F-Theory and G-bundles over Elliptic Curves

摘要：本课题主要研究椭圆曲线上的全纯G束，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中两个交换元的分类，直到同时共轭。在与J. Morgan和E. Witten的合作中，Friedman对椭圆曲线上的全纯g束进行了分类。本文还研究了椭圆型纤维上g束的相对情况。该项目的下一阶段是找出这些结构之间的关系，在这种情况下，G是特殊群E6， E7， E8之一，以及相应的K3或del Pezzo表面族。对于紧李群中的可交换元族，Friedman在与A. Borel和J. Morgan的联合工作中给出了可交换对和三元组的正规形式，并且该方法在原则上扩展到处理N个互交换元。紧李群是自然界的对称群，可能存在的群已经被分类和认识了一百年。除了这类群的四个无穷级数之外，还有少数例外群。最近，这样的群在粒子物理学中变得很重要，因为可能的亚原子粒子及其相互作用似乎可以用这样的对称群来描述。特别是试图将量子力学与引力统一起来的弦理论，认为宇宙是10维的，所涉及的对称群是一个特殊的群。此外，在弦理论的某些版本中，宇宙中缺失的6个维度可以用所谓的椭圆纤曲来解释。因此，该项目试图研究与弦理论相关的一部分数学，同时也是一个有趣的数学领域。

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.