Classification of Lagrangian fibrations

拉格朗日纤维的分类

• 批准号: 1206309
• 负责人: Justin Sawon
• 金额: $ 15.06万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1206309&HistoricalAwards=false
• 关键词: Classification; Lagrangian; fibrations

Hyperkahler manifolds arise in many different areas of high energy theoretical physics, for example as spaces parametrizing Yang-Mills instantons, magnetic monopoles, and Higgs fields. In their algebro-geometric manifestation, hyperkahler manifolds appear as holomorphic symplectic manifolds. Fibrations on holomorphic symplectic manifolds are known as Lagrangian fibrations, since the fibers must be Lagrangian with respect to the holomorphic symplectic structure. In addition, the generic fibres must be abelian varieties, and Lagrangian fibrations may be viewed as higher-dimensional analogues of elliptic K3 surfaces. The Hitchin system is a celebrated non-compact example. In this project the P.I. will study the classification of (compact) Lagrangian fibrations. The goals are to classify Lagrangian fibrations in some particular cases: when the fibers are Jacobians of curves or principally polarized, when the fibration is locally isotrivial, in dimension four (abelian surfaces as fibers). The P.I. will also explore new constructions of Lagrangian fibrations, and address the question of finiteness of deformation classes.This project will develop some particular kinds of geometry that are known to arise in theoretical physics, particularly in quantum field theory and string theory. The main focus will be on hyperkahler geometry, a class of spaces admitting Ricci-flat metrics, i.e., solutions to Einstein's equations in a vacuum. The P.I. will also study generalized complex geometry, which was invented by mathematicians to bridge complex and symplectic geometry, on which type IIA and IIB string theory are based, and to help unravel the Mirror Symmetry phenomenon predicted by string theorists. In short, these are all innovative new kinds of geometry developed largely in response to the demands of theoretical physicists, but also of inherent mathematical interest. The P.I. will actively promote the exchange of ideas between mathematicians and physicists by organizing and participating in cross-disciplinary conferences and workshops, and is also involved in the training of graduate students in these topics.

超卡勒流形出现在高能理论物理的许多不同领域，例如杨-米尔斯瞬子、磁单极子和希格斯场的空间参数化。在它们的代数几何表现中，超卡勒流形表现为全纯辛流形。全纯辛流形上的纤维化称为拉格朗日纤维化，因为纤维必须是关于全纯辛结构的拉格朗日纤维化。此外，一般的纤维必须是阿贝尔变种，拉格朗日纤维化可以被看作是椭圆K3曲面的高维类似物。希钦系统是一个著名的非紧性例子。在这个项目中，PI。将研究（紧）拉格朗日纤维化的分类。目标是对某些特定情况下的拉格朗日纤维化进行分类：当纤维是曲线的雅可比量或主要极化时，当纤维化是局部等平凡时，在四维（阿贝尔表面作为纤维）。私家侦探也将探索拉格朗日纤维化的新结构，并解决变形类的有限性问题。这个项目将发展一些已知在理论物理中出现的特殊几何，特别是在量子场论和弦理论中。主要重点将是超卡勒几何，一类空间承认里奇平坦度量，即，爱因斯坦方程的解私家侦探还将研究广义复几何，这是由数学家发明的桥梁复杂和辛几何，其中IIA型和IIB弦理论的基础上，并帮助解开弦理论家预测的镜像对称现象。简而言之，这些都是创新的新几何类型，主要是为了响应理论物理学家的需求而发展起来的，但也有内在的数学兴趣。私家侦探将通过组织和参加跨学科会议和讲习班，积极促进数学家和物理学家之间的思想交流，并参与培训这些专题的研究生。