在过去的几年里，向量束的模空间（包括点和曲线的Hilbert格式）、Donaldson-Thomas理论和Gromov-Witten理论之间的优雅关系被揭示出来。在本课题中，秦教授拟在代数几何的大背景下研究向量束的模、Donaldson-Thomas理论和Gromov-Witten理论及其与表示理论和弦理论的相互作用等几个问题。主要的工具有定域虚基类、顶点代数技术、量子上同调、曲面和三折上的稳定束。代数几何研究用多项式方程描述的几何对象。它一直处于近代数学与物理融合的中心阶段。许多这些相互作用导致了对数学和物理的理解的深刻改进。秦教授的研究有助于加强这些相互作用。