Logarithmic Geometry and the Gauged Linear Sigma Model

代数变量是将多项式方程的一组解粘合在一起而得到的一类几何对象。弦理论是理论物理学的一个分支，在弦理论中，代数变量被用来描述宇宙的各个部分。在这个理论中，一切都是由微小的弦组成的，这些弦在时空中穿行，并在一些代数变体中勾勒出代数曲线。Gromov-Witten不变量起源于物理学，是满足规定关联约束的代数变量中代数曲线的虚计数。它们在物理学中被用来描述我们宇宙的结构。它们还为代数几何中的经典问题提供了新的方法和见解。尽管这些不变量很重要，但它们很难计算。本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来研究物理学中测量线性sigma模型的边界。本项目为研究生提供研究训练机会。更详细地说，该项目侧重于使用Abramovich-Chen-Gross-Siebert的稳定对数图研究测量线性西格玛模型（GLSM）的几何形状。Witten在20世纪90年代提出的GLSM可以看作是对所有格中的Gromov-Witten不变量的超平面性质的深度推广。然而，在GLSM中，采用完美阻碍理论来定义GLSM不变量的模栈通常是非固有的。这给计算GLSM不变量带来了很大的困难。近年来，Chen、Janda和阮等人利用稳定的测井图构造了混合型GLSM的测井紧化。这些紧化提供了适当的模栈，携带了一个简化的完全阻塞理论，其相关的虚环恢复了GLSM虚环。本项目是一项综合研究，旨在通过研究这些对数紧化的虚拟循环的结构来计算Gromov-Witten不变量的新计算方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.