FRG: Collaborative Research: Crossing the Walls in Enumerative Geometry

这个项目涉及代数几何领域，这是一个研究多项式方程解的几何结构的数学分支。这个项目中研究的许多问题都是由弦理论推动的，弦理论是与基本粒子结构有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的前沿研究和理论物理之间的密集和富有成效的互动。这项研究旨在扩展验证和推广源自物理学的猜想的数学发展，这项工作预计也将对物理理论的发展产生重大影响。通过会议、暑期班、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，以获得在这一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的枚举不变量及其对各种稳定性条件的依赖，以及与不同的枚举不变量相关的对偶。研究人员计划进一步发展规范线性西格玛模型(GLSM)的理论，并将在所有类别中研究ε墙交叉猜想和Zeta墙交叉猜想；Gromov-Witten(GW)和准映射不变量通过一系列ε墙交叉联系在一起，而Calabi-Yau/Landau-Ginzburg对应(相关的GW不变量和FJRW不变量)和Pfaffian/Grassmanian对应是Zeta壁交叉的例子。研究人员正在发展混合自旋P(MSP)场理论，插值五次三重GW理论和费马五次多项式的FJRW理论，并研究更高亏格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致Calabi-Yau三重数的高亏格GW不变量这一中心和长期存在的问题。研究人员一直在研究三重的K-理论Donaldson-Thomas不变量，以及中岛箭图簇的GW和拟映射不变量。由于一些受理论物理启发的猜想只能用K-理论计数不变量来恰当地表述，他们计划研究与不同几何的K-理论计数不变量有关的对偶，并将传统计数不变量的结果提升到K-理论环境中。

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.