FRG: Collaborative Research: Crossing the Walls in Enumerative Geometry

这个项目涉及代数几何领域，这是数学的一个分支，研究多项式方程解的几何结构。在这个项目中，许多正在研究的问题都是由弦理论激发的，弦理论是理论物理学中与基本粒子结构有关的分支。该项目旨在显著增强枚举代数几何和理论物理前沿研究之间的密集和富有成效的互动。该研究旨在扩展数学发展，以验证和推广源自物理学的理论，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，该项目提供了独特的机会，为新一代的数学家，以获得跨学科的知识和技能，需要在这个令人兴奋的研究领域工作。该项目的目的是研究枚举不变量在广义上和它们的依赖于各种稳定性条件，以及对偶相关的不同枚举不变量。研究人员计划进一步发展规范线性西格玛模型（GLSM）的理论，并将在所有属中研究ε-壁穿越猜想和ζ-壁穿越猜想; Gromov-Witten（GW）和拟映射不变量通过ε-壁交叉序列相关联，而卡拉比-丘/朗道-金兹伯格对应（与GW不变量和FJRW不变量相关）和普夫兰/格拉斯曼对应是zeta壁交叉的例子。研究者们正在发展混合自旋P（MSP）场的理论，插值五次三重的GW理论和费马五次多项式的FJRW理论，研究更高亏格的GW和FJRW不变量的代数结构。GLSM和MSP域的新理论将为求解紧致Calabi-Yau三重域的高亏格GW不变量提供新的工具。研究人员一直在研究三倍的K理论Donaldson-Thomas不变量，以及Nakajima的GW和准映射不变量。由于理论物理学中的一些命题只能用K-理论枚举不变量来表示，他们计划研究不同几何的K-理论枚举不变量的对偶性，并将传统枚举不变量的结果提升到K-理论设置。

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.