Holomorphic G-bundles On Elliptic Fibrations

摘要DMS-0200810这个项目的一个动机来自F理论/杂种严格二元性。在这个对偶中，给定一个椭圆纤维的Calabi-Yau流形的数据，它上的一个主丛具有适当的结构群和连接，以及一个大致类似于全纯2-形式的B-场，人们期望找到一个新的Calabi-Yau流形，它在相同的基上纤维在椭圆K3曲面上。第二个动机是在纯粹的数学框架下理解第一个问题提出的一些数学问题。这些问题导致了关于紧李群中两个或多个交换元素、半单代数或复李代数中的共轭类、椭圆曲线上半稳定主丛的模空间、Del Pezzo曲面和K3曲面的几何等一系列丰富的问题。这项研究的目标之一是从数学上理解由弦理论产生的两种基本粒子物理理论之间的关系。这种关系最好用几个几何结构之间的关系来数学表达。从数学的观点来看，这些关系和各种推广也是非常有趣的，因为它们与任何维度的空间的可能对称以及与这些对称相关联的几何结构有关。

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.