Holomorphic G-bundles On Elliptic Fibrations

椭圆纤维上的全纯 G 丛

• 批准号: 0200810
• 负责人: Robert Friedman
• 金额: $ 22.84万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200810&HistoricalAwards=false
• 关键词: Holomorphic; bundles; Elliptic; Fibrations

ABSTRACT DMS - 0200810.One motivation for this project comes from F theory/heterotic stringduality. In this duality, given the data of an elliptically fiberedCalabi-Yau manifold, a principal bundle over it with the appropriatestructure group and connection, and a B-field, which is roughly like aholomorphic 2-form, one expects to find a new Calabi-Yau manifold fibered inelliptic K3 surfaces over the same base. A second motivation is tounderstand some of the mathematical issues raised by the first question, ina purely mathematical framework. These issues lead to a rich set ofquestions concerning two or more commuting elements in compact Lie groups,conjugacy classes in semisimple algebraic groups or complex Lie algebras,the moduli space of semistable principal bundles over elliptic curves, andthe geometry of del Pezzo surfaces and K3 surfaces.This project is concerned with the study of certain geometric structuresarising in mathematics and physics. One goal of this study is a mathematicalunderstanding of the relation between two physical theories of elementaryparticles arising from string theory. The relationship is best expressedmathematically by relationships between several geometric structures. Theserelationships, and various generalizations, are also very interesting from amathematical point of view, because they are connected with the possiblesymmetries of spaces in any number of dimensions and with geometricstructures connected with these symmetries.

本项目的一个动机来自F理论/杂种优势弦对偶性。在这种对偶中，给定椭圆纤维化的Calabi-Yau流形，其上具有适当结构群和联络的主丛，以及一个近似于全纯2-形式的B-域的数据，人们期望找到一个新的在同一基上纤维化的非椭圆K3曲面的Calabi-Yau流形。第二个动机是在纯粹的数学框架内理解第一个问题提出的一些数学问题。这些问题引出了关于紧李群中两个或多个交换元、半单代数群或复李代数中的共轭类、椭圆曲线上半稳定主丛的模空间以及del Pezzo曲面和K3曲面的几何等一系列丰富的问题。本研究的目标之一是对源自弦理论的基本粒子的两种物理理论之间的关系进行数学理解。这种关系在数学上最好用几种几何结构之间的关系来表达。从数学的观点来看，这些关系和各种推广也是非常有趣的，因为它们与任意维数空间的可能对称性以及与这些对称性相关的几何结构有关。