Wall-crossings in quasimap theory and applications

该项目的目的是扩大调查员的研究紧的模空间的映射曲线到一大类GIT商目标。这些紧化，称为模空间的稳定准映射，产生新的曲线计数不变量，这是预期相关的跨墙公式与Gromov-Witten不变量。在最近与Kim的合作中，PI在亏格零中发现了这种跨壁公式，并且它们给出了Givental复曲面镜像定理的重要推广。PI将努力在几个方向上扩展该理论。首先，过壁公式应推广到更高亏格的曲线，由此得到的Calabi-Yau三重曲线的理论在性质上与镜像Calabi-Yau的BCOV B模型理论是等价的，但没有通过镜像映射改变坐标。PI计划至少对局部卡-丘目标证明这一猜想。其他项目将提供扩展orbifold目标（应用于Crepant分辨率猜想），并Landau-Ginzburg模型（应用于Calabi-Yau/LG对应）。进一步的应用，如Gromov-Witten理论中的一般Abel/Non-Abel对应的证明也是预期的。这项研究是在代数几何领域，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。模量理论关注的是这些形状如何变形。研究者所研究的模空间与弦论中发现的镜像对称现象有着深刻的联系，弦论是理论物理学中一个非常活跃的领域。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，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假设，同时将未解决的老问题带入新的视野，开辟了数学研究的新方向。该项目将继续这一富有成效的互动，提供新的见解镜像对称在更高的属。这个奖项是共同资助的代数和数论和拓扑程序。

The project aims to expand on 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 are expected to be related by wall-crossing formulas with Gromov-Witten invariants. In recent work with Kim, the PI has found such wall-crossing formulas in genus zero and they turn out to give a significant generalization of Givental's toric mirror theorems. The PI will work to extend the theory in several directions. First, the wall-crossing formulas should extend to curves of higher genus, and the resulting theory for Calabi-Yau three-folds is conjecturally equal to the BCOV B-model theory of the mirror Calabi-Yau, but WITHOUT changing coordinates by the mirror map. The PI plans to prove this conjecture at least for local Calabi-Yau targets. Other projects will provide extensions to orbifold targets (with applications to the Crepant Resolution Conjecture), and to Landau-Ginzburg models (with applications to the Calabi-Yau/LG correspondence). Further applications, such as a proof of the general Abelian/Non-abelian Correspondence in Gromov-Witten theory are also expected.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. The moduli spaces studied by the investigator 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. The project will continue this fruitful interaction by offering new insights on mirror symmetry at higher genus.This award is co-funded by the Algebra and Number Theory and the Topology programs.