Quasimap Theory and Gromov-Witten Invariants of Complete Intersections

这项研究是在代数几何领域进行的，这是一个古老而高度发展的数学分支，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方式变化时，这些形状是如何变形的，紧凑的模空间特别描述了当执行变形时可能出现的退化极限形状的种类。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进行足够的限制，通常更容易分析。“穿墙现象”泛指通过不允许某些极限形状而代之以不同的极限形状来改变紧致的模空间。这个项目中研究的紧致模空间与弦理论中发现的镜像对称现象有很深的联系，弦理论是理论物理中非常活跃的一个领域。在过去的二十年里，代数几何的结果和技巧，特别是模空间理论，被成功地应用于弦理论。另一方面，弦理论的思想开辟了数学研究的新方向，它提出了引人注目的猜想，同时也让旧的悬而未决的问题有了新的认识。这个项目将通过研究模空间之间的墙交叉来继续这种卓有成效的相互作用，通过提供关于更高亏格上的镜像对称性的新见解。本项目旨在继续研究从曲线到一大类Git商目标的映射的模空间的紧致化。这些紧化被称为稳定拟映射的模空间，它产生了新的曲线计数不变量，这些不变量应该通过跨越墙的公式与Gromov-Witten不变量相联系。事实上，这种零亏格中的跨墙公式是由Pi和Kim在最近几年建立的，它们证明了Givental环镜定理的重要推广。这个项目的主要目标之一是通过在更高的亏格和虚拟类的水平上为许多紧凑的目标建立跨越墙公式来极大地扩展它们。这些公式将产生许多后果，我们将对此进行调查。一个重要的应用是关于完全交的Calabi-Yau簇的高亏格的Mirror猜想，如五次三重数。在这种情况下，穿墙公式可以被看作是对物理学家作为准映射不变量的生成函数的“B模型配分函数的全纯极限”给出了数学上的严格解释。墙交叉研究中出现的进一步应用和推广涉及到所谓的Landau-Ginzburg/Calabi-Yau对应，更广泛地涉及到更高亏格中的Fan-Jarvis-Ruan的规范线性西格玛模型的新理论。

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.