Quasimap Theory and Gromov-Witten Invariants of Complete Intersections

拟映射理论和完全交集的 Gromov-Witten 不变量

• 批准号: 1601771
• 负责人: Ionut Ciocan-Fontanine
• 金额: $ 16.75万
• 依托单位: University of Minnesota-Twin Cities
• 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: 2016-08-15 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601771&HistoricalAwards=false
• 关键词: Quasimap; Theory; Gromov; Witten; Invariants

This research is in the field of algebraic geometry, an old and highly developed branch of mathematics, which at its core is the study of geometric shapes defined by polynomial equations. Moduli theory is concerned with how these shapes deform when parameters are varied in a continuous fashion, and compactified moduli spaces describe in particular the kind of degenerate limiting shapes that may appear when deformations are performed. Various geometric properties of interest do not change under deformations and are often easier to analyze if limits adequate for the problem at hand are allowed. The "wall-crossing phenomenon" refers loosely to changing the compactified moduli spaces by disallowing certain limiting shapes and replacing them with different ones. The compactified moduli spaces studied in this project have deep connections with the mirror symmetry phenomenon discovered in string theory, a very active area of theoretical physics. In the last two decades, the results and techniques from algebraic geometry, especially the theory of moduli spaces, have been successfully employed in string theory. On the other hand, ideas from string theory have opened up new directions of research in mathematics by suggesting striking conjectures and at the same time putting old unsolved problems into a new light. This project will continue this fruitful interaction by offering new insights on mirror symmetry at higher genus, via the study of wall-crossing between moduli spaces.This project aims to continue the investigator's study of compactifications of moduli spaces of maps from curves to a large class of GIT quotient targets. These compactifications, called moduli spaces of stable quasimaps, produce new curve-counting invariants, which should be related to Gromov-Witten invariants by wall-crossing formulas. Indeed, such wall-crossing formulas in genus zero were established by the PI with Kim in recent years, and they turn out to provide significant generalizations of Givental's toric mirror theorems. One of the main goals of this project is to vastly extend the wall-crossing formulas by establishing them in higher genus and at the level of virtual classes for many compact targets. These formulas will then have many consequences which will be investigated. An important application is to the Mirror Conjecture at higher genus for complete intersection Calabi-Yau varieties, such as the quintic threefold. In this case, the wall-crossing formula may be viewed as giving a mathematically rigorous interpretation of the physicist's "`holomorphic limit of the B-model partition function" as the generating function for quasimap invariants. Further applications and generalizations that emerge from the study of wall-crossing relate to the so-called Landau-Ginzburg/Calabi-Yau correspondence, and more generally to the new theory of the gauged linear sigma model of Fan-Jarvis-Ruan in higher genus.

这项研究属于代数几何领域，代数几何是数学的一个古老而高度发展的分支，其核心是研究由多项式方程定义的几何形状。模理论关注的是当参数以连续的方式变化时，这些形状是如何变形的，而紧化模空间特别描述了当变形发生时可能出现的退化极限形状。各种感兴趣的几何性质在变形下不会改变，如果允许对手头的问题有足够的限制，通常更容易分析。“过墙现象”是指通过不允许某些极限形状而用不同的极限形状来改变紧化模空间。本项目研究的紧化模空间与弦理论中发现的镜像对称现象有着深刻的联系，弦理论是理论物理中一个非常活跃的领域。近二十年来，代数几何的成果和技术，特别是模空间理论，已经成功地应用于弦理论。另一方面，弦理论的思想通过提出惊人的猜想，开辟了数学研究的新方向，同时使旧的未解决的问题有了新的认识。该项目将通过研究模空间之间的壁交叉，为更高属的镜像对称提供新的见解，从而继续这种富有成效的互动。本项目旨在继续研究者从曲线映射到一大类GIT商目标的模空间的紧化研究。这些紧化被称为稳定拟映射的模空间，产生了新的曲线计数不变量，这些不变量应该通过壁交公式与Gromov-Witten不变量联系起来。事实上，近年来PI和Kim建立了这种零属的过墙公式，它们提供了Givental的环面镜像定理的重要推广。该项目的主要目标之一是通过在许多紧凑目标的更高属和虚拟类级别上建立它们来极大地扩展过墙公式。然后，这些公式将产生许多结果，这些结果将被研究。对于完全交的Calabi-Yau变种，如五次三重，在高属处的镜像猜想是一个重要的应用。在这种情况下，隔墙公式可以被看作是对物理学家“b模型配分函数的全纯极限”作为准映射不变量的生成函数给出了数学上严格的解释。从wall-crossing研究中产生的进一步应用和推广涉及到所谓的Landau-Ginzburg/Calabi-Yau对应关系，以及更普遍的van - jarvis -阮氏高等属的测量线性sigma模型的新理论。