Elliptic Fibrations and Applications to Particle Physics

摘要奖：DMS-1007414首席研究员：大卫·R·莫里森弦理论为粒子物理理论模型提供了一致的高能外推，并自然地将这些模型与重力统一起来。在过去的两年里，一种被称为“F理论”的粒子物理弦理论模型重新引起了人们的兴趣。这项由这笔拨款资助的研究将调查F理论模型(在数学界被称为“椭圆纤颤”)的数学基础，目的是回答一些关键的数学问题，这些问题将澄清模型的物理性质。目标是确定F理论结构可以在多大程度上被局部研究；确定局部Picard群的数学结构如何允许物理理论中的超电荷规范对称性在低能量下存活；以及研究可能影响F理论模型数量的椭圆纤颤的各个具体方面(余维三现象，正则丛公式)。粒子物理学家已经使用了各种复杂的数学工具来帮助在亚原子水平上解释我们世界的行为和结构。为了准备即将从欧洲核子研究组织(CERN)的大型强子对撞机上获得的数据，理论粒子物理学家一直在改进他们的预测；在某些情况下，只有在数学工具本身得到改进的情况下，才能进行这些改进。由这笔赠款资助的研究旨在改进目前正在使用的重要数学工具之一，这是一种被称为“椭圆纤维”的工具，它来自对多项式方程的解的几何研究，多项式方程的解是代数几何数学领域的一部分。理论粒子物理学家最近对这一工具的使用将注意力集中在它需要改进的一些领域，以便能够对粒子物理进行必要的计算；这项研究将做出这些改进。

AbstractAward: DMS-1007414Principal Investigator: David R. MorrisonString theory provides consistent high-energy extrapolations of theoretical models of particle physics, as well as naturally unifying those models with gravity. One promising type of string theory model for particle physics known as "F-theory" has seen a resurgence of interest in the past two years. The research funded by this grant will investigate the mathematical underpinnings of F-theory models (which are known as "elliptic fibrations" in the mathematical world), with the goal of answering some key mathematical questions which will clarify the physical properties of the models. The goals are the determination of the extent to which the F-theory constructions can be studied locally; determining how the mathematical structure of the local Picard group allows the hypercharge gauge symmetry in the physical theory to survive to low energy; and studying various specific aspects of elliptic fibrations (codimension three phenomena, canonical bundle formulae) which may affect how numerous the F-theory models are.Particle physicists have employed a wide variety of sophisticated mathematical tools to help explain the behavior and structure of our world at a subatomic level. In preparation for the data which will soon be available from the Large Hadron Collider at the European Organization for Nuclear Research (CERN), theoretical particle physicists have been refining their predictions; in some cases, these refinements can only be made if the mathematical tools themselves are improved. The research funded by this grant aims to improve one of the important mathematical tools currently being used, a tool known as ``elliptic fibrations'' which comes from the geometric study of solutions of polynomial equations, a part of the mathematical field of algebraic geometry. Recent uses of this tool by theoretical particle physicists have focussed attention on some areas where it needs improvement in order to be able to make the necessarily calculations for particle physics; this research will make those improvements.