Logarithmic Gauged Linear Sigma Models

在过去的几十年里，纯数学和物理学之间的联系已经发展起来。弦理论是理论物理学中与基本粒子结构有关的分支，而计数几何学研究几何问题的解的个数（例如，有多少条线穿过空间中的四条固定线）。本计画将发展一种新的技术，其灵感来自于物理学中的规范线性sigma模型（GLSM），以解决数个计数问题。更详细地说，本计画将发展一种新的技术（log GLSM）在更高亏格的Gromov-Witten理论中。目标是从物理学的角度证明一些命题，如五次三重和更一般的完全相交的Calabi-Yau三重的全纯异常方程。为此，将构造一个新的模空间，其局部化公式给出了从其周围空间的Gromov-Witten不变量计算凸完全交的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。

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.