F-Theory and G-bundles over Elliptic Curves

椭圆曲线上的 F 理论和 G 丛

• 批准号: 9970437
• 负责人: Robert Friedman
• 金额: $ 9.27万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970437&HistoricalAwards=false
• 关键词: Theory; bundles; over; Elliptic; Curves

Proposal: DMS-9970437Principal Investigator: Robert D. FriedmanAbstract:This project involves the study of holomorphic G-bundles over an elliptic curve, where G is a reductive algebraic group, together with various generalizations. If G is the complexification of a compact group K, a closely related question is the classification of two commuting elements in K up to simultaneous conjugation. In joint work with J. Morgan and E. Witten, Friedman has classified holomorphic G-bundles over an elliptic curve. The relative situation for G-bundles over an elliptic fibration has also been studied. The next stage of the project is to work out the relation between these constructions, in the case where G is one of the exceptional groups E6, E7, E8, and corresponding families of K3 or del Pezzo surfaces. In terms of families of commuting elements in compact Lie groups, in joint work with A. Borel and J. Morgan, Friedman has given normal forms for commuting pairs and triples, and the method extends in principle to handle N mutually commuting elements. Compact Lie groups are the symmetry groups of nature, and the possible such groups have been classified and understood for one hundred years. Aside from four infinite series of such groups, there are also a small number of exceptional groups. Recently, such groups have become important in particle physics, because the possible subatomic particles and their interactions seem to be described by such symmetry groups. In particular string theory, which is an attempt to unify quantum mechanics with gravity, has suggested that the universe is 10-dimensional and that the symmetry group involved is one of the exceptional groups. Moreover, in some versions of string theory, the missing 6 dimensions of the universe are accounted for by a so-called elliptic fibration. Thus, the project attempts to study a part of mathematics which is relevant to string theory as well as being an interesting piece of mathematics in its own right.

提案：DMS-9970437主要研究者：Robert D. Friedman摘要：该项目涉及椭圆曲线上的全纯G-丛的研究，其中G是一个约化代数群，以及各种推广。如果G是紧群K的复化，一个密切相关的问题是K中两个交换元素的分类直到同时共轭。在与J.摩根和E.维滕和弗里德曼对椭圆曲线上的全纯G-丛进行了分类。本文还研究了椭圆纤维化上G-丛的相对情形。该项目的下一阶段是在G是例外群E6、E7、E8和相应的K3或del Pezzo曲面族之一的情况下，确定这些构造之间的关系。利用紧李群中的交换元族，与A. Borel和J. Morgan，Friedman给出了交换对和三元组的标准形，并且该方法在原则上扩展到处理N个相互交换的元素。 紧李群是自然界的对称群，一百年来，人们对可能的李群进行了分类和理解。除了四个无限系列的此类群之外，还存在少量的特殊群。最近，这种群在粒子物理学中变得很重要，因为可能的亚原子粒子和它们的相互作用似乎可以用这种对称群来描述。特别是试图统一量子力学和引力的弦理论，它提出宇宙是10维的，所涉及的对称群是例外群之一。此外，在某些版本的弦理论中，宇宙中缺失的6维可以用所谓的椭圆纤维化来解释。因此，该项目试图研究与弦理论相关的数学部分，同时也是一个有趣的数学部分。