Logarithmic Geometry and the Gauged Linear Sigma Model

对数几何和测量线性西格玛模型

• 批准号: 2001089
• 负责人: Qile Chen
• 金额: $ 18万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001089&HistoricalAwards=false
• 关键词: Logarithmic; Geometry; Gauged; Linear; Sigma

Algebraic varieties are a class of geometric objects obtained by gluing together sets of solutions of polynomial equations. In string theory, a branch of theoretical physics, algebraic varieties are used to describe fine pieces of our universe. In this theory, everything is made of tiny strings which travel through spacetime, and trace out algebraic curves in some algebraic varieties. Gromov-Witten invariants, originating from physics, are virtual counts of algebraic curves in algebraic varieties satisfying prescribed incidence constraints. They are used in physic to describe the structures of our universe. They also provide new approaches and insights to classical problems from algebraic geometry. Despite their importance, these invariants are very difficult to compute. The primary goal of this project is to develop a new method to calculate Gromov-Witten invariants by investigating the boundary of the gauged linear sigma model from physics using tools of logarithmic structures from algebraic geometry. This project provides research training opportunities for graduate students.In more detail, this project focuses on studying the geometry of the gauged linear sigma model (GLSM) using stable log maps of Abramovich-Chen-Gross-Siebert. The GLSM proposed by Witten in the 1990s can be viewed as a deep generalization of the hyper-plane property of Gromov-Witten invariants in all genus. However, the moduli stacks in GLSM which carry the perfect obstruction theory for defining GLSM invariants are in general non-proper. This presents a major difficulty in calculating GLSM invariants. Recently, log compactifications of hybrid-type GLSM were constructed by Chen, Janda, and Ruan using stable log maps. These compactifications provide proper moduli stacks carrying a reduced perfect obstruction theory whose associated virtual cycles recover the GLSM virtual cycles. This project is an integrated study aiming at a new computational method for calculating Gromov-Witten invariants by investigating the structures of the virtual cycles of these log compactifications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

代数品种是通过将多项式方程溶液组粘合在一起而获得的一类几何对象。在弦理论中，理论物理学的一个分支，代数品种用于描述我们宇宙的细碎片。在这一理论中，一切都是由小弦制成的，这些弦穿过时空，并在某些代数品种中追踪代数曲线。源自物理学的Gromov-witten不变性是代数曲线的虚拟曲线计数，这些代数曲线满足规定的发生率约束。它们在物理中用于描述我们宇宙的结构。他们还提供了代数几何形状的经典问题的新方法和见解。尽管它们的重要性，但这些不变性很难计算。该项目的主要目标是通过使用代数几何形状的对数结构工具来研究物理学的测量线性Sigma模型的边界来开发一种新方法来计算Gromov-inten的不变性。该项目为研究生提供了研究培训机会。此项目，该项目着重于研究测量线性Sigma模型（GLSM）的几何形状，并使用Abramovich-Chen-Gross-Siebert的稳定日志图。 Witten在1990年代提出的GLSM可以被视为对所有属Gromov-witten不变性的超平面特性的深刻概括。但是，GLSM中具有完美阻塞理论的模量堆栈，用于定义GLSM不变性的构造。这在计算GLSM不变性方面存在一个主要的困难。最近，使用稳定的日志图构建了混合型GLSM的对数压实。这些压缩提供了具有减少完美阻塞理论的适当模量堆栈，其相关的虚拟周期恢复了GLSM虚拟循环。该项目是一项综合研究，旨在通过研究这些对数紧凑型的虚拟周期的结构来计算Gromov-witten不变性的新计算方法。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估来通过评估来获得支持的。

项目成果

The logarithmic gauged linear sigma model

• DOI: 10.1007/s00222-021-01044-2
• 发表时间: 2019-06
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Qile Chen;F. Janda;Y. Ruan

Virtual cycles of stable (quasi-)maps with fields

• DOI: 10.1016/j.aim.2021.107781
• 发表时间: 2019-11
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Qile Chen;F. Janda;Rachel Webb