Applications of Equivariant Lifts in Algebraic and Symplectic Geometry

等变升力在代数和辛几何中的应用

• 批准号: 1510518
• 负责人: Eduardo Gonzalez
• 金额: $ 13.05万
• 依托单位: University of Massachusetts Boston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1510518&HistoricalAwards=false
• 关键词: Applications; Equivariant; Lifts; Algebraic; Symplectic

This is a project in symplectic geometry, a branch of pure mathematics. It focuses on Gromov-Witten theory. Gromov-Witten invariants are numbers that appropriately count curves in spaces satisfying certain conditions; just as the fact that there is exactly one straight line passing through any two given distinct points. These invariants have been a central object of study in several fields, from their origins in theoretical physics (string theory) to algebraic and symplectic geometries. Computing such numbers for general spaces is a difficult task. For spaces with symmetries, gauged Gromov Witten theory has proven to be a far-reaching tool used to systematically understand the behavior of Gromov-Witten invariants as well as to compute them. One of the major accomplishments of gauged Gromov-Witten theory is the discovery of several explicit formulas for the invariants, which have been recently used to tackle open questions. This research project aims to advance understanding in this area. Due to the multidisciplinary nature of this field, new results will find direct applications in other areas.One of the major accomplishments of gauged Gromov-Witten theory is the discovery of wall-crossing formulas, which in turn proved to be decisive in the proof of the so-called crepant conjecture. The present project undertakes work to further advance the research program in gauged Gromov-Witten theory for algebraic and symplectic quotients. The Principal Investigator and collaborators will work on an explicit computation of quantum K-theory of toric varieties and Grassmannians by means of the K-theoretic quantum Kirwan map, and extend wall crossing formulas to quantum K-theory. The Principal Investigator and collaborators will also study the moduli of parabolic gauged maps and the gauged Gromov-Witten theory for hypersurfaces. They will investigate relations to other constructions such as stable maps with p-fields and the Landau-Ginzburg A-model. The Principal Investigator will continue his joint work on a degeneration formula for Seidel elements and its relation to mirror symmetry and potential function of disc counting. He will work on a construction of invariants for toric orbifolds generalizing a Calabi quasi-morphism for projective spaces, and plans to use this to study non-displaceability. Finally, he will investigate the possibility of extending other equivariant tools into quantum cohomology and quantum K-theory.

这是辛几何的一个项目，辛几何是纯数学的一个分支。本文主要介绍Gromov-Witten理论。 Gromov-Witten不变量是在满足某些条件的空间中适当计数曲线的数;正如恰好有一条直线通过任何两个给定的不同点。这些不变量一直是多个领域研究的中心对象，从理论物理学（弦理论）到代数几何和辛几何。 在一般空间中计算这样的数是一项困难的任务。对于具有对称性的空间，规范Gromov维滕理论已被证明是一个意义深远的工具，用于系统地理解Gromov-维滕不变量的行为以及计算它们。规范Gromov-Witten理论的主要成就之一是发现了几个不变量的显式公式，这些公式最近被用来解决开放问题。 该研究项目旨在促进对这一领域的了解。 由于该领域的多学科性质，新的结果将在其他领域找到直接的应用。规范Gromov-Witten理论的主要成就之一是发现了跨壁公式，这反过来又被证明是决定性的，在证明所谓的crepant猜想。本项目的工作是进一步推进规范Gromov-Witten理论的代数和辛代数的研究计划。主要研究者和合作者将通过K理论量子Kirwan映射对复曲面簇和Grassmannian的量子K理论进行显式计算，并将跨壁公式扩展到量子K理论。 首席研究员和合作者还将研究抛物规范映射的模和超曲面的规范Gromov-Witten理论。他们将研究与其他结构的关系，如p场稳定映射和Landau-Ginzburg A模型。首席研究员将继续他的合作工作的退化公式赛德尔元素及其关系镜像对称性和潜在的功能光盘计数。他将致力于建设不变量的环面orbifolds推广卡拉比准态射的射影空间，并计划利用这一点来研究non-performability。最后，他将研究将其他等变工具扩展到量子上同调和量子K理论的可能性。