FRG: Collaborative Research: Crossing the Walls in Enumerative Geometry

FRG：协作研究：跨越枚举几何的墙壁

• 批准号: 1564497
• 负责人: Andrei Okounkov
• 金额: $ 23.4万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2022-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1564497&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Crossing; Walls

This project concerns the field of algebraic geometry, a branch of mathematics studying the geometric structure of solutions of polynomial equations. Many of the questions under study in this project are motivated by string theory, a branch of theoretical physics connected with the structure of elementary particles. This project aims to significantly enhance the intensive and fruitful interaction between cutting edge research in enumerative algebraic geometry and theoretical physics. The research aims to extend mathematical developments that verify and generalize conjectures originating from physics, and the work is expected to significantly impact development of the physical theory as well. Through conferences, a summer school, seminars, and research involvement, this project provides unique opportunities for a new generation of mathematicians to obtain the interdisciplinary knowledge and skills needed to work in this exciting research area.The aim of the project is to study enumerative invariants in the broad sense and their dependence on various stability conditions, as well as dualities relating different enumerative invariants. The investigators plan to further develop the theory of Gauged Linear Sigma Models (GLSM) and will study the epsilon-wall-crossing conjecture and zeta-wall-crossing conjecture at all genera; Gromov-Witten (GW) and quasimap invariants are related by a sequence of epsilon-wall-crossing, whereas the Calabi-Yau/Landau-Ginzburg correspondence (relating GW invariants and FJRW invariants) and Pfaffian/Grassmannian correspondence are examples of zeta-wall-crossing. The investigators are developing the theory of Mixed-Spin-P (MSP) fields, to interpolate GW theory of quintic threefolds and FJRW theory of Fermat quintic polynomials, and to study algebraic structures of higher genus GW and FJRW invariants. The new theories of GLSM and MSP fields will provide new tools to attack the central and longstanding problem of computing higher genus GW invariants of compact Calabi-Yau threefolds. The investigators have been investigating K-theoretic Donaldson-Thomas invariants of threefolds, as well as GW and quasimap invariants of Nakajima quiver varieties. Because some of conjectures motivated by theoretical physics can only be properly formulated in terms of K-theoretic enumerative invariants, they plan to study dualities relating K-theoretic enumerative invariants of different geometries, and to lift results on traditional enumerative invariants to the K-theoretic setting.

该项目涉及代数几何领域，代数几何是研究多项式方程解的几何结构的数学分支。该项目研究的许多问题都是由弦理论激发的，弦理论是与基本粒子结构相关的理论物理学的一个分支。该项目旨在显着加强枚举代数几何和理论物理领域前沿研究之间的密集和富有成效的互动。该研究旨在扩展验证和概括源自物理学的猜想的数学发展，预计这项工作也将对物理理论的发展产生重大影响。通过会议、暑期学校、研讨会和研究参与，该项目为新一代数学家提供了独特的机会，以获得在这个令人兴奋的研究领域工作所需的跨学科知识和技能。该项目的目的是研究广义上的枚举不变量及其对各种稳定性条件的依赖性，以及与不同枚举不变量相关的对偶性。研究人员计划进一步发展测量线性西格玛模型（GLSM）理论，并将研究所有属的epsilon穿墙猜想和zeta穿墙猜想； Gromov-Witten (GW) 和准图不变量通过一系列 epsilon 穿墙相关，而 Calabi-Yau/Landau-Ginzburg 对应（涉及 GW 不变量和 FJRW 不变量）和 Pfaffian/Grassmannian 对应是 zeta 穿墙的示例。研究人员正在发展混合自旋-P (MSP) 场理论，对五次三重的 GW 理论和费马五次多项式的 FJRW 理论进行插值，并研究更高属 GW 和 FJRW 不变量的代数结构。 GLSM 和 MSP 场的新理论将提供新的工具来解决计算紧凑 Calabi-Yau 三重的更高属 GW 不变量的核心和长期存在的问题。研究人员一直在研究三倍的 K 理论唐纳森-托马斯不变量，以及中岛箭袋变种的 GW 和准映射不变量。由于一些由理论物理引发的猜想只能用 K 理论枚举不变量来正确表述，因此他们计划研究与不同几何的 K 理论枚举不变量相关的对偶性，并将传统枚举不变量的结果提升到 K 理论环境。