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.

代数簇是一类通过将多项式方程的解集合粘合在一起而得到的几何对象。在理论物理学的一个分支弦理论中，代数簇被用来描述我们宇宙的细微部分。在这个理论中，一切都是由微小的弦组成的，弦穿过时空，在某些代数簇中描绘出代数曲线。Gromov-Witten不变量源于物理学，是满足给定关联约束的代数簇中代数曲线的虚计数。它们被用来描述我们宇宙的结构。他们还提供了新的方法和见解，从代数几何的经典问题。尽管它们很重要，但这些不变量很难计算。该项目的主要目标是开发一种新的方法来计算Gromov-Witten不变量，通过使用代数几何中的对数结构工具从物理学角度研究规范线性sigma模型的边界。本计画提供研究训练的机会给研究生。更详细地说，本计画的重点是使用Abramovich-Chen-Gross-Siebert的稳定对数图来研究规范线性sigma模型（GLSM）的几何。维滕在20世纪90年代提出的广义线性空间模型可以看作是Gromov-维滕不变量超平面性质在所有亏格中的一个深入推广。然而，GLSM中的模栈，携带的理想障碍理论定义GLSM不变量一般是不适当的。这是计算GLSM不变量的一个主要困难。最近，Chen，Janda和Ruan利用稳定的log映射构造了混合型GLSM的log紧化。这些紧化提供了适当的模栈携带一个简化的完美障碍理论，其相关的虚拟循环恢复GLSM虚拟循环。该项目是一个综合性的研究，旨在通过调查这些日志紧致化的虚拟循环的结构，计算Gromov-Witten不变量的新的计算方法。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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