Higher genus categorical Gromov-Witten invariants

高属分类 Gromov-Witten 不变量

基本信息

批准号：
1811925
负责人：
Andrei Caldararu
金额：
$ 24.99万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811925&HistoricalAwards=false
关键词：
Higher genus categorical Gromov Witten

项目摘要

This project concerns research in algebraic geometry, the geometric study of solutions of polynomial equations. This subject has seen major developments in the last few years. One of the most striking is the invention of modern invariants counting the number of curves with certain properties in spaces that are expected to be related to our current universe's space-time. A major breakthrough came from understanding that the computation of some of these invariants can be understood in terms of solutions of certain equations related to the geometry of so-called mirror spaces. While the initial description of these invariants was geometric in nature, work of Kontsevich and Costello suggests that an algebraic approach would lead to increased flexibility and more efficient computation. The goal of this project is to expand that work and define the invariants in such a way that existing results on the Homological Mirror Symmetry conjecture will automatically imply that existing numerical predictions on curve-counts are correct. Kevin Costello introduced in 2005 a categorical generalization of Gromov-Witten invariants, defined for all genera. Despite considerable interest, very little is known about these invariants. The first calculation of them, for the universal family of elliptic curves, was achieved by the PI in 2017, in joint work with Junwu Tu. In this project the PI expand the current understanding of these invariants. The PI will complete ideas originally proposed by K. Costello, and then use the resulting theory to compute B-model Gromov-Witten invariants of positive genus for higher dimensional varieties, including the quintic threefold. This will verify the validity of the mirror symmetry predictions in higher genus. The main approach will be to replace the geometric spaces with algebraic structures called categories of matrix factorizations. This is of independent interest in itself: the invariants of these categories should be closely related to the Fan-Jarvis-Ruan-Witten invariants, but such a relationship is not explicitly known.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.

该项目涉及代数几何研究，即多项式方程解的几何研究。这个主题在过去几年中取得了重大发展。最引人注目的之一是现代不变量的发明，它计算了空间中具有某些属性的曲线的数量，这些曲线预计与我们当前宇宙的时空相关。一个重大突破来自于理解，其中一些不变量的计算可以用与所谓的镜像空间几何相关的某些方程的解来理解。虽然这些不变量的最初描述本质上是几何的，但 Kontsevich 和 Costello 的工作表明，代数方法将带来更大的灵活性和更高效的计算。该项目的目标是扩展这项工作并定义不变量，使得同调镜像对称猜想的现有结果将自动暗示现有的曲线计数数值预测是正确的。 Kevin Costello 在 2005 年引入了为所有属定义的 Gromov-Witten 不变量的分类概括。尽管人们对此很感兴趣，但人们对这些不变量知之甚少。 2017 年，PI 与屠俊武共同完成了针对椭圆曲线万能族的首次计算。在这个项目中，PI 扩展了当前对这些不变量的理解。 PI 将完成 K. Costello 最初提出的想法，然后使用所得理论计算高维簇（包括五次三重）的正属的 B 模型 Gromov-Witten 不变量。这将验证更高属中镜像对称预测的有效性。主要方法是将几何空间替换为称为矩阵分解类别的代数结构。这本身就具有独立的利益：这些类别的不变量应该与 Fan-Jarvis-Ruan-Witten 不变量密切相关，但这种关系尚不清楚。该奖项反映了 NSF 的法定使命，并被认为值得支持通过使用基金会的智力优点和更广泛的影响审查标准进行评估。