Logarithmic Gauged Linear Sigma Models

在过去的几十年里，纯数学和物理学之间发展了深刻的联系。其中一个卓有成效的互动领域是弦理论和枚举几何之间的互动。弦理论是与基本粒子结构有关的理论物理学的一个分支，而枚举几何研究几何问题的解的数量(例如，有多少条线穿过空间中的四条固定线)。这个项目将开发一种新的技术，其灵感来自物理学中的规范线性西格玛模型(GLSM)，以解决几个计数问题。更详细地，该项目将开发更高亏格Gromov-Witten理论的新技术(LOG GLSM)。其目的是证明物理猜想，如五次三重全纯反常方程和更一般的完全交Calabi-Yau三重全纯反常方程。为此，构造了一个新的模空间，它的局部化公式给出了一种计算凸完全交与其周围空间的Gromov-Witten不变量的方法，以及新的“有效不变量”。不同的应用程序将利用本地化公式的结构以及有效不变量之间的对应关系。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Over the past few decades, deep connections between pure mathematics and physics have developed. One such area of fruitful interaction is between string theory, a branch of theoretical physics connected with the structure of elementary particles, and enumerative geometry, which studies counts of the number of solutions to geometric problems (how many lines, for example, pass through four fixed lines in space). This project will develop a new technique, which is inspired from the gauged linear sigma model (GLSM) in physics, to solve several enumerative problems.In more detail, the project will develop a new technique (log GLSM) in higher genus Gromov-Witten theory. The goal is to prove conjectures from physics such as the holomorphic anomaly equations for quintic threefolds and more general complete intersection Calabi-Yau 3-folds. For this, a new moduli space will be constructed whose localization formula gives a way of computing Gromov-Witten invariants of a convex complete intersection from those of its ambient space, and new "effective invariants". Different applications will exploit the structure of the localization formula, and correspondences between effective invariants.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.